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We explain how the moments of the (weight function of the) Askey Wilsonpolynomials are related to the enumeration of the staircase tableaux introducedby the first and fourth authors. This gives us a direct combinatorial formulafor these moments. Then we use techniques developed by Ismail and the thirdauthor to give explicit formulae for these moments and for the enumeration ofstaircase tableaux. Finally we study the enumeration of staircase tableaux atvarious specializations of the parameterizations; for example, we obtain theCatