Formulae for Askey-Wilson moments and enumeration of staircase tableaux

TL;DR

This paper establishes a direct combinatorial formula connecting Askey-Wilson polynomial moments with staircase tableaux enumeration, providing explicit formulas and exploring special cases related to well-known combinatorial sequences.

Contribution

It introduces a new combinatorial approach to compute Askey-Wilson moments and links these to staircase tableaux enumeration, including explicit formulas and special case analyses.

Findings

Derived a direct combinatorial formula for Askey-Wilson moments.

Obtained explicit formulas for moments and staircase tableaux enumeration.

Connected staircase tableaux enumeration to classical combinatorial numbers like Catalan and Fibonacci numbers.

Abstract

We explain how the moments of the (weight function of the) Askey Wilson polynomials are related to the enumeration of the staircase tableaux introduced by the first and fourth authors. This gives us a direct combinatorial formula for these moments. Then we use techniques developed by Ismail and the third author to give explicit formulae for these moments and for the enumeration of staircase tableaux. Finally we study the enumeration of staircase tableaux at various specializations of the parameterizations; for example, we obtain the Catalan numbers, Fibonacci numbers, Eulerian numbers, the number of permutations, and the number of matchings.