README.md

---
license: cc-by-2.0
pretty_name: the mHeight of permutations of size 9
---

# The mHeight Function of a Permutation of Size 9

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 \\(3412\\)-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](https://github.com/pnnl/ML4AlgComb/tree/master/kl-polynomial_coefficients)) 
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)\\) patterns and the mHeight of a permutation

A \\(3412\\) *pattern* in a permutation \\(\sigma  = a_1 \ldots a_n \in S_n\\) is a 
quadruple \\((a_i,a_j,a_k,a_\ell)\\) such that \\(i < j < k < \ell\\) but 
\\(a_k < a_\ell < a_i < a_j\\). Patterns have deep connections to algebra 
and geometry [3]. Suppose \\(\sigma\\) contains at least 
one occurrence of a \\(3412\\) pattern, \\((a_i,a_j,a_k,a_\ell)\\). 
The *height* of \\((a_i,a_j,a_k,a_\ell)\\) is \\(a_i - a_\ell\\). The *mHeight* of 
\\(\sigma\\) is then the minimum height over all \\(3412\\) patterns in \\(\sigma\\). 
If \\(\sigma\\) contains no \\(3412\\) permutations then the mHeight is set to 0.

## Dataset 

This dataset contains permutations of \\(9\\) elements labeled by their mHeight. Permutations are written in 
1-line notation.

For \\(n = 9\\), mHeight takes values 0, 1, 2, 3, 4, 5, so we frame this as a classification task.

| mHeight value  | 0 | 1 | 2 | 3 | 4 | 5 |  Total number of instances | 
|----------|----------|----------|----------|----------|----------|----------|----------|
| Train | 49,092 | 3,161 | 524 | 77 | 9 | 1 | 52,864 |
| Test  | 12,317 | 759 | 118 | 19 | 3 | 0 | 13,216 |

## Data Generation

The datasets generation scripts can be found at [here](https://github.com/pnnl/ML4AlgComb/tree/master/mheight_function).

## 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= 9\\) | \\(93.2\%\\) | \\(99.8\% \pm 0.6\%\\) | \\(99.9\% \pm 0.4\%\\) | \\(93.2\%\\) |

The \\(\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](https://www.sagemath.org/). Data generation scripts can be found [here](https://github.com/pnnl/ML4AlgComb/tree/master/mheight_function).

- **Repository:** [ACD Repo](https://github.com/pnnl/ML4AlgComb/tree/master/mheight_function)
  
## 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] Gaetz, Christian, and Yibo Gao. "On the minimal power of \\(q\\) 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.