Q-analogues of Ehrhart polynomials

Abstract

AbstractWe consider weighted sums over points of lattice polytopes, where the weight of a point v is the monomial qλ(v) for some linear form λ. We propose a q-analogue of the classical theory of Ehrhart series and Ehrhart polynomials, including Ehrhart reciprocity and involving evaluation at the q-integers. The main novelty is the proposal to consider q-Ehrhart polynomials. This general theory is then applied to the special case of order polytopes associated with partially ordered sets. Some more specific properties are described in the case of empty polytopes.