diff --git "a/7dE0T4oBgHgl3EQfwQEF/content/tmp_files/load_file.txt" "b/7dE0T4oBgHgl3EQfwQEF/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/7dE0T4oBgHgl3EQfwQEF/content/tmp_files/load_file.txt" @@ -0,0 +1,633 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf,len=632 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='02628v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='CO] 6 Jan 2023 PINNACLE SETS OF SIGNED PERMUTATIONS NICOLLE GONZ´ALEZ, PAMELA E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' HARRIS, GORDON ROJAS KIRBY, MARIANA SMIT VEGA GARCIA, AND BRIDGET EILEEN TENNER Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Pinnacle sets record the values of the local maxima for a given family of permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' They were introduced by Davis-Nelson-Petersen-Tenner as a dual concept to that of peaks, previ- ously defined by Billey-Burdzy-Sagan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In recent years pinnacles and admissible pinnacles sets for the type A symmetric group have been widely studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In this article we define the pinnacle set of signed permutations of types B and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' We give a closed formula for the number of type B/D admissible pinnacle sets and answer several other related enumerative questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Introduction The study of permutation statistics is an active subdiscipline of combinatorics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Given a per- mutation w = w(1)w(2) · · · w(n), two particularly well-studied statistics are descents and peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Respectively, these statistics refer to indices i such that w(i) > w(i + 1), and indices i such that w(i − 1) < w(i) > w(i + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The collection of a permutation’s descent indices is its descent set, with a permutation’s peak set being similarly defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Two fundamental goals in the study of these particular statistics are (1) understanding which subsets can arise as descent sets or peak sets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=', which sets are admissible as descent or peak sets), and (2) enumerating the permutations that have a given admissible descent or peak set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For descents of permutations in the (type A) symmetric group Sn, this question was answered by Stanley [15, Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='4] and is well known to give rise to the Eulerian numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Inspired by Stembridge’s study of peaks in the context of poset partitions [16], Billey, Burdzy, and Sagan [1] introduced the study of admissible peak sets for Sn with an interest in probabilistic applications, and established that the number of permutations with peak set I is given by 2n−|I|−1p(n), where p(n) is a polynomial of degree max(I) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Shortly thereafter, their results were extended to permutations in type B by Castro-Velez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' [2] where it was shown that the number of permutations with a given peak set I is 22n−|I|−1p(n), with p(n) the same as in [1] above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The second author and various collaborators went further by extending these results to types C and D [7], using peaks to study properties of the descent polynomial [6], and then initiating the study of peaks in the context of graphs [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' A notion that is closely related to peaks is the pinnacle set of a permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Pinnacles are the set of values held by the permutation at the peak indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' More precisely, given a permutation w = w(1)w(2) · · · w(n) with peak set Peak(w), the pinnacle set of w is Pin(w) = {w(i) : i ∈ Peak(w)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Given a subset I ⊆ [n], if there exists a permutation w whose pinnacle set is I, we say that I is an admissible pinnacle set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In [3], Davis, Nelson, Petersen, and the last author pioneered the study of pinnacles for permutations in Sn and gave a complete characterization of admissible pinnacle sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' They provided a closed formula for the number of admissible pinnacle sets with a given maximum value, as well as a refinement to those appearing in Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In particular, Davis et Date: January 9, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' was partially supported through a Karen Uhlenbeck EDGE Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='G was partially supported by the NSF grant DMS 2054282.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' was partially supported by the NSF grant DMS-2054436.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 1 al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' gave a recursive formula for the number of permutations in Sn with a given pinnacle set p(n) and asked whether a more efficient expression could be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' This paper led to a sequence of articles in recent years, many focused on improved and faster formulas for p(n), by realizing permutations with given pinnacle sets as invariants under certain modified Sn-actions [5, 9] or via more traditional enumerative methods [8, 10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In related work, Rusu [13] and Rusu-Tenner [14] deepened the knowledge of pinnacles in Sn by investigating further properties of these statistics and characterizing admissible pinnacle orderings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In this article we look beyond type A and study pinnacles and admissible pinnacle sets for the type B and type D signed symmetric groups, SB n and SD n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Our main results are the following, where we write APSX n to denote the admissible pinnacle sets in SX n for X ∈ {A, B, D}: (1) Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='12 gives a closed formula for the number of admissible pinnacle sets in SB n , |APSB n | = ⌊ n−1 2 ⌋ � k=0 �n k ��n − 1 − k � n−1 2 � − k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' (2) Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='2 proves that any admissible pinnacle set in SB 2k is also admissible in SD 2k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' that is, APSD 2k = APSB 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' (3) In counterpoint to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='2, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='9 counts the admissible pinnacle sets of type B that are not in type D when n = 2k + 1, |APSB 2k+1 \\ APSD 2k+1| = �2k − 1 k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' (4) Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='11 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='12 count the all-positive admissible pinnacle sets of type B that are not admissible in type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Namely, defining APS+ n := {S ∈ APSB n : S ⊂ N};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' we prove that the sets APS+ n \\ APSn are enumerated by, ��APS+ n \\ APSn �� = � 4k − �2k k � if n = 2k + 1, and 22k−1 − �2k k � if n = 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' This article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In Section 2, we introduce all the necessary background and notation, defining pinnacles and related notions in type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In Section 3, we give a characterization of admissible signed pinnacle sets and a formula for their enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In Section 4, we provide relations between admissible pinnacle sets of type A, B, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Lastly, in Section 5, we describe some future directions and open conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The authors thank Patrek K´arason Ragnarsson for the coding and data that facilitated the research in this project, and Freyja K´arad´ottir Ragnarsson for the key insight to the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The authors also thank the American Institute of Mathematics and the National Science Foundation for sponsoring the Latinx Mathematicians Research Community, which brought together a subset of the authors initially for collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Background Let N = {1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='} and for n ∈ N we write [n] := {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For any set X, typically of positive values, although we make the definition more generally, we define −X := {−x : x ∈ X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Finally, we define ±X = X ∪ −X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 2 Throughout this paper, we let Sn denote the (type A) symmetric group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' That is, Sn is the group of bijections from [n] → [n] under function composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' We often write w ∈ Sn using one-line notation, as w = w(1)w(2) · · · w(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The type B symmetric group (that is, the hyperoctahedral group) is the group of signed permutations SB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' These are bijections ±[n] → ±[n] such that w(−i) = −w(i) for all i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In particular, any w ∈ SB n satisfies the property that {|w(1)|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' , |w(n)|} = [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The type D symmetric group is the subgroup SD n of SB n consisting of signed permutations with an even number of signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Namely, these are the signed permutations w for which |{i ∈ [n] : w(i) < 0}| is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' As in type A, we use one-line notation to denote signed permutations w ∈ SB n , where we may write only w = w(1)w(2) · · · w(n) since this uniquely determines w(−i) for all positive i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Following convention, we write −i = ¯i to ease notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For example, w = ¯12¯3 is the signed permutation with w(1) = −1, w(2) = 2, and w(3) = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Recall that a permutation w ∈ Sn has a peak at index i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' , n − 1} if w(i − 1) < w(i) > w(i + 1), and the value w(i) is a pinnacle of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' We denote by Peak(w) the set of all peaks of w ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The pinnacle set of w ∈ Sn is Pin(w) = {w(i) : i ∈ Peak(w)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' A set P ⊆ [n] is an n-admissible pinnacle set in type A if there exists a permutation w ∈ Sn such that Pin(w) = P, and we call the permutation w a witness for the set P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For example, the identity permutation is a witness for the admissible pinnacle set ∅ (as is any peak-less permutation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Denote by APSn the set of all n-admissible pinnacle sets in type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In order to facilitate our discussions about pinnacles, we introduce terminology about their minimal counterparts: a permutation w ∈ Sn has a valley at index i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' , n − 1} if w(i − 1) > w(i) < w(i + 1), and the value w(i) is a vale of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The graph of the permutation 23715648 ∈ S8 with the pinnacles/peaks circled in red and the vales/valleys in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Consider the permutation w = 23715648 ∈ S8 shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' We have Peak(w) = {3, 6} and Pin(w) = {6, 7}, and valleys and vales {4, 7} and {1, 4}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Pinnacles in types B and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Pinnacles were defined in [3] for unsigned permutations, but they could just as easily have been defined for signed permutations—or, in fact, for arbitrary strings of distinct real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' We now expand the type A definitions to type B, and note that since SD n ⊂ SB n , these definitions also hold for type D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Let w be a signed permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' A pinnacle of w is a value w(i) that is larger than both w(i − 1) and w(i + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The pinnacle set of w is the set of its pinnacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In order to define admissible pinnacle sets, it is important to establish which subsets could even appear among the one-line notation of a signed permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' A signed set (or signed subset, depending on context) is a set I such that x ∈ I implies −x ̸∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Throughout this paper, we assume that all subsets of ±[n] are signed subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' A signed subset S ⊂ ±[n] is an admissible pinnacle set if S is the pinnacle set of some signed permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' That permutation is a witness for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Note that when we study sets that are admissible as pinnacle sets in type D, any witness permutation will be required to be in SD n for some n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' As before, we denote by APSB n (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=', APSD n ) the set of all n-admissible pinnacle sets in type B (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=', type D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Once again, we have ∅ ∈ APSD n ⊆ APSB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For example, 123 · · · n and ¯2¯134 · · · n are both witnesses for ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' While there can be multiple witness permutations for a given admissible pinnacle set, we will often refer to a particular witness permutation that we call “canonical.” Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For S ∈ APSB n , write S = {s1 < s2 < · · · < sk}, and set S′ := −[n] \\ {−|s| : s ∈ S} = {s′ 1 < s′ 2 < · · · < s′ n−k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Then the canonical witness permutation is w := s′ 1 s1 s′ 2 s2 · · · s′ k sk s′ k+1 · · · s′ n−k ∈ SB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' If S ∈ APSD n , then its canonical (type D) witness permutation is w as defined above if w is in SD n , and otherwise its canonical witness is obtained from w by replacing s′ n−k with |s′ n−k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Next we establish that the “canonical witness permutations” are, in fact, witnesses and follow this by providing canonical witness permutations in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The canonical witness permutation for an admissible set S is a witness for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The set S is admissible, so there is some permutation whose pinnacle set is S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The canonical witness has been constructed to have minimal possible non-pinnacle values, and to position the smallest non-pinnacle values beside the smallest pinnacle values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Therefore, if any permutations were to have S as a pinnacle set (and we know that some permutation does), the permutation given in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='6 would be among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ Although SB n contains both Sn and SD n as subgroups, there are interesting subtleties to the pinnacle sets that become admissible when witness permutations can be signed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' First, some sets will be admissible with type B permutations, but not with type D permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' And second, some sets of all-positive values will be admissible with type B permutations, but not with type A (unsigned) permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' We demonstrate each of these scenarios below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 4 (a) 1 2 3 4 5 6 7 1 2 3 4 5 6 7 0 −1 −2 −3 −4 −5 −6 −7 (b) (b) 1 2 3 4 5 6 7 1 2 3 4 5 6 7 0 −1 −2 −3 −4 −5 −6 −7 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' (a) The graph of the permutation ¯7¯4¯61¯52¯3 ∈ SB 7 with the pinna- cles/peaks circled in red and the vales/valleys in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' (b) The graph of the permu- tation ¯63¯54¯17¯2 ∈ SB 7 with the pinnacles/peaks circled in red and the vales/valleys in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The set {¯4, 1, 2} is admissible in SB 7 , with canonical witness permutation ¯7¯4¯61¯52¯3 as shown in Figure 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' However, there is no element of SD 7 having this pinnacle set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' That is, {¯4, 1, 2} ̸∈ APSD 7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The set {3, 4, 7} is admissible in SB 7 , with canonical witness permutation ¯63¯54¯17¯2, as shown in Figure 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' However, despite its pinnacle set having all positive values, there is no type A permutation having this pinnacle set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' That is, {3, 4, 7} ̸∈ APSn for any n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Admissible signed pinnacle sets in type B In this section, we characterize and enumerate the admissible pinnacle sets among signed permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' This expands on the work begun in [3] for unsigned permutations, but, as we show, the results for signed permutations are subtly different from those in type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Characterization of admissible pinnacle sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For the remainder of the article, we will often use the fact that given an admissible pinnacle set S ∈ APSB n , we can always write S = P(S) ⊔ N(S) with P(S) := S ∩ [n] and N(S) := S ∩ −[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' When no confusion will arise, we simply write P := P(S) and N := N(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' To give a first inkling of how admissible pinnacle sets in type B are fundamentally different from those in type A, we note that there are some sets of positive integers that are never in APSn for any n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For example, any set containing 1 or 2 will never be the pinnacle set of any permutation in Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' On the other hand, such a statement is not true in type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Every finite signed subset S is admissible in SB n , for some n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' That is, there exists w ∈ SB n such that S = PinB(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Write S = {s1 < · · · < sk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Let m = max{|s| : s ∈ S} (that is, m = max{|s1|, |sk|}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Define the set S′ := −[2m + 1] \\ {−|s| : s ∈ S}, which we write as S′ = {s′ 1 < · · · < s′ 2m+1−k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Then w = s′ 1 s1 s′ 2 s2 · · · s′ k sk s′ k+1 s′ k+2 · · · s′ 2m+1−k ∈ SB 2m+1, and PinB(w) = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ Using a similar argument as the one proving Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='1, it follows that any finite set of all positive values is admissible in some SB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Any subset P ⊂ [n] with |P| ≤ n−1 2 is admissible in SB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Let P = {p1 < · · · < pk}, and set P ′ := −([n] \\ P) = {p′ 1 < · · · < p′ n−k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Then w = p′ 1 p1 p′ 2 p2 · · · p′ k pk p′ k+1 p′ k+2 · · · p′ n−k ∈ SB n and PinB(w) = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ This can be particularly interesting when the set P was not admissible in Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Consider P = {1, 2} with n = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The permutation ¯51¯42¯3 ∈ SB 5 is a witness permutation for P, so P ∈ APSB 5 , while P ̸∈ APSn for any n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Our goal is to establish a characterization and formula for the number of admissible pinnacle sets in SB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' We