On q-integrals over order polytopes

A combinatorial study of multiple q-integrals is developed. This includes a q-volume of a convex polytope, which depends upon the order of q-integration. A multiple q-integral over an order polytope of a poset is interpreted as a generating function of linear extensions of the poset. Specific modifications of posets are shown to give predictable changes in q-integrals over their respective order polytopes. This method is used to combinatorially evaluate some generalized q-beta integrals. One such application is a combinatorial interpretation of a q-Selberg integral. New generating functions for generalized Gelfand–Tsetlin patterns and reverse plane partitions are established. A q-analogue to a well known result in Ehrhart theory is generalized using q-volumes and q-Ehrhart polynomials.