Mapping Uncharted Symmetries: Machine Discovery in Combinatorics

TL;DR

This paper demonstrates how machine learning can aid in mathematical discovery, specifically in algebraic combinatorics, by introducing methods that find symbolic formulas and interpret polynomials.

Contribution

It introduces two novel machine learning methods, MapSeek-Functional and MapSeek-Symbolic, for discovering mathematical functions and formulas under strict constraints.

Findings

Discovered a new combinatorial interpretation of q,t-Narayana polynomials

Provided a combinatorial proof of polynomial symmetry in an unsolved case

Released code and formalizations in Lean 4 for reproducibility

Abstract

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.