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README.md

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-license: cc-by-2.0
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+---
+license: cc-by-2.0
+pretty_name: the mHeight of permutations of size 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 \\(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\\). 
+
+## Dataset 
+
+This dataset contains permutations of \\(8\\) elements labeled by their mHeight. Permutations are written in 
+1-line notation.
+
+For $n = 8$, 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](https://github.com/pnnl/ML4AlgComb/tree/master/mheight_function).
+
+## Task
+
+**ML task:** Given a permutation, predict the mHeight. 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= 8\\) | \\(91.4\%\\) | \\(99.4\% \pm 0.3\%\\) | \\(99.7\% \pm 0.4\%\\) | \\(91.4\%\\) |
+
+The $\pm$ signs indicate 95% confidence intervals from random weight initialization and training.
+
+## 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.