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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: Coefficients on Kazhdan–Lusztig polynomials for permutations of size 5
+---
+
+# The Coefficients of Kazhdan-Lusztig Polynomials for Permutations of Size 5
+
+Kazhdan-Lusztig (KL) polynomials are polynomials in a variable \\(q\\) and 
+with integer coefficients that (for our purposes) are indexed by a pair of permutations [1]. 
+We will write the KL polynomial associated with permutations \\(\sigma\\) and \\(\nu\\) as 
+\\(P_{\sigma,\nu}(q)\\). For example, the KL polynomial associated with permutations 
+\\(\sigma = 1 \; 4 \; 3 \; 2 \; 7 \; 6 \; 5 \; 10 \; 9 \; 8 \; 11\\) and 
+\\(\nu = 4 \; 6 \; 7 \; 8 \; 9 \; 10 \; 1 \; 11 \; 2 \; 3 \; 5\\) is  
+
+
+\\(P_{\sigma,\nu}(q) = 1 + 16q + 103q^2 + 337q^3 + 566q^4 + 529q^5 + 275q^6 + 66q^7 + 3q^8\\)  
+
+(see [here](https://gswarrin.w3.uvm.edu/research/klc/klc.html) for efficient software to compute
+these polynomials). KL polynomials have deep connections throughout several areas of mathematics. For example, 
+KL polynomials are related to the dimensions of intersection homology in Schubert calculus, 
+the study of the Hecke algebra, and representation theory of the symmetric group. They 
+can be computed via a recursive formula [[1]](https://link.springer.com/article/10.1007/BF01390031), 
+nevertheless, in many ways they remain mysterious. For instance, there is no known closed 
+formula for the degree of \\(P_{\sigma,\nu}(q)\\).
+
+One family of questions revolve around the coefficients of \\(P_{\sigma,\nu}(q)\\). 
+For instance, it has been hypothesized that the coefficient on the largest possible monomial term 
+\\(q^{\ell(\sigma) - \ell(\nu)-1/2}\\) (where \\(\ell(x)\\) is a statistic of the 
+permutation \\(x\\) called the *length* of the permutation), which is known as the 
+\\(\mu\\)-coefficient, has a combinatorial interpretation but currently this is not 
+known. Better understanding this and other coefficients is of significant 
+interest to mathematicians from a range of fields.
+
+## Dataset details
+
+Each instance in this dataset consists of a pair of permutations of \\(n,x \in S_n\\) 
+along with the coefficients of the polynomial \\(P_{x,w}(q)\\). If \\(x = \;1 \;2 \;3\; 4\; 5\; 6\\),
+\\(w=4 \;5\; 6\; 1 \;2 \;3\\) and \\(P_{v,w}(q) = 1 + 4q + 4q^2 + q^3\\) 
+then the coefficients field is written as `1, 4, 4, 1`. Note that coefficients are listed 
+by increasing degree of the power of \\(q\\) (e.g., the coefficient on \\(1\\) comes first, 
+then the coefficient on \\(q\\), then the coefficient on \\(q^2\\), etc.)
+
+We summarize the limited number of values of coefficients on \\(P_{x,w}(q)\\) take when \\(x, w \in S_5\\). 
+
+**Constant Terms:**
+
+|  | 0 | 1 | Total number of instances | 
+|----------|----------|----------|----------|
+| Train | 8,496 | 3,024 | 11,520 |
+| Test  | 2,123 | 757 | 2,880 |
+
+**Coefficients on \\(q\\):**
+
+|  | 0 | 1 | 2 | Total number of instances | 
+|----------|----------|----------|----------|----------|
+| Train | 11,219 | 267 | 34 | 11,520 |
+| Test  | 2,793 | 77 | 10 | 2,880 |
+
+**Coefficient on \\(q^2\\):**
+
+|  | 0 | 1 | Total number of instances | 
+|----------|----------|----------|----------|
+| Train | 11,514 | 6 | 11,520 |
+| Test  | 2,876 | 4 | 2,880 |
+
+### Kazhdan-Lusztig Polynomials for Permutations of \\(6\\) elements
+
+We summarize the limited number of values coefficients on \\(P_{x,w}(q)\\) take when \\(x, w \in S_6\\).
