The ${1/k}$-Eulerian Polynomials

Abstract

We use the theory of lecture hall partitions to define a generalization of the Eulerian polynomials, for each positive integer $k$. We show that these ${1}/{k}$-Eulerian polynomials have a simple combinatorial interpretation in terms of a single statistic on generalized inversion sequences. The theory provides a geometric realization of the polynomials as the $h^*$-polynomials of $k$-lecture hall polytopes. Many of the defining relations of the Eulerian polynomials have natural ${1}/{k}$-generalizations. In fact, these properties extend to a bivariate generalization obtained by replacing ${1}/{k}$ by a continuous variable. The bivariate polynomials have appeared in the work of Carlitz, Dillon, and Roselle on Eulerian numbers of higher order and, more recently, in the theory of rook polynomials.