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arXiv · 1608.03342

On q-integrals over order polytopes

Jang Soo Kim·Dennis Stanton
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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.

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Jang Soo Kim, Dennis Stanton. 2016-08-11. On q-integrals over order polytopes. https://arxiv.org/abs/1608.03342

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