diff --git "a/DNFJT4oBgHgl3EQfBCxz/content/tmp_files/load_file.txt" "b/DNFJT4oBgHgl3EQfBCxz/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/DNFJT4oBgHgl3EQfBCxz/content/tmp_files/load_file.txt" @@ -0,0 +1,1282 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf,len=1281 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='11423v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='CO] 26 Jan 2023 Improved Permutation Arrays for Kendall Tau Metric∗ Sergey Bereg William Bumpass Mohammadreza Haghpanah Brian Malouf I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Hal Sudborough Abstract Permutation arrays under the Kendall-τ metric have been considered for error-correcting codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Given n and d ∈ [1, �n 2 � ], the task is to find a large permutation array of permutations on n symbols with pairwise Kendall-τ distance at least d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Let P(n, d) denote the maximum size of any permutation array of permutations on n symbols with pairwise Kendall-τ distance d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Using new recursive techniques, new automorphisms, and programs that combine randomness and greedy strategies, we obtain several improved lower bounds for P(n, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 1 Introduction In [1, 2, 5, 8, 10, 11], permutation arrays under the Kendall-τ metric were studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' This comple- mented many studies of permutation arrays under other metrics, such as the Hamming metric [3] [4] [6], Chebyshev metric [9] and several others [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' The use of the Kendall-τ metric was motivated by applications of error correcting codes and rank modulation in flash memories [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Let σ and π be two permutations (or strings) over an alphabet Σ ⊆ [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='n] = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=', n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' The Kendall-τ distance between σ and π, denoted by d(σ, π), is the minimum number of adjacent trans- positions (bubble sort operations) required to transform σ into π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' For an array (set) A of permuta- tions (strings), the pairwise Kendall-τ distance of A, denoted by d(A), is min{ d(σ, π) | σ, π ∈ A }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' An array A of permutations on [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='n] with d(A) = d will be called a (n, d)-PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Let P(n, d) denote the maximum cardinality of any (n, d)-PA A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Vijayakumaran [10] showed several lower bounds for P(5, d) and P(6, d) using integer linear programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Buzaglo and Etzion [5] showed many new bounds, including that P(7, 3) ≥ 588 using two permutation representatives and a set of permutations generated by specific automorphism operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' We also show results using automorphisms, namely those given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Details of these automorphisms are shown in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' We also used other programs to compute good lower bounds: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Programs which find a maximum size clique in a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Programs which combine randomness with a Greedy approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' That is, the first constructs a graph with a node for each permutation on n symbols and an edge connecting two nodes whose permutations are at Kendall-τ distance at least d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' The set of nodes (permutations) in a maximum size clique in this graph is a (n, d)-PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' The second initially chooses randomly a specified size set of permutations at pairwise Kendall-τ distance d, and then proceeds through all remaining permutations in lexicographic order and adds them to the set if they have Kendall-τ distance at least d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 1 In Tables 1 and 2 are given sporadic results obtained by these techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Blank positions in our tables signify other papers have the best lower bounds known e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' [5], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' All other lower bounds we give are larger than previous lower bounds, except for the two noted in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' n:d 3 4 5 6 7 8 9 6 102(∗) 7 588(∗) 336 126 84 42 8 3,752 2,240 672 448 168 9 1,008 288 Table 1: Improved lower bounds on P(n, d) by automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' (The bounds for P(6, 3) and P(7, 3) are from [10] and [5], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=') n:d 3 4 5 6 7 8 9 8 115 57 9 26,831 15,492 3,882 2,497 608 10 233,421 133,251 29,113 18,344 5,629 3,832 1,489 11 247,014 153,260 42,013 28,008 9,747 12 73,068 n:d 10 11 12 13 14 15 7 13 8 7 4 8 43 26 21 15 12 8 9 195 100 77 46 37 24 10 1,066 491 370 195 152 89 11 6,890 2,861 2,108 1,005 768 409 12 50,649 19,227 13,935 6,087 4,564 2,239 Table 2: Improved lower bounds by random Greedy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' In [2] Barg and Mazumdar described their Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='5, which is given below: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' [2] Let m = ((n − 2)t+1 − 1)/(n − 3), where n − 2 is a prime power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Then P(n, 2t + 1) ≥ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' t(t + 1)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' This was improved by Wang, Zhang, Yang, and Ge in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' [11] Let m = ((n − 2)t+1 − 1)/(n − 3), where n − 2 is a prime power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Then P(n, 2t + 1) ≥ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' (2t + 1)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' For example, by choosing t = 1 and n = 11, one obtains, by Theorem 2, P(11, 3) ≥ 1,330,560.