begin with some preliminary steps, from which those results will follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The first of these is a bijection between admissible pinnacle sets in Sn and those admissible pinnacle sets in SB n that have no positive values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' There exists a bijection between APSn and {S ∈ APSB n : S ⊂ −N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Given T ∈ APSn, define T ′ := {t − (n + 1) : t ∈ T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The set T ′ has no positive elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Let w ∈ Sn be the canonical witness for T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Then w′ := (w(1) − (n + 1)) · · · (w(n) − (n + 1)) ∈ SB n has pinnacle set T ′, and so T ′ ∈ APSB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' This process can be inverted: given S ∈ APSB n with P(S) = ∅, map this S to S′ := {s + n + 1 : s ∈ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' It follows that S′ ∈ APSn, as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ We illustrate Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='4 with an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The set {3, 6, 7, 10} ∈ APS10 is in correspondence with {¯8, ¯5, ¯4, ¯1} ∈ APSB 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The permutations described in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='4, which exhibit these sets as pinnacle sets, are shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' We have defined admissible pinnacle sets in types A, B, and D, referring to permutations in Sn, SB n , or SD n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' However, as suggested earlier, there is a natural generalization of admissible pinnacle sets to permutations of any totally ordered set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For any totally ordered set X, let APS(X) be the set of admissible pinnacle sets of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The definitions of witness and canonical witness permutations in this general setting are analogous to their definitions in the symmetric groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Because they arise so often, we have been easing notation by writing APS(Sn) as APSn, APS(SB n ) as APSB n , and APS(SD n ) as APSD n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The set X = {−2, π, 4, 5, 100} has six admissible pinnacle sets: ∅, {4}, {5}, {100}, {4, 100}, and {5, 100}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 6 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 −1 −2 −3 −4 −5 −6 −7 −8 −9 −10 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The left-hand figure shows the canonical witness for {3, 6, 7, 10} in S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The right-hand figure shows the corresponding witness permutation for {¯8, ¯5, ¯4, ¯1}, as defined in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Note that if we are only interested in how many admissible pinnacle sets X has, as opposed to the sets themselves, then the size of X is what matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For any totally ordered finite set X, |APS(X)| = |APS ([|X|]) | = |APS|X||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' This calculation will be useful in the enumeration appearing in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' We are now are able to characterize admissible pinnacle sets for signed permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The sets in APSB n are exactly the sets S = P(S) ⊔ N(S) for which |P(S)| + |N(S)| ≤ (n − 1)/2, P(S) ⊂ [n], N(S) ⊂ −([n] \\ P(S)), and N(S) ∈ APS(−([n] \\ P(S))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' First of all, it is clear that any admissible pinnacle set in SB n must satisfy the listed require- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Now suppose that a set S satisfies the listed requirements, with P := P(S) = {p1 < · · · < pk} and N := N(S) = {n1 < · · · < nr}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In light of the last requirement, let w be the canonical witness permutation of the set (−([n] \\ P)), having pinnacle set N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' That is, w = i1 n1 i2 n2 · · · ir nr ir+1 ir+2 ir+3 · · · in−k−r where ij < ij+1 and {i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' , in−k−r} = −([n] \\ P) \\ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Then i1 n1 i2 n2 · · · ir nr ir+1 p1 ir+2 p2 ir+3 · · · pk ir+k+1 ir+k+2 · · · in−r−k is a canonical witness for S = P ⊔ N in SB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Hence S ∈ APSB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Enumeration of admissible pinnacle sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The conditions listed in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='9 inform our enumeration of the admissible pinnacle sets in SB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In particular, we will construct these sets by 7 first fixing a collection P of positive pinnacles and then determining how many sets N of negative pinnacles exist for which P ∪ N is admissible in SB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In order not to have too many pinnacles (that is, not more than ⌊(n − 1)/2⌋), we need to understand the following value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Let pn(d) be the number of admissible pinnacle sets in Sn having cardinality at most d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' That is, pn(d) := |{S ∈ APSn : |S| ≤ d}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' This statistic has a particularly nice formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For all integers d ∈ [0, ⌊(n − 1)/2⌋], pn(d) = �n − 1 d � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The admissible pinnacle sets in Sn having cardinality at most d can be partitioned into two sets: those that contain n, and those that do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' We claim that the first set is counted by pn−1(d − 1), and the second set is counted by pn−1(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Suppose, first, that S ∈ APSn such that n ∈ S and |S| = k ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Let w ∈ Sn be the canonical witness for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Deleting n from the one-line notation of w will produce a permutation v ∈ Sn−1 with Pin(v) = S \\ {n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Conversely, given T ∈ APSn−1 with |T| = k − 1, let u ∈ Sn−1 be the canonical witness for T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Inserting n between the non-pinnacles u(2k − 1) and u(2k) will produce a permutation in Sn whose pinnacle set is T ∪ {n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' This establishes the first part of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For the second part of the claim, suppose that S ∈ APSn with n ̸∈ S and |S| = k ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Let w ∈ Sn be the canonical witness