+
+**Constant Terms:**
+
+|  | 0 | 1 | Total number of instances | 
+|----------|----------|----------|----------|
+| Train | 336,071 | 78,649 | 414,720 |
+| Test  | 83,922 | 19,758 | 103,680 |
+
+**Coefficients on \\(q\\):**
+
+|  | 0 | 1 | 2 | 3 | 4 | Total number of instances | 
+|----------|----------|----------|----------|----------|----------|----------|
+| Train | 397,386 | 13,253 | 3,483 | 535 | 63 | 414,720 |
+| Test  | 99,354 | 3,311 | 883 | 117 | 15 | 103,680 |
+
+**Coefficient on \\(q^2\\):**
+
+|  | 0 | 1 | 2 | 3 | 4 | Total number of instances | 
+|----------|----------|----------|----------|----------|----------|----------|
+| Train | 412,707 | 1,705 | 242 | 40 | 26 | 414,720 |
+| Test  | 103,177 | 441 | 46 | 8 | 8 | 103,680 |
+
+**Coefficient on \\(q^3\\):**
+
+|  | 0 | 1 | Total number of instances | 
+|----------|----------|----------|----------|
+| Train | 414,688 | 32 | 414,720 |
+| Test  | 103,670 | 10 | 103,680 |
+
+### Kazhdan-Lusztig Polynomials for Permutations of \\(7\\) elements
+
+We summarize the limited number of values coefficients on \\(P_{x,w}(q)\\) take when \\(x, w \in S_7\\). 
+
+**Constant Terms:**
+
+|  | 0 | 1 | Total number of instances | 
+|----------|----------|----------|----------|
+| Train | 17,479,910 | 2,841,370 | 20,321,280 |
+| Test  | 4,370,771 | 709,549 | 5,080,320 |
+
+**Coefficients on \\(q\\):**
+
+|  | 0 | 1 | 2 | 3 | 4 | 5 | 6 | Total number of instances | 
+|----------|----------|----------|----------|----------|----------|----------|----------|----------|
+| Train | 19,291,150 | 660,600 | 266,591 | 80,173 | 18,834 | 3,221 | 711 | 20,321,280 |
+| Test  | 4,822,214 | 165,768 | 66,593 | 19,963 | 4,762 | 819 | 201 | 5,080,320 |
+
+**Coefficient on \\(q^2\\):**
+
+|  | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | Total number of instances | 
+|----------|----------|----------|----------|----------|----------|----------|----------|----------|----------|----------|----------|----------|----------|----------|----------|----------|
+| Train | 20,072,738 | 170,412 | 46,226 | 16,227 | 7,621 | 4,023 | 1,287 | 1,153 | 785 | 350 | 152 | 139 | 121 | 42 | 4 | 20,321,280 |
+| Test  | 5,017,962 | 42,748 | 11,568 | 4,021 | 1,905 | 1,065 | 349 | 287 | 183 | 86 | 40 | 37 | 47 | 22 | 5,080,320 |
+
+**Coefficient on \\(q^3\\):**
+
+|  | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 15 | Total number of instances | 
+|----------|----------|----------|----------|----------|----------|----------|----------|----------|----------|----------|----------|----------|----------|----------|
+| Train | 20,291,535 | 22,094 | 4,779 | 1,660 | 590 | 195 | 206 | 115 | 34 | 26 | 24 | 18 | 4 | 20,321,280 |
+| Test  | 507,2831 | 5,498 | 1,213 | 442 | 146 | 61 | 50 | 37 | 14 | 6 | 8 | 14 | 5,080,320 |
+
+**Coefficient on \\(q^4\\):**
+
+|  | 0 | 1 | Total number of instances | 
+|----------|----------|----------|----------|
+| Train | 17,479,910 | 2,841,370 | 20,321,280 |
+| Test  | 4,370,771 | 709,549 | 5,080,320 |
+
+
+## Data Generation
+
+Datasets were generated using C code from Greg Warrington's 
+[website](https://gswarrin.w3.uvm.edu/research/klc/klc.html). The code we used can be found 
+[here](https://github.com/pnnl/ML4AlgComb/tree/master/kl-polynomial_coefficients).