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Theorem 2 applies only when n is two greater than a power of a prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' To compute good lower bounds for P(n, d) when n is not two greater than a power of a prime, one needs other techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' The lower bounds given by Theorem 2 are close to corresponding upper bounds when the Kendall-τ 2 distance is small, but not so close when the Kendall-τ distance is close to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Our Theorems 6 and 7, described below, give better lower bounds when the Kendall-τ distance is close to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' The following theorem from [8] allows one to obtain good lower bounds for even Kendall-τ distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' [8] For all n ≥ 1 and even d ≥ 2, we have P(n, d) ≥ 1 2P(n, d − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' [8] For all n, d ≥ 1 we have P(n + 1, d) ≤ (n + 1) · P(n, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Using Theorems 4 and 2 we have P(14, 11) ≥ P(15, 11)/15 ≥ 15!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='/(11 · 402234 · 15) ≈ 19, 703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='2 Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' [8] For all n, d > 1 we have P(n + 1, d) ≥ ⌈n+1 d ⌉P(n, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' For example, to compute a lower bound for P(14, 11) one can use, iteratively, Theorem 5 to obtain P(14, 11) ≥ ⌈14 11⌉·⌈13 11⌉·P(12, 11) = 4·P(12, 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' By computation (using the random greedy algorithm) we have P(12, 11) ≥ 19, 277, so P(14, 11) ≥ 76, 908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' We next give generalizations of Theorem 5 that yield improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Let Sn,m be the set of permutations on [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='n] with the restriction that the first n−m symbols are in sorted order, for a given m < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' A set A ⊆ Sn,m with Kendall-τ distance d is called a (n, m, d)-PA or (n, m, d)-array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Let P(n, m, d) denote the maximum cardinality of any (n, m, d)-array A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' For any m < n and d, P(n, d) ≥ P(n, m, d) · P(n − m, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Let A be a (n, m, d)-array and B be a (n − m, d)-array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' For each permutation π in A and each permutation τ in B, form the permutation (π, τ) by substituting the n − m symbols in the order given by τ for the first n − m symbols, given in order, in π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' It is easily seen that d((π, τ), (ρ, σ)) ≥ d, if either π ̸= ρ or σ ̸= τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' That is, for π, ρ ∈ A, if π ̸= ρ, then d(π, ρ) ≥ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Clearly, changing the order of the other n − m symbols, which appear in order in permutations in A, does not make the distance smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' A symmetric argument applies when σ, τ are different permutations in the (n − m, d)-array B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' In [8] Theorem 5 was proved using the set {1, d + 1, 2d + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' , ⌈n+1 d ⌉d + 1}, which corresponds to a (n + 1, 1, d)-array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' In general, a (n, m, d)-array can be much larger than one obtained by the iterative use of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' For example, for all n, we give (n, 2, 3)-arrays with n(n+1) 6 permutations, when n−1 is not divisible by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Also, for n = 14 we computed a (14, 2, 11)-array with 5 permutations τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' , τ5 shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Thus, using Theorem 6 we obtain P(14, 11) ≥ 5·P(12, 11) ≥ 5·19, 277 = 96, 135 which is a better lower bound than obtained by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' One can also improve on Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' For each permutation, say τ in a (n, m, d)-array A, one can generally find a larger set of permutations than in the best (n−m, d)-array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Let Pτ(n, d) denote the maximum cardinality of any (n, d) PA with the highest m symbols in the same positions as in τ, but where the other n − m symbols can be in any order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' We also denote it by P(n, d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' , im), where i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' , im are the fixed positions of symbols n − m + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' , n, not necessarily in that