for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Because n ̸∈ S, we have w(n) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Thus the permutation w(1) · · · w(n − 1) ∈ Sn−1 has pinnacle set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Conversely, if v ∈ Sn−1 has pinnacle set S, then appending n to the end of v will produce a permutation in Sn that also has pinnacle set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' This gives the binomial recurrence pn(d) = pn−1(d − 1) + pn−1(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' To complete the argument, notice that pn(0) = 1 and pn(1) = 1 + (n − 2) = n − 1, for all positive integers n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ Combining Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='9, which characterizes admissible pinnacle sets for signed permutations, with the enumeration in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='11, we now count the admissible pinnacle sets for signed permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' If n ≥ 2, then ��APSB n �� = ⌊ n−1 2 ⌋ � k=0 �n k �� n − 1 − k � n−1 2 � − k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The main idea of the proof will be to construct admissible pinnacle sets in SB n following the requirements of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' First, we will select a set P of positive pinnacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In other words, P ⊂ [n] and |P| ≤ (n − 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Then we add to it any set N ⊂ −([n] \\ P) that is in APS(−([n] \\ P)), so long as |P| + |N| ≤ (n − 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' We are interested in the number of such sets, and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='8 says that we only need to care about the size of P in this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' This and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='4 mean that such sets N can be counted in terms of admissible pinnacle sets of Sn−|P |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 8 Fix an integer k ∈ [0, (n − 1)/2], and choose a k-element subset P ⊂ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' There are �n k � ways to do this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' We can supplement P with any r-element admissible pinnacle set N ⊂ −([n] \\ P), as long as k + r ≤ ⌊(n − 1)/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The number of ways to do this is pn−k ��n − 1 2 � − k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Therefore, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='11, the number of admissible pinnacle sets in SB n is ⌊ n−1 2 ⌋ � k=0 �n k �� n − 1 − k � n−1 2 � − k � , as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ In Table 1, we give the number of signed admissible pinnacle sets in type B for 3 ≤ n ≤ 15, while permutations in SB 1 and SB 2 have no pinnacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' This appears in the OEIS as sequence [12, A359066].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The even-indexed terms in the table appear in [12, A240721] and the odd-indexed terms appear in [12, A178792].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' n 3 4 5 6 7 8 9 10 11 12 13 14 15 ��APSB n �� 5 7 31 49 209 351 1471 2561 10625 18943 78079 141569 580865 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The number of admissible pinnacle sets in SB n , for 3 ≤ n ≤ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In the next Section, we will be able to answer the analogous enumerative question in type D (see Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Relating admissible pinnacle sets in types A, B, and D There is a natural embedding of Sn in SD n , and of SD n in SB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Having spent Section 3 analyzing pinnacle sets that are admissible in SB n , it is natural to wonder how these sets are related to those that are admissible in SD n or, for those elements of APSB n without negative values, to those that are admissible in Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' We now give complete characterization of each of these relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Comparing admissible pinnacle sets in types B and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' As mentioned before, SD n ⊂ SB n , thus it is natural to investigate the relationship between those sets that are admissible as pinnacle sets in type B and those that are in type D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' It is, perhaps, not surprising that this relationship depends on the parity of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' As a first step in this analysis, we identify a technique that will be handy in proving that a set is admissible for type D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Suppose that w ∈ SB n is a witness for a pinnacle set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' If w(n − 1) > ±w(n) or if w(n − 1) < ±w(n), then the permutation w′, defined by w′(i) = � w(i) i < n and −w(i) i = n, is also a witness for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Moreover, S ∈ APSD n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' First note that w′ is an element of SB n because changing the sign of the last letter does not alter the fact that this is a signed permutation on ±[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Next observe that the pinnacle set has not changed from w to w′ because none of the inequalities between consecutive letters has been altered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Finally, note that the numbers of negative values in w and in w′ differ by 1, meaning that one of these permutations is in SD n while the other is in SB n \\ SD n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ We will call on the previous result often throughout our arguments in this section, beginning with a description of the simple relationship between APSB n and APSD n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For k ≥ 1, APSB 2k = APSD 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Certainly anything admissible in type D is also admissible in type B, because signed per- mutations include the signed permutations in type D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' It remains to show that any pinnacle set that is admissible in SB 2k is also admissible in SD 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Fix S := {s1 < · · · < sl} ∈ APSB 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Because l ≤ ⌊(2k − 1)/2⌋, we have l ≤ k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Then the canonical witness w for S satisfies the hypotheses of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='1, and so in fact S ∈ APSD 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ The equality shown in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='2 relies on the fact that there are always at least two more non-pinnacles than there are pinnacles in signed permutations on 2k letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' This not necessarily true for signed permutations