+
+## Task 
+
+**Math question:** Generate conjectures around the properties of coefficients appearing on KL polynomials.
+
+**Narrow ML task:** Predict the coefficients of \\(P_{x,w}(q)\\) given \\(x\\) and \\(w\\). 
+We break this up into a separate task for each coefficient though one could 
+choose to predict all simultaneously. Since there are generally very few 
+different integers that arise as coefficients (at least in these small examples), 
+we frame this problem as one of classification. 
+
+While the classification task as framed does not capture the broader math question exactly, 
+illuminating connections between \\(x\\), \\(w\\), and the coefficients of \\(P_{x,w}(q)\\)
+has the potential to yield critical insights.
+
+## Small model performance
+
+Since there are many possible tasks here, we did not run exhaustive hyperparameter searches. 
+Instead, we ran ReLU MLPs with depth 4, width 256, and learning rate 0.0005.
+
+### Kazhdan-Lusztig Polynomials for Permutations of \\(5\\) elements
+
+Accuracy predicting coefficients for permutations of 5 elements:
+
+| Coefficient | MLP | Transformer | Guessing largest class | 
+|----------|----------|-----------|------------|
+| \\(1\\) | \\(99.8\% \pm 0.2\%\\) | \\(99.9\% \pm 0.1\%\\) | \\(73.7\%\\) | 
+| \\(q\\) | \\(99.5\% \pm 0.4\%\\) | \\(99.2\% \pm 1.0\%\\) | \\(97.0\%\\) |
+| \\(q^2\\) | \\(99.9\% \pm 0.1\%\\) | \\(100.0\% \pm 0.0\%\\) | \\(99.9\%\\) |
+
+The associated macro F1-scores are:
+
+| Coefficient | MLP | Transformer | 
+|----------|----------|-----------|
+| \\(1\\) | \\(99.7\% \pm 0.1\%\\) | \\(99.9\% \pm 0.4\%\\) | 
+| \\(q\\) | \\(93.9\% \pm 3.7\%\\) | \\(92.7\% \pm 7.6\%\\) | 
+| \\(q^2\\) | \\(50.0\% \pm 0.0\%\\) | \\(100.0\% \pm 0.0\%\\) | 
+
+### Kazhdan-Lusztig Polynomials for Permutations of \\(6\\) elements
+
+Accuracy predicting coefficients for permutations of 6 elements:
+
+| Coefficient | MLP | Transformer | Guessing largest class | 
+|----------|----------|-----------|------------|
+| \\(1\\) | \\(99.9\% \pm 0.0\%\\) | \\(100.0\% \pm 0.0\%\\) | \\(80.9\%\\)| 
+| \\(q\\) | \\(99.9\% \pm 0.0\%\\) | \\(99.9\% \pm 0.0\%\\) | \\(95.8\%\\) |
+| \\(q^2\\) | \\(99.9\% \pm 0.0\%\\) | \\(99.9\% \pm 0.0\%\\) | \\(99.5\%\\) |
+| \\(q^3\\) | \\(99.9\% \pm 0.0\%\\) | \\(99.9\% \pm 0.0\%\\) | \\(99.9\%\\) |
+
+The associated macro F1-scores are:
+
+| Coefficient | MLP | Transformer | 
+|----------|----------|-----------|
+| \\(1\\) | \\(99.9\% \pm 0.0\%\\) | \\(100.0\% \pm 0.0\%\\) | 
+| \\(q\\) | \\(99.0\% \pm 1.5\%\\) | \\(98.0\% \pm 3.7\%\\) | 
+| \\(q^2\\) | \\(97.4\% \pm 5.2\%\\) | \\(98.0\% \pm 3.7\%\\) | 
+| \\(q^3\\) | \\(87.9\% \pm 4.5\%\\) | \\(88.3\% \pm 17.1\%\\) | 
+
+The \\(\pm\\) signs indicate 95% confidence intervals from random weight initialization and training.
+
+## References
+
+\[1\] Kazhdan, David, and George Lusztig. "Representations of Coxeter groups and Hecke algebras." Inventiones mathematicae 53.2 (1979): 165-184.
+
+\[2\] Warrington, Gregory S. "Equivalence classes for the μ-coefficient of Kazhdan–Lusztig polyno