Rational lecture hall polytopes and inflated Eulerian polynomials

Abstract

For a sequence of positive integers s=(s ₁,…,s _n ), we define the rational lecture hall polytope $\mathbf{R}_{n}^{(\mathbf{s})}$. We prove that its h ^∗-polynomial, $Q_{n}^{(\mathbf{s})}(x)$, has nonnegative integer coefficients that count certain statistics on s-inversion sequences. The polynomial $Q_{n}^{(\mathbf{s})}(x)$ can be viewed as an inflated version of the s-Eulerian polynomial, $A_{n}^{(\mathbf{s})}(x)$, associated with the integral lecture hall polytope, $\mathbf {P}_{n}^{(\mathbf{s})}$, introduced by Savage and Schuster. The result is applied in three ways: (1) in the theory of s-lecture hall partitions, introduced by Bousquet-Mélou and Eriksson, the generating function, refined to include the size of the last part, now has an explicit description in terms of the inflated s-Eulerian polynomial, $Q_{n}^{(\mathbf{s})}(x)$; (2) for special sequences, s, we get an explicit formula for $Q_{n}^{(\mathbf{s})}(x)$ by computing the Ehrhart quasi-polynomial of $\mathbf{R}_{n}^{(\mathbf {s})}$; and (3) for many sequences, s, the coefficients the inflated s-Eulerian polynomial form a symmetric unimodal sequence, even when the coefficients of the (uninflated) s-Eulerian polynomial, $A_{n}^{(\mathbf {s})}(x)$, do not.