of an odd number of letters, and hence it is not surprising that the relationship between APSB 2k+1 and APSD 2k+1 has more nuance than the relationship presented in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Indeed, we will show that APSD 2k+1 is a strict subset of APSB 2k+1, and we will describe the elements of the latter that are not elements of the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' If S ∈ APSB 2k+1 \\ APSD 2k+1, then |S| = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Fix S ∈ APSB 2k+1 and let w ∈ SB 2k+1 be the canonical witness for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' If |S| < k, then both w(2k) and w(2k + 1) are non-pinnacles and w(2k) < w(2k + 1) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In particular, the hypotheses of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='1 are satisfied by w, and so S ∈ APSD 2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Hence, if S ∈ APSB 2k+1 \\ APSD 2k+1, then |S| = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ One implication of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='3 is that if w ∈ SB 2k+1 is a witness for S ∈ APSB 2k+1\\APSD 2k+1, then w(3), w(5), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' , w(2k−1) are all vales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' With Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='3 providing a first step toward understanding elements of APSB 2k+1 \\ APSD 2k+1, we now proceed to describe these sets more clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Fix S ∈ APSB 2k+1 \\ APSD 2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In every witness permutation for S, the non-pinnacle values are all negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Fix S ∈ APSB 2k+1 \\ APSD 2k+1 and w ∈ SB 2k+1 a witness for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Following Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='3, the non- pinnacles of w are precisely w(1), w(3), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' , w(2k + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In particular, each w(2i + 1) is less than its immediate neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Suppose, for the purpose of obtaining a contradiction, that w(2j +1) > 0 for some j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Let w′ ∈ SB 2k+1 be the permutation obtained from w by replacing w(2j +1) by −w(2j +1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Then w′ is still a witness for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Either w or w′ is in SD 2k+1, meaning that S must be an element of APSD 2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' This is a contradiction, so there is no such j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ In fact, the negative values of S ∈ APSB 2k+1 \\ APSD 2k+1 are enough to determine all of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Suppose that S ∈ APSB 2k+1 \\ APSD 2k+1, with P := S ∩ N and N := S ∩ −N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Then the elements of P are the smallest k − |N| values in the set [2k + 1] \\ −N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In particular, N determines P, and hence all of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 10 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Fix S ∈ APSB 2k+1 \\ APSD 2k+1, with P and N as defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='3, we have |S| = k, so let S = {s1 < s2 < · · · < sk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' If |N| = k, then there is nothing to check, so assume that |N| < k and hence sk > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Suppose, for the purpose of obtaining a contradiction, that there exists q ∈ ([2k + 1] \\ −N) \\ P with q < sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Let w be the canonical witness permutation for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' By definition, w(2k) = sk and w(2k + 1) = −q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' But then w′, which agrees with w everywhere except w′(2k + 1) = q, is also a witness for S, contradicting Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Therefore P consists precisely of the smallest k − |N| values in the set [2k + 1] \\ −N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='5 gives a necessary condition for elements of APSB 2k+1 \\ APSD 2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The next result establishes that the set N ⊔ P constructed in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='5 is, in fact, an admissible signed pinnacle set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Suppose that N ⊂ −N and N ∈ APSB 2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Let P be the smallest k − |N| values in [2k + 1] \\ −N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Then N ⊔ P ∈ APSB 2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' This follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ Maintaining the terminology of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='6, note that for any set N ⊂ −N, all witness permutations for N ⊔ P are forced by construction of P to have the same number of negative values: k + 1 + |N|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' This yields the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Suppose S ∈ APSB 2k+1 \\ APSD 2k+1, with N := S ∩ −N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The sets |N| and |S| have the same parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' To have S ∈ APSB 2k+1 \\ APSD 2k+1, we need |S| = k, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Moreover, as discussed above, the number of negative values is k + 1 + |N|, and this must be odd because S /∈ APSD 2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Thus k + |N| = |S| + |N| is even, completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ The consequence of this collection of results is that if we have a set N ⊂ −N that is, itself, admissible in SB 2k+1, and for which |N| has the same parity as k, then there is a unique ((k − |N|)- element) set P ⊂ N for which N ⊔ P ∈ APSB 2k+1 \\ APSD 2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Therefore, to enumerate APSB 2k+1 \\ APSD 2k+1, it suffices to count the elements of APSB 2k+1 that have no positive values and that have size of the form k − 2i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Because we want to look at the elements of APSB 2k+1 having no positive values, we can take advantage of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='4 to look, instead, at APS2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' That is, it will suffice to count � i≥0 ����{S ∈ APS2k+1 : |S| = k − 2i} ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The last step of this enumeration requires a parity result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For k ≥ 0, ����{S ∈ APS2k+1 : |S| is even} ���� = ����{S ∈ APS2k+1 : |S| is odd} ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Fix S ⊂ [2k + 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' If 2k + 1 ∈ S, then set S′ := S \\ {2k + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Clearly if S ∈ APS2k+1 then also S′ ∈ APS2k+1, and the sets |S| and |S′| have different parities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Now consider S ∈ APS2k+1 with 2k + 1 ̸∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' By [3, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='8], max(S) > 2|S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' We have max(S) < 2k + 1, so |S| < k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Consequently, S has a witness permutation w using at most k vales, so there are at least (2k + 1) − (k − 1 + k) = 2 non-pinnacle/non-vale values in this witness 11 permutation, and one of these is 2k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' We can create a new permutation w′ by inserting 2k + 1 immediately to the right of the largest vale in w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Thus the pinnacle set of w′ is S ∪ {2k + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Therefore there is a bijection between even-sized elements of APS2k+1 and odd-sized ones, obtained by adding/removing the element 2k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' This partitions APS2k+1 into two evenly sized parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ We have now completed all of the steps necessary to give the desired enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For k ≥ 1, ��APSB 2k+1 \\ APSD 2k+1 �� = �2k − 1 k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Following Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='5 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='7, we can enumerate APSB 2k+1 \\ APSD 2k+1 by counting elements of APS2k+1 that have size {k − 2i : i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' These are either all of the odd-sized sets in APS2k+1 or all of the even-sized ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='8, then, ��APSB 2k+1 \\ APSD 2k+1 �� = 1 2 |APS2k+1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' It was shown in [3, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='8] that |APS2k+1| = �2k k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Finally, it is straightforward to check that 1 2 �2k k � = �2k−1 k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ We can now use Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='12, which enumerated APSB n , and Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='9 to enu- merate APSD n for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For k ≥ 1, ��APSD 2k �� = ��APSB 2k �� and ��APSD 2k+1 �� = � k � i=0 �2k + 1 i ��2k − i k − i �� − �2k − 1 k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In Table 2, we give the number of signed admissible pinnacle sets in type D for 3 ≤ n ≤ 15, while permutations in SD 1 and SD 2 have no pinnacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' This appears in the OEIS as sequence A359067.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The even-indexed terms are identical to even terms in Table 1 and the odd-indexed terms are �2k−1 k � less than the corresponding odd-indexed terms in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' n 3 4 5 6 7 8 9 10 11 12 13 14 15 ��APSD n �� 4 7 28 49 199 351 1436 2561 10499 18943 77617 141569 579149 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The number of admissible pinnacle sets in SD n , for 3 ≤ n ≤ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Comparing admissible pinnacle sets in types B and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Some elements of APSB n have no negative values, and so one could ask if those sets might also be admissible in Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In this section we consider how those elements of APSB n are related to the admissible pinnacle sets in APSn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' To make this discussion precise, we introduce: APS+ n := {S ∈ APSB n : S ⊂ N};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' in other word, APS+ n consists of the pinnacle sets that are admissible in SB n and that contain no negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 12 For example, {1, 3} ∈ APS+ 5 , with canonical witness 51432 ∈ SB 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In fact, by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='2, any subset of [n] having at most (n − 1)/2 elements is admissible in SB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Contrast this with APSn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' for example, APS+ 5 \\ APS5 = � {1}, {2}, {1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Our goal in this section is to understand APS+ n \\ APSn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' As with the comparison of APSB n and APSD n , this will depend on the parity of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For k ≥ 0, |APS+ 2k+1 \\ APS2k+1| = 4k − �2k k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Because APS2k+1 ⊂ APS+ 2k+1, the desired value is equal to ��APS+ 2k+1 �� − |APS2k+1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Following Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='2, we can compute ��APS+ 2k+1 �� by counting i-element subsets of [2k +1] for all i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The result follows by recognizing that this yields a sum that is half of a row-sum of Pascal’s triangle, and combining this with the enumeration of APS2k+1 from [3]: |APS+ 2k+1 \\ APS2k+1| = ��APS+ 2k+1 �� − |APS2k+1| = �2k + 1 0 � + �2k + 1 1 � + · · · + �2k + 1 k � − �2k k � = 1 222k+1 − �2k k � = 4k − �2k k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ We now complete this analysis by considering the even-indexed case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' For k ≥ 1, ��APS+ 2k \\ APS2k �� = 22k−1 − �2k k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' This calculation is almost identical to that from the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='11, except that we will also have to subtract the central term from a row of Pascal’s triangle: |APS+ 2k \\ APS2k| = ��APS+ 2k �� − |APS2k| = �2k 0 � + �2k 1 � + · · · + � 2k k − 1 � − �2k − 1 k − 1 � = 1 2 � 22k − �2k k �� − �2k − 1 k − 1 � = 22k−1 − �1 2 �2k k � + �2k − 1 k − 1 �� = 22k−1 − �2k k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' □ We combine the enumerations of Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='11 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='12 in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Specifically, we list ��APS+ n \\ APSn �� for 3 ≤ n ≤ 15, while permutations in SB 1 and SB 2 have no pinnacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The nth term of this appears in the OEIS as double the (n − 1)st term of [12, A294175].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Moreover, the odd-indexed terms, enumerated in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='11, appear in [12, A068551] and the even-indexed terms are double the terms of [12, A008549].