On q-integrals over order polytopes

TL;DR

This paper develops a combinatorial framework for multiple q-integrals over order polytopes, linking them to generating functions of linear extensions and applying this to evaluate generalized q-beta and q-Selberg integrals.

Contribution

It introduces a novel combinatorial interpretation of q-integrals over order polytopes, connecting them to poset linear extensions and deriving new generating functions and q-analogues.

Findings

Interpreted q-integrals as generating functions of linear extensions.

Evaluated generalized q-beta and q-Selberg integrals combinatorially.

Established new q-analogues of Ehrhart theory results.

Abstract

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.