ACDRepo/weaving_patterns_7

Dataset Card for Weaving Patterns of Size $7\times 67\; \backslash times\; 6$

Weaving patterns are $(n\times n-1)(n\; \backslash times\; n-1)$-matrices with $\{1,2,\dots ,n\}\backslash \{1,\; 2,\; \backslash dots\; ,\; n\backslash \}$- entries introduced by [1] to study the number of reduced decompositions of the longest permutation (which swaps $nn$ and $11$, $nn$ - $11$ and $22$, etc.) up to commutation equivalence. The number of such objects counts a wide range of combinatorial phenomena, including the number of parallel sorting networks, the number of rhombic tilings of regular polygons, and is connected to the study of the higher Bruhat orders [2]. An $O(n^2)O(n^2)$ algorithm for determining if a given $\{1,2,\mathrm{.}\mathrm{.}\mathrm{.},n\}\backslash \{1,\; 2,\; .\; .\; .\; ,\; n\backslash \}$-matrix is a valid weaving pattern exists but gives no additional insight into the structure of weaving patterns and correspondingly the asymptotics of reduced decompositions. The enumeration of reduced decompositions up to commutation equivalence has been studied by many including Knuth and Stanley. An exact formula is likely out of reach, so asymptotic upper and lower bounds are of great interest. ML models that can detect necessary or sufficient conditions for a matrix to be a valid weaving pattern have the potential to lead to substantial improvements in the upper bound.

This dataset is a mixture of enriched weaving patterns and non-weaving pattern matrices with $\{1,2,\dots ,7\}\backslash \{1,\; 2,\; \backslash dots,\; 7\backslash \}$-entries.

Dataset Details

Each matrix is stored on a single line in row-major format. For instance,

[ 3, 5, 4, 2, 6, 7, 7, 5, 6, 1, 4, 3, 6, 4, 1, 5, 7, 2, 6, 3, 1, 5, 7, 2, 2, 3, 1, 4, 7, 6, 2, 3, 4, 1, 7, 5, 2, 3, 4, 1, 6, 5 ].

Labels are 0 (not a weaving pattern) and 1 (a weaving pattern).

Data loaders can be found here.

Statistics

	Weaving patterns	Non-weaving patterns	Total instances
Train	17,388	96,012	113,400
Test	7,310	41,290	48,600

Math question: Find necessary or sufficient conditions to distinguish between weaving pattern matrices and non-weaving pattern matrices. These should be more efficient than the $O(n^2)O(n^2)$ algorithm that can be found in the references above.

ML task: Train a model to classify whether a $\{1,2,\mathrm{.}\mathrm{.}\mathrm{.},7\}\backslash \{1,\; 2,\; .\; .\; .\; ,\; 7\backslash \}$- matrix is a weaving pattern or not. This task is framed as binary classification. Extract mathematical insights from a performant model.

Small model performance

We provide some basic baselines for this task. Benchmarking details can be found in the associated paper.

Size	Logistic regression	MLP	Transformer	Guessing largest class
$7\times 67\; \backslash times\; 6$	$85.8\mathrm{\%}85.8\backslash \%$	$99.3\mathrm{\%}\pm 0.2\mathrm{\%}99.3\backslash \%\; \backslash pm\; 0.2\backslash \%$	$99.9\mathrm{\%}\pm 0.4\mathrm{\%}99.9\backslash \%\; \backslash pm\; 0.4\backslash \%$	$85.0\mathrm{\%}85.0\backslash \%$

The $\pm \backslash pm$ signs indicate 95% confidence intervals from random weight initialization and training.

• Curated by: Herman Chau
• Funded by: Pacific Northwest National Laboratory
• Language(s) (NLP): NA
• License: CC-by-2.0

Dataset Sources

Data generation scripts can be found here.

• Repository: ACD Repo

Citation

BibTeX:

@article{chau2025machine,
    title={Machine learning meets algebraic combinatorics: A suite of datasets capturing research-level conjecturing ability in pure mathematics},
    author={Chau, Herman and Jenne, Helen and Brown, Davis and He, Jesse and Raugas, Mark and Billey, Sara and Kvinge, Henry},
    journal={arXiv preprint arXiv:2503.06366},
    year={2025}
}

APA:

Chau, H., Jenne, H., Brown, D., He, J., Raugas, M., Billey, S., & Kvinge, H. (2025). Machine learning meets algebraic combinatorics: A suite of datasets capturing research-level conjecturing ability in pure mathematics. arXiv preprint arXiv:2503.06366.

Dataset Card Contact

Henry Kvinge, [email protected]

References

[1] Felsner, Stefan. "On the number of arrangements of pseudolines." Proceedings of the twelfth annual Symposium on Computational Geometry. 1996.
[2] Chau, Herman. "On enumerating higher bruhat orders through deletion and contraction." arXiv preprint arXiv:2412.10532 (2024).