Ehrhart series of lecture hall polytopes and Eulerian polynomials for inversion sequences

Abstract

For a sequence s=(s1,…,sn) of positive integers, an s-lecture hall partition is an integer sequence λ satisfying 0⩽λ1/s1⩽λ2/s2⩽⋯⩽λn/sn. In this work, we introduce s-lecture hall polytopes, s-inversion sequences, and relevant statistics on both families. We show that for any sequence s of positive integers: (i) the h⁎-vector of the s-lecture hall polytope is the ascent polynomial for the associated s-inversion sequences; (ii) the ascent polynomials for s-inversion sequences generalize the Eulerian polynomials, including a q-analog that tracks a generalization of major index on s-inversion sequences; and (iii) the generating function for the s-lecture hall partitions can be interpreted in terms of a new q-analog of the s-Eulerian polynomials, which tracks a “lecture hall” statistic on s-inversion sequences. We show how four different statistics are related through the three s-families of partitions, polytopes, and inversion sequences. Our approach uses Ehrhart theory to relate the partition theory of lecture hall partitions to their geometry.