机器学习发现组合数学新对称性，首次给出q,t- Narayana多项式的新组合解释。
Mapping Uncharted Symmetries: Machine Discovery in Combinatorics
- 用伪标签与监督训练交替的函数学习方法，自动构造满足严格比例约束的数学函数。
- 成功发现q,t-Narayana多项式的非交叉划分新组合解释，并证明其对称性在未解情形下成立。
- 代码与形式化证明开源，支持数学结论可验证、可复现，适合组合数学与形式化推理研究者。
受代数组合学中长期未解问题启发，我们表明现代机器学习可促成可验证的数学发现。聚焦于在精确分布约束下构造简单数学函数的问题，我们形式化为「简单学习下的刚性比例」（SLURP）。提出两种方法：MapSeek-Functional 通过伪标签与监督训练交替建模目标函数；MapSeek-Symbolic 直接生成符号公式。将二者应用于代数组合学中的研究问题，首次发现基于非交叉划分的 q,t-Narayana 多项式的新组合解释。利用其中一个发现的统计量，给出了该多项式对称性在先前未解情况下的组合证明。为保障可验证性与可复现性，本文发布全部代码，并在 Lean 4 中形式化所有数学发现。
原文摘要 · Abstract (English)
Inspired by long-standing open problems in algebraic combinatorics, we show that modern machine learning can meaningfully contribute to verifiable mathematical discoveries. In particular, we focus on the construction of simple mathematical functions under exact distributional constraints, a setting we formalize as Simple Learning Under Rigid Proportions (SLURP). We tackle this problem by introducing two methods: MapSeek-Functional, which models the desired function alternating pseudo-labeling and supervised training steps; and MapSeek-Symbolic, designed to directly produce symbolic formulas. We successfully apply both methods to a research problem in algebraic combinatorics, discovering a new combinatorial interpretation of the $q,t$-Narayana polynomials arising from representation theory. To our knowledge, this is the first such interpretation based on noncrossing partitions. Using one discovered statistic, we find a combinatorial proof of the symmetry of these polynomials in a previously unsolved case. To streamline verification and reproducibility, we release all code, including a formalization of all the mathematical discoveries of this paper in Lean 4.
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