order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' For any (n, m, d)-array A, P(n, d) ≥ � τ∈A Pτ(n, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Let A be a (n, m, d)-array and, for each permutation π ∈ A, let τ be a permutation in an (n, d)-PA with the highest m symbols in the same position as in π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Form the new permutation (π, τ) by substituting the n − m symbols in the order given by τ for the first n − m symbols, given in order, in π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' It is easily seen, as in the proof of Theorem 6, that d((π, τ), (ρ, σ)) ≥ d, if either π ̸= ρ or σ ̸= τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Pτi(14, 11) τ1 0 0 0 0 0 0 13 14 0 0 0 0 0 0 47,851 τ2 0 0 14 0 0 0 0 0 0 0 0 0 0 13 36,250 τ3 0 13 0 0 0 0 0 0 0 0 0 0 0 14 19,227 τ4 13 14 0 0 0 0 0 0 0 0 0 0 0 0 19,227 τ5 0 0 0 0 0 0 0 0 0 0 0 0 14 13 19,227 Table 3: (14, 2, 11)-array with 5 permutations τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' , τ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Since the first 12 symbols in all τi are sorted, they are replaced by zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' The last column contains lower bounds for Pτi(14, 11), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' , 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' For example, we saw the result P(14, 11) ≥ 96, 125 using Theorem 6, with a (14, 2, 11)-array with five permutations τi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' , 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' We computed lower bounds for Pτi(14, 11), see the last column in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' By Theorem 7, we obtain the improved lower bound of P(14, 11) ≥ �5 i=1 Pτi(14, 11) ≥ 141, 782.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 2 Bounds for P(n, m, d) There are n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' (n−m)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' permutations in Sn,m for finding P(n, m, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' When m is small, this is relatively small compared to the n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' permutations to explore for finding P(n, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Also, P(n, m, d) generalizes P(n, d) as P(n, d) = P(n, n, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Finding exact values or bounds for P(n, m, d) is an interesting problem in its own right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Clearly, P(n, 1, d) = ⌈n/d⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' In general, by Theorem 5 P(n, m, d) ≥ �n d � �n − 1 d � · · · · �n − m + 1 d � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' (1) We denote by ε the identity permutation (1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' , n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' P(n, m, d) ≥ 2 if d ≤ mn−m(m+1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' The bound for d is tight for all n > m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Let π = (n, n−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' , n−m+1, 1, 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' , n−m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' The bubble sort for π uses n−1 transpositions for symbol n, n − 2 transpositions for symbol n − 1, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Then d(ε, π) = (n − 1) + (n − 2) + · · · + (n − m) = nm − (1 + 2 + · · · + m) = mn − m(m + 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' The bound is tight since for any permutation σ ̸= π, d(ε, σ) < mn − m(m + 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' We improve the bound in Equation 1 for m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' For any d ≥ 1, (a) P(n, 2, d) ≥ 3 if d ≤ n + ⌊n/3⌋ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' (b) P(n, 2, d) ≥ 5 if d ≤ n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' (a) Let τ1 = (n − 1, n, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' , n − 2), τ2 = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' , x − 1, n − 1, x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' , n − 2, n) and τ3 = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' , x, n, x + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' , n − 1) where x = ⌊n/3⌋, see an example in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Transformation of τ1 to τ2 requires n − 1 transpositions for symbol n − 1 and x − 1 transpositions for symbol n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Then d(τ1, τ2) = n+x−2 ≥ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Similarly d(τ1, τ3) = (n−2)+x ≥ d, and d(τ2, τ3) = (n−x)+(n−x−2) = 2n − 2x − 2 ≥ n + x − 2 ≥ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 4 1 2 3 4 5 6 7 8 9 τ1 8 9 1 2 3 4 5 6 7 τ2 1 2 9 3 4 5 6 7 8 τ3 1 2 3 8 4 5 6 7 9 Table 4: P(9, 2, 10) ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' (b) Suppose n = 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Consider 5 permutations τi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' , 5 where symbols n − 1 and n are placed at positions 1 and 2 for τ1, n − 1 and n for τ2, k and k + 1 for τ3, 1 and n for τ4, n and 1 for τ5, see an example in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' We show that d(τi, τj) ≥ n − 2 if 1 ≤ i < j ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' For all pairs i, j ∈ {1, 2, 4, 5} with i < j, transformation of τi to τj requires n − 2 transpositions for only one of two symbols n − 1 or n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Transformation of τ3 to any τi, i = 1, 2, 4 requires k − 1 transpositions for each symbol n − 1 and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Transformation