Rational lecture hall polytopes and inflated Eulerian polynomials

For a sequence of positive integers s = (s1, . . . , sn), we define the rational lecture hall polytope R n . We prove that its h∗-polynomial, Q (s) n (x), has nonnegative integer coefficients that count certain statistics on s-inversion sequences. The polynomial Q n (x) can be viewed as an inflated version of the s-Eulerian polynomial, A n (x), associated with the integral lecture hall polytope, P n , 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 (x); (2) for special sequences, s, we get an explicit formula for Q (s) n (x) by computing the Ehrhart quasi-polynomial of R n ; 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 (x), do not.