An algebraic interpretation of Eulerian polynomials, derangement polynomials, and beyond, via Gröbner methods

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.