of τ3 to any τ5 requires k − 1 transpositions for each symbol n − 1 and n after transposition of n − 1 and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 11 12 τ1 11 12 0 0 0 0 0 0 0 0 0 0 τ2 0 0 0 0 0 0 0 0 0 0 11 12 τ3 0 0 0 0 0 11 12 0 0 0 0 0 τ4 11 0 0 0 0 0 0 0 0 0 0 12 τ5 12 0 0 0 0 0 0 0 0 0 0 11 Table 5: P(12, 2, 10) ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' The first 10 symbols in all τi are in the sorted order and replaced by zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 11 12 13 τ1 12 13 0 0 0 0 0 0 0 0 0 0 0 τ2 0 0 0 0 0 0 0 0 0 0 0 12 13 τ3 0 0 0 0 0 13 12 0 0 0 0 0 0 τ4 12 0 0 0 0 0 0 0 0 0 0 0 13 τ5 13 0 0 0 0 0 0 0 0 0 0 0 12 Table 6: An example for P(13, 2, 11) ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Similarly, a (n, 2, n − 2)-array can be constructed for n = 2k + 1 where symbols n and n − 1 are placed at positions k and k + 1 for τ3, see an example in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' We have constructed a program for computing P(n, m, d) for various values of n, m, and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' For each of the � n m � positions for m symbols out of n, and each of the possible m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' orders of the m symbols, the program uses the random/Greedy strategy described earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' That is, it chooses a specified number of random choices first and then tries adding all remaining possible permutations in increasing order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' When m is small, the program finds solutions quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' It allows one to compute P(15, 12), for example, without examining all 15!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' permutations of 15 symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' That is, by Theorem 6 one can first compute, for example, P(15, 3, 12), which as shown in Table 9 is at least 12, and then compute P(12, 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' As shown in Table 11 these are useful for obtaining improved lower bounds for P(n, d) when 5 the Kendall-τ distance d is close to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' We give lower bounds for P(n, m, d), for 8 ≤ d ≤ 15 and 10 ≤ n ≤ 20 in Tables 7, 8, 9, and 10.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='4571 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='Table 8: Lower bounds for P(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 10) (left) and P(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 11) (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 3 Improved Lower Bounds by Theorems 5, 6, and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Each of the improved lower bounds given in Table 11 is explained in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Many of the computations described took weeks on Apple MacBook Air computers with an M1 or M2 processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' By Theorem 7, P(12, 5) ≥ P(12, 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 2) + P(12, 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 7) + P(12, 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 12) ≥ 318, 641 + 334, 200 + 246, 968 = 899, 809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' By Theorem 7, P(12, 7) ≥ P(12, 7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 3) + P(12, 7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 10) ≥ 2 · 64, 649 = 129, 298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' By Theorem 7, P(12, 8) ≥ P(12, 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 3) + P(12, 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 11) ≥ 44, 042 + 41049 = 85, 091.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' By Theorem 7, P(13, 9) ≥ P(13, 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 3) + P(13, 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 12) ≥ 124,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 047 + 112,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 717 = 236,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='109 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='385 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='41 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='138 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='530 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='41 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='174 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='720 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='46 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='220 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='961 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='Table 10: Lower bounds for P(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 14) (left) and P(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 15) (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' By Theorem 7, P(14, 9) ≥ P(14, 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 2, 6, 8)+P(14, 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 2, 3, 5)+P(14, 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 1, 5, 6)+P(14, 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 1, 6, 8)+ P(14, 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 4, 7, 8)+P(14, 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 4, 9, 10)+P(12, 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 8)+P(12, 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 9)+P(13, 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 4, 5)+P(12, 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 3)+P(14, 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 