An algebraic interpretation of Eulerian polynomials, derangement polynomials, and beyond, via Gr\"obner methods

TL;DR

This paper uses Gr"obner basis methods to explore algebraic structures underlying Eulerian and derangement polynomials, connecting them to Chow rings of matroids and ascent statistics on inversion sequences.

Contribution

It introduces a new Gr"obner basis approach to study Chow rings of matroids, linking classical polynomials to algebraic and combinatorial structures.

Findings

Reestablishes the connection between derangement polynomials and Chow rings of corank 1 uniform matroids.

Expresses Hilbert series of Chow rings for uniform matroids via ascent statistics.

Provides a new algebraic interpretation of Eulerian and derangement polynomials.

Abstract

Motivated by the question of whether Chow polynomials of matroids have only real roots, this article revisits the known relationship between Eulerian polynomials and the Hilbert series of Chow rings of permutohedral varieties. This is done using a quadratic Gr\"obner basis associated to a new presentation of those rings, which is obtained by iterating the semi-small decomposition of Chow rings of matroids. This Gr\"obner basis can also be applied to compute certain principal ideals in these rings, and ultimately reestablish the known connection between derangement polynomials and the Hilbert series of Chow rings for corank 1 uniform matroids. More broadly, this approach enables us to express the Hilbert series of Chow rings for any uniform matroid as polynomials related to the ascent statistics on particular sets of inversion sequences.