ACDRepo/mheight_function_8

The mHeight Function of Permutations of Size 8

Truly challenging open problems in mathematics often require the development of new mathematical constructions (or even entire new areas of mathematics). This dataset represents a modest example of this. The mHeight function is a statistic associated with a permutation that relates to all $34123412$-patterns in the permutation. It was developed and plays a crucial role in the proof by Gaetz and Gao [1] which resolved a long-standing conjecture of Billey and Postnikov [2] about the coefficients on Kazhdan-Lusztig polynomials (see our Kazhdan-Lusztig polynomial dataset) which carry important geometric information about certain spaces, Schubert varieties, that are of interest both to mathematicians and physicists. The task of predicting the mHeight function represents an interesting opportunity to understand whether a non-trivial intermediate step in an important proof can be learned by machine learning.

$(3412)(3412)$ patterns and the mHeight of a permutation

A $34123412$ pattern in a permutation $\sigma =a_1\dots a_n\in S_n\backslash sigma\; =\; a\_1\; \backslash ldots\; a\_n\; \backslash in\; S\_n$ is a quadruple $(a_i,a_j,a_k,a_{\mathrm{\ell }})(a\_i,a\_j,a\_k,a\_\backslash ell)$ such that but . Patterns have deep connections to algebra and geometry [3]. Suppose $\sigma \backslash sigma$ contains at least one occurrence of a $34123412$ pattern, $(a_i,a_j,a_k,a_{\mathrm{\ell }})(a\_i,a\_j,a\_k,a\_\backslash ell)$. The height of $(a_i,a_j,a_k,a_{\mathrm{\ell }})(a\_i,a\_j,a\_k,a\_\backslash ell)$ is $a_i-a_{\mathrm{\ell }}a\_i\; -\; a\_\backslash ell$. The mHeight of $\sigma \backslash sigma$ is then the minimum height over all $34123412$ patterns in $\sigma \backslash sigma$. If $\sigma \backslash sigma$ contains no $34123412$ permutations then the mHeight is set to 0.

Dataset

This dataset contains permutations of $88$ elements labeled by their mHeight. Permutations are written in 1-line notation.

For $n=8n\; =\; 8$, mHeight takes values 0, 1, 2, 3, 4, so we frame this as a classification task.

mHeight value	0	1	2	3	4	Total number of instances
Train	6,716	508	78	9	1	7,312
Test	1,672	136	18	3	0	1,829

Data Generation

The datasets generation scripts can be found at here.

Task

ML task: Re-discover the notation of mHeight 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 0
$n=8n=\; 8$	$91.4\mathrm{\%}91.4\backslash \%$	$99.4\mathrm{\%}\pm 0.3\mathrm{\%}99.4\backslash \%\; \backslash pm\; 0.3\backslash \%$	$99.7\mathrm{\%}\pm 0.4\mathrm{\%}99.7\backslash \%\; \backslash pm\; 0.4\backslash \%$	$91.4\mathrm{\%}91.4\backslash \%$

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

Further information

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

Dataset Sources

The dataset was generated using SageMath. 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, acdbenchdataset@gmail.com

References

[1] Gaetz, Christian, and Yibo Gao. "On the minimal power of $qq$ in a Kazhdan-Lusztig polynomial." arXiv preprint arXiv:2303.13695 (2023).
[2] Billey, Sara, and Alexander Postnikov. "Smoothness of Schubert varieties via patterns in root subsystems." Advances in Applied Mathematics 34.3 (2005): 447-466.
[3] Billey, Sara C. "Pattern avoidance and rational smoothness of Schubert varieties." Advances in Mathematics 139.1 (1998): 141-156.