3, 5, 7)+ P(14, 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 3, 5, 14)+P(14, 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 2, 4, 14)+P(14, 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 2, 7, 9)+P(13, 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 4, 5)+P(12, 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 7)+2∗P(12, 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 4)+ P(13, 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 3, 5)+P(13, 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 2, 3)+P(12, 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 3)+P(13, 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 3, 4)+ ≥ 51, 871+26, 347+19, 878+31, 130+ 39, 622 + 42, 132 + 18, 649 + 18, 397 = 19, 914 + 17, 294 + 48, 029 + 28, 367 + 25, 367 + 52, 958 + 19, 915 + 18, 807 + 36, 794 + 28, 348 + 16, 073 + 17, 294 + 18, 542 = 575, 728 By Theorem 7, P(13, 10) ≥ P(13, 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 2) + P(13, 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 12) ≥ 2 ∗ 79, 104 = 158, 208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' By Theorem 7, P(14, 10) ≥ P(14, 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 5, 14)+P(14, 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 8, 14)+P(14, 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 6, 7)+P(14, 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 1, 11)+ P(14, 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 1, 12)+P(14, 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 1, 2) ≥ 94, 643+95, 052+102, 965+93, 157+89, 021+50, 649 ≥ 525, 427 By Theorem 7, P(13, 11) ≥ P(13, 11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 2) + P(13, 11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 13) ≥ 31, 809 + 19, 227 = 51, 046.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' By Theorem 7, P(14, 11) ≥ P(14, 11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 7, 8)+P(14, 11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 14, 3)+P(14, 11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 13, 14)+P(14, 11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 1, 2)+ P(14, 11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 1, 14) ≥ 47, 851 + 36, 250 + 3 ∗ 19, 227 = 141, 782.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' By Theorem 7, P(15, 11) ≥ P(15, 11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 1, 7, 9) + P(15, 11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 9, 10, 15) + P(15, 11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 11, 14, 15) + 7 n:d 5 7 8 9 10 11 12 13 14 15 12 899,809 129,298 85,091 13 236,764 158,208 51,046 29,859 14,158 10,756 5,527 14 595,728 525,427 141,782 100,813 52,565 41,673 15,674 15 1,049,633 524,817 105,130 83,346 37,104 16 2,099,266 1,049,634 267,828 173,432 74,208 17 244,051 Table 11: Improved lower bounds using Theorems 5, 6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Blanks indicate other methods have the best lower bounds known e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' [11] or, for n=12, the best lower bounds are in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' P(15, 11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 8, 9, 11)+P(15, 11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 6, 10, 15)+P(15, 11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 5, 7, 13)+P(15, 11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 5, 6, 15)+P(15, 11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 4, 12, 14)+ P(15, 11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 4, 5, 6)+P(15, 11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 3, 14, 15)+P(15, 11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 2, 7, 11)+P(15, 11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 1, 13, 15)+P(15, 11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 1, 2, 3)+ P(15, 11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 1, 2, 15)+P(15, 11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 1, 2, 15)+P(15, 11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 1, 2, 13)+P(15, 11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 1, 9, 11) ≥ 70, 509+47, 069+ 36, 430+93, 986+85, 010+138, 475+47, 027+107, 707+45, 837+145, 804+3∗19227+31, 861+ 69, 377 ≥ 1, 049, 633 By Theorem 4, P(16, 11) ≥ 2 ∗ P(15, 11) ≥ 2 ∗ 1, 049, 633 = 2, 099, 266 By Theorem 7, P(13, 12) ≥ P(13, 12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 7) ≥ 29, 859.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' By Theorem 7, P(14, 12) ≥ P(14, 12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 7, 8)+P(14, 12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 13, 14)+P(14, 12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 14, 2)+P(14, 12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 1, 14)+ P(14, 12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 1, 2) ≥ 35, 709 + 13, 935 + 23, 299 + 19, 227 + 19, 227 = 100, 813.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' By Theorem 3, P(15, 12) ≥ 1 2P(15, 11) ≥ 524, 817.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' By Theorem 4, P(16, 12) ≥ 2 ∗ P(15, 12) ≥ 1, 049, 634 By Theorem 7, P(13, 13) ≥ P(13, 13;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 7) ≥ 14, 158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' By Theorem 7 P(14, 13) ≥ P(14, 13;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 7, 13) + P(14, 13;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 6, 14) + P(14, 13;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 3, 4) ≥ 23, 388 + 14, 073 + 15, 104 ≥ 52, 565.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' By Theorem 5, P(15, 13) ≥ 2 ∗ P(14, 13) ≥ 2 ∗ 52, 565 = 105, 130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' By Theorem 6, P(16, 13) ≥ P(12, 13) ∗ P(16, 4, 13) ≥ 6, 087 ∗ 44 = 267, 828.