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 13 n 3 4 5 6 7 8 9 10 11 12 13 14 15 ��APS+ n \\ APSn �� 2 2 10 12 44 58 186 260 772 1124 3172 4760 12952 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The number of all-positive pinnacle sets that are admissible in SB n but not in Sn, for 3 ≤ n ≤ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Future directions As demonstrated by the results in this paper, admissible pinnacle sets have rich structure and properties even beyond the symmetric group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' There are many directions for further research on this topic, including broad questions about pinnacle sets for families of permutations with certain properties, and enumerative specializations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' As a complement to those large questions, we conclude this work by pointing out that we uncovered a possible connection between ��APSB n �� and sequence [12, A119258].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' In particular, we have the following conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Conjecture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Consider the sequence [12, A119258], given by T(n, 0) = T(n, n) = 1 and T(n, k) = 2T(n − 1, k − 1) + T(n − 1, k) for 0 < k < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Then ��APSB n �� = T � n, �n − 1 2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Data Patrek Ragnarsson’s code for computing the data in Tables 1, 2, and 3 can be found at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='com/PatrekR/Signed-pinnacle-sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Note that the data in Table 2 is the difference between the enumerations given in two of the files posted at this GitHub link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' References [1] Sara Billey, Krzysztof Burdzy, and Bruce E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Sagan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Permutations with given peak set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Integer Seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=', 6(16), 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' [2] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Castro-Velez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Diaz-Lopez, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Orellana, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Pastrana, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Zevallos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Number of permutations with same peak set for signed permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Journal of Combinatorics, 8(4):631–652, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' [3] Robert Davis, Sarah A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Nelson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Kyle Petersen, and Bridget E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Tenner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' The pinnacle set of a permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Discrete Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=', 341(11):3249–3270, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' [4] Alexander Diaz-Lopez, Lucas Everham, Pamela E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Harris, Erik Insko, Vincent Marcantonio, and Mohamed Omar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Counting peaks on graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Australas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' J Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=', 75:174–189, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' [5] Alexander Diaz-Lopez, Pamela E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Harris, Isabella Huang, Erik Insko, and Lars Nilsen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' A formula for enumerating permutations with a fixed pinnacle set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Discret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=', 344:112375, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' [6] Alexander Diaz-Lopez, Pamela E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Harris, Erik Insko, Mohamed Omar, and Bruce E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Sagan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Descent polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Discrete Mathematics, 342(6):1674–1686, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' [7] Alexander Diaz-Lopez, Pamela E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Harris, Erik Insko, and Darleen Perez-Lavin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Peak sets of classical coxeter groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Involve, 10(2):263–290, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' [8] Rachel Domagalski, Jinting Liang, Quinn Minnich, Bruce E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Sagan, Jamie Schmidt, and Alexander Sietsema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Pinnacle set properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Discrete Mathematics, 345(7):112882, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' [9] Justine Falque, Jean-Christophe Novelli, and Jean-Yves Thibon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Pinnacle sets revisited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Preprint arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='05248, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' [10] Wenjie Fang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' [11] Quinn Minnich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Further results on pinnacle sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Discrete Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=', 346(4):Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' 113296, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' [12] OEIS Foundation Inc.' metadata={'source': 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address: nicolle@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='edu (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Harris) Department of Mathematical Sciences, University of Wisconsin, Milwaukee, WI 53211 Email address: peharris@uwm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='edu (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Rojas Kirby) Department of Mathematics and Statistics, San Diego State University, CA 92182 Email address: gkirby@sdsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='edu (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Smit Vega Garcia) Department of Mathematics, Western Washington University, Bellingham, WA 98225 Email address: smitvem@wwu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='edu (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content=' Tenner) Department of Mathematical Sciences, DePaul University, Chicago, IL 60614 Email address: bridget@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='depaul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'} +page_content='edu 15' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQfwQEF/content/2301.02628v1.pdf'}