README.md

metadata

license: cc-by-2.0
pretty_name: Characterizing weaving patterns, n = 7

Dataset Card for Weaving Patterns of Size, $n=7n\; =\; 7$

Weaving patterns are $n\times n-1n\; \backslash times\; n-1$-matrices with $\{1,2,\mathrm{.}\mathrm{.}\mathrm{.},n\}\backslash \{1,\; 2,\; .\; .\; .\; ,\; n\backslash \}$- entries introduced by [1] to study the number of reduced decompositions of the permutation $\sigma =nn-1\dots 1\backslash sigma\; =\; n\backslash ;\; n\; -\; 1\; \backslash ;\backslash dots\backslash ;\; 1$ up to commutation equivalence. The number of such objects also counts 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.

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

Dataset Details

Weaving patterns of size $n\times n-1n\; \backslash times\; n\; -\; 1$ are a special type of matrix containing entries in $\{1,2,\mathrm{.}\mathrm{.}\mathrm{.},n\}\backslash \{1,\; 2,\; .\; .\; .\; ,\; n\backslash \}$. They correspond to representations of the longest word permutation of $nn$ elements (the permutation that sends $1\mapsto n1\; \backslash mapsto\; n$, $2\mapsto n-12\; \backslash mapsto\; n\; -\; 1$, etc.). This task involves trying to identify weaving patterns among matrices that look like weaving patterns but are not. Each matrix is stored on a single line in row-major format. For instance,

(0, 1, 2, 3, 3, 2, 3, 4, 2, 3, 2, 1, 5, 4, 3, 2)

The datasets can also be found here. Data loaders can be found here.

Statistics

	Weaving patterns	Non-weaving patterns
Train	17,388	96,012
Test	7,310	41,290

This dataset is small, we encourage users to also look at the dataset for $n=7n\; =\; 7$.

Math question: Find an algorithm or set of rules that can efficiently distinguish between weaving pattern matrices and non-weaving pattern matrices. This 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 $\{1,2,\mathrm{.}\mathrm{.}\mathrm{.},n\}\backslash \{1,\; 2,\; .\; .\; .\; ,\; n\backslash \}$- matrix is a weaving pattern or not. This task is framed as binary classification. Extract mathematical insights from a performant model.

If a successful model is trained, it would be interesting to understand whether the model has learned an existing algorithm or whether it has discovered something new.

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
$n=7n=\; 7$	$85.8\mathrm{\%}85.8\backslash \%$	$99.3\mathrm{\%}\pm 0.2\mathrm{\%}99.3\backslash \%\; \backslash pm\; 0.2\backslash \%$	$99.9\mathrm{\%}\pm 0.4\mathrm{\%}99.9\backslash \%\; \backslash pm\; 0.4\backslash \%$	$85.0\mathrm{\%}85.0\backslash \%$

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

Uses

This dataset was generated to study ML model's ability yield insight on an open problem in algebraic combinatorics, specifically, the problem of better understanding commutation equivalent representations of reduced words coresponding to the longest permutation.

Direct Use

We use this dataset for a classification task distinguishing between weaving and non-weaving patterns.

Out-of-Scope Use

None.

Dataset Structure

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

(0, 1, 2, 3, 3, 2, 3, 4, 2, 3, 2, 1, 5, 4, 3, 2)

Dataset Creation

Data generation scripts can be found here.

Curation Rationale

This dataset was generated as a test of current AI system's ability to advance research mathematics.

Who are the source data producers?

Herman Chau wrote code to generate this dataset.

Bias, Risks, and Limitations

We only provide data for weaving patterns of size $6\times 56\; \backslash times\; 5$ and $7\times 67\; \backslash times\; 6$ in this repository. We are happy to generate (subsets) of datasets for larger values of \(n \times n-1)).

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).