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' By Theorem 7, P(13, 14) ≥ P(13, 14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 7) ≥ 10, 756.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' By Theorem 7, P(14, 14) ≥ P(14, 14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 1, 3) + P(14, 14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 4, 14) + P(14, 14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 6, 11) ≥ 8, 036 + 10, 060 + 23, 577 = 41, 673 By Theorem 5, P(15, 14) ≥ 2 ∗ P(14, 14) ≥ 2 ∗ 41, 673 = 83, 346.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' By Theorem 6, P(16, 14) ≥ P(12, 14) ∗ P(16, 4, 14) ≥ 4, 564 ∗ 38 = 173, 432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' By Theorem 7, P(13, 15) ≥ P(13, 15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 7) ≥ 5, 527.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' By Theorem 7, P(14, 15) ≥ P(14, 15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 6, 14) + P(14, 15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 14, 6) + P(14, 15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 2, 3) ≥ 5, 493 + 5, 493 + 4, 688 = 15, 674.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' By Theorem 7, P(15, 15) ≥ P(15, 15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 3, 4, 7, 8) + 3 ∗ P(11, 15) + P(15, 15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 4, 5, 6, 7) + 3 ∗ P(14, 15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 6, 7, 8) + P(13, 15 : 2, 10) + P(15, 15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 2, 3, 4, 13) + P(3, 5, 7, 11) + P(15, 15, 2, 4, 10, 11) + 2P ∗ (13, 15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 2, 3) + P(12, 15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 3) + 4 ∗ P(12, 15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 2) + P(14, 15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 3, 4, 5) ≥ 4, 279 + 3 ∗ 409 + 1, 787 + 3 ∗ 1, 848 + 1, 738 + 1, 964 + 7, 798 + 5, 773 + 2 ∗ 879 + 895 + 4 ∗ 743 + 1, 369 ≥ 37, 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 8 By Theorem 5, P(16, 15) ≥ 2 ∗ P(16, 15) ≥ 74, 208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' By Theorem 6, P(17, 15) ≥ P(12, 15) ∗ P(17, 5, 15) ≥ 2, 239 ∗ 109 = 244, 051.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 4 Automorphism Lower Bounds It is known that for a permutation π(x) : Fq → Fq, where Fq denotes a finite field of order q, the operations of multiplying by a non-zero constant a, adding a constant c, and adding to the argument a constant b, each yield another permutation on Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' That is, aπ(x + b) + c, for all non-zero a and all b, c ∈ Fq, is again a permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' We use this to search for sets of permutations at specified Kendall-τ distance d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' That is, the search can be done for a set of representative permutations and expanded into a full set of permutations using operations on the representatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Our program verifies that the full set of permutations has the stipulated Kendall-τ distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Use the operation π(x) + c on the following 17 representatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' This gives 102 permutations for P(6, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 0 1 2 3 5 4 0 1 2 4 5 3 0 1 3 5 4 2 0 1 5 4 2 3 0 2 3 4 1 5 0 2 4 5 1 3 0 2 5 3 4 1 0 3 1 4 2 5 0 3 2 5 1 4 0 3 4 2 5 1 0 3 5 4 1 2 0 4 1 5 3 2 0 4 2 1 3 5 0 4 5 3 2 1 0 5 2 1 3 4 0 5 3 1 2 4 0 5 4 2 1 3 Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Use the operations aπ(x) + c on the following 14 representatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' This gives 1, 008 permutations for P(9, 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 0 1 2 4 8 3 7 5 6 0 1 2 7 8 5 3 4 6 0 1 3 4 7 2 8 6 5 0 1 3 8 2 6 7 4 5 0 1 3 8 4 6 5 7 2 0 1 4 5 6 7 3 8 2 0 1 4 5 8 2 7 6 3 0 1 6 2 3 4 7 8 5 0 1 6 2 8 7 5 4 3 0 1 6 4 5 2 3 8 7 0 1 6 7 3 4 8 5 2 0 1 7 2 4 6 8 5 3 0 1 7 4 8 3 5 2 6 0 1 8 5 7 4 6 3 2 Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Use the operations aπ(x) + c on the following 8 representatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' This gives 576 permutations for P(9, 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 0 1 2 3 8 4 6 5 7 0 1 2 5 8 6 3 7 4 0 1 4 5 2 8 6 7 3 0 1 5 3 2 4 6 8 7 0 1 5 6 4 8 3 7 2 0 1 6 4 7 2 5 8 3 0 1 6 7 3 2 8 5 4 0 1 8 3 6 5 7 2 4 Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Use the operations aπ(x) + c on the following four representatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' This gives 288 permutations for P(9, 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 0 1 2 6 5 8 7 4 3 0 1 3 8 4 5 2 6 7 0 1 4 6 5 3 7 2 8 0 1 5 2 4 7 3 6 8 Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Use the operations π(x) + c on the following 12 representatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' This gives 84 permutations for P(7, 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 0 1 3 6 5 4 2 0 1 4 2 3 6 5 0 1 6 2 5 4 3 0 2 3 4 1 5 6 0 2 3 6 5 1 4 0 3 4 6 1 2 5 0 3 5 4 1 2 6 0 4 5 6 3 1 2 0 5 2 4 3 6 1 0 5 3 6 1 2 4 0 6 3 5 4 2 1 0 6 4 2 1 3 5 Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Use the operation aπ(x) + c on 8 permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' This gives 448 permutations for P(8, 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 9 0 1 7 4 5 6 2 3 0 2 1 5 3 4 6 7 0 2 6 4 7 3 1 5 0 3 7 5 4 2 1 6 0 5 4 6 7 1 2 3 0 7 3 1 2 6 5 4 0 7 5 4 3 6 1 2 0 7 6 4 2 1 3 5 Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Use the operation aπ(x) + c