Lecture hall tableaux

TL;DR

This paper introduces lecture hall tableaux, a new combinatorial object that generalizes existing structures and connects to multivariate little q-Jacobi polynomials, providing new formulas and properties.

Contribution

It defines lecture hall tableaux, relates them to multivariate little q-Jacobi polynomials, and derives new generating functions and product formulas for their moments.

Findings

Lecture hall tableaux generalize several known combinatorial objects.

Coefficients in Schur expansion are generating functions for lecture hall tableaux.

Moments of multivariate little q-Jacobi polynomials have a product formula.

Abstract

We introduce lecture hall tableaux, which are fillings of a skew Young diagram satisfying certain conditions. Lecture hall tableaux generalize both lecture hall partitions and anti-lecture hall compositions, and also contain reverse semistandard Young tableaux as a limit case. We show that the coefficients in the Schur expansion of multivariate little $q$-Jacobi polynomials are generating functions for lecture hall tableaux. Using a Selberg-type integral we show that moments of multivariate little $q$-Jacobi polynomials, which are equal to generating functions for lecture hall tableaux of a Young diagram, have a product formula. We also explore various combinatorial properties of lecture hall tableaux.