on 67 permutation representatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' This gives 3,752 permutations for P(8, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 1 2 3 4 5 6 7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 1 2 5 3 6 7 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 1 3 5 7 2 6 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 1 5 4 3 6 2 7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 1 6 2 7 3 4 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 1 6 3 4 2 7 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 1 6 7 4 5 2 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 1 7 3 2 5 6 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 1 7 5 3 2 4 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 1 7 5 6 3 4 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 2 3 5 1 4 7 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 2 3 5 7 6 4 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 2 3 6 5 4 7 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 2 4 1 6 5 7 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 2 4 5 6 3 1 7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 2 5 1 7 4 3 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 2 5 3 4 6 7 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 2 5 4 3 1 6 7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 2 5 6 4 1 7 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 3 7 4 5 6 2 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 3 7 5 4 2 1 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 4 1 6 2 3 5 7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 4 2 7 3 1 5 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 4 2 7 5 6 1 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 4 5 6 2 1 3 7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} 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+page_content='0 6 7 5 3 2 1 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 7 1 2 3 4 5 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 7 1 3 5 4 6 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 7 1 4 3 6 2 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 7 3 4 2 1 5 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 7 3 6 1 4 2 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 7 4 6 3 1 2 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 7 4 6 5 2 3 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 7 5 1 2 3 6 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Use the operation aπ(x) + c on 12 permutation representatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' This gives 672 permutations for P(8, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 0 2 3 6 5 4 7 1 0 2 4 3 1 5 6 7 0 3 2 1 6 4 7 5 0 3 5 1 6 2 7 4 0 5 7 2 4 6 1 3 0 6 3 4 5 2 1 7 0 6 3 7 1 5 2 4 0 6 5 4 7 3 1 2 0 7 1 5 4 6 2 3 0 7 3 6 4 2 1 5 0 7 4 1 2 6 5 3 0 7 5 6 4 1 3 2 Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Use the operation aπ(x) + c on 40 permutation representatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' This gives 2,242 permutations for P(8, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 1 4 5 7 6 3 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 1 7 3 2 5 6 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 2 1 3 7 4 5 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 2 1 5 7 4 6 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 2 1 6 7 5 4 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 2 3 6 1 5 4 7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 2 4 3 5 6 1 7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 2 5 3 7 4 6 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 5 6 2 1 7 4 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 5 6 4 1 3 2 7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 5 7 1 6 4 2 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 5 7 3 2 4 6 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 6 1 2 4 3 5 7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 6 7 2 4 3 1 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 7 1 2 6 3 5 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 7 2 5 1 4 6 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 7 2 5 3 6 4 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 7 4 3 1 5 2 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 7 5 1 4 2 3 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 7 5 6 2 1 4 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Use the operation aπ(x) + c on 3 permutation representatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' This gives 168 permu- tations for P(8, 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 0 5 3 1 4 6 2 7 0 6 1 3 2 5 7 4 0 7 3 1 2 6 5 4 Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Use the operation π(x) + c on the following 48 permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' This gives 336 permu- tations for P(7, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 1 2 4 3 6 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 1 2 5 4 6 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 1 3 2 6 5 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 1 3 4 2 5 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 1 4 5 6 2 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 1 5 3 6 2 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 1 6 2 3 4 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 1 6 5 2 4 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 2 1 3 5 4 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 2 3 4 1 6 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 2 3 6 5 4 1 ' metadata={'source': 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+page_content='0 4 6 1 3 2 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 4 6 5 1 2 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 5 1 2 3 4 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 5 2 4 1 6 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 5 3 1 4 6 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 5 3 6 2 1 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 5 4 1 6 3 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 5 4 3 6 2 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 5 6 1 4 2 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 6 1 4 2 5 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 6 1 5 3 4 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 6 2 5 3 1 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 6 3 1 4 2 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 6 3 2 4 5 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 6 3 5 1 2 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 6 4 2 3 1 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 6 4 5 3 2 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='0 6 5 2 4 1 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Use the operation π(x) + c on the following 18 permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' This gives 126 permu- tations for P(7, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 0 1 4 2 5 3 6 0 1 4 6 3 2 5 0 1 5 2 6 4 3 0 2 1 3 5 4 6 0 2 4 5 6 3 1 0 2 6 4 1 3 5 0 3 1 5 6 4 2 0 3 2 4 5 1 6 0 3 2 6 1 4 5 0 3 5 4 6 2 1 0 4 3 1 5 2 6 0 4 3 6 2 1 5 0 4 5 1 6 3 2 0 5 1 3 4 2 6 0 5 3 2 6 1 4 0 6 1 2 5 3 4 0 6 5 2 4 1 3 0 6 5 3 4 1 2 5 Patterns for P(n, m, d) In this section, let us, for convenience, describe general patterns for strings (permutations) in P(n, 2, d) and P(n, 3, d), by replacing the symbols [1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' n−2] ([1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' n−3], respectively), which are in order, by blank symbols, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' ’-’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' For example, for P(5, 2, 3), we have the set { 4 5 - - - , 5 4 - - , - 4 5 - , - - 5 4 , 4 - - - 5 , 5 - - - 4 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' It is easy to verify that the Kendall-τ distance between any two strings in this set is at least 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' This set agrees with that found by our program, namely P(5, 2, 3) ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Also, for P(10, 2, 3), we have the set { 9 10 - - - - - - - - , 10 9 - - - - - - -, - 9 10 - - - - - - , - - 10 9 - - - - -, - - - 9 10 - - - -, - - - - 10 9 - - -, - - - - - 9 10 - -, - - - - - - 10 9 -, - - - - - - - 9 10, 9 - - - 10 - - - - -, 10 - - - 9 - - - - -, - 9 - - - 10 - - -, - 10 - - - 9 - - - - - 9 - - - - 10 -, - - 10 - - - - 9 -, - - - - 9 - - - 10, - - - - 10 - - - - 9, 9 - - - - - 10 - - -, 10 - - - - - 9 - - -, 9 - - - - - - - 10, 10 - - - - - - - 9 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' It is easy to verify that the Kendall-τ distance between any two strings in this set is at least 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' This set agrees with that found by our program, namely P(10, 2, 3) ≥ 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' These examples show that sets of strings that form a (n, 2, 3)-array contain easily recognized patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' It is an interesting open question if such patterns can be determined for other choices of n, m, and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Along these lines, for d = 3, consider π1(a, b, c) = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' , n − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' , n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' and π2(a, b, c) = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' , n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' , n − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=', where a, b, c denote the number of symbols in the 3 gaps represented by the “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' We will use π1(a, b, c) for a = 0, 2, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' and b = 0, 3, 6, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=', and π2(a, b, c) for a = 1, 3, 5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 11 and b = 0, 3, 6, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=', for each choice of a and b for which the resulting string has length at most n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Using π1(a, b, c) and π2(a, b, c), it can be observed that P(n, 2, 3) ≥ n(n+1) 6 , for n ̸≡ 1 mod 3 and P(n, 2, 3) ≥ (n+2)(n−1) 6 for n ≡ 1 mod 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Similarly, for Kendall-τ distance 4 and for n = 2k +1, use π1(a, b, c) for a = 0, 2, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' and b = 0, 4, 8, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' π2(a, b, c) for a = 0, 2, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' and b = 3, 7, 11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content='. Using these patterns, it can be observed that P(4k+1, 2, 4) ≥ 2k2+k for k ≥ 1 and P(4k+3, 2, 4) ≥ 2k2 + 3k + 1 for k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' 6 Conclusions and Open Questions Theorems 6 and 7 improve many lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' All of the bounds shown in Tables 1, 2, and 11 are improvements on previous results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' The techniques described can be used to obtain other improvements, with sufficient time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFJT4oBgHgl3EQfBCxz/content/2301.11423v1.pdf'} +page_content=' Many of our computations required weeks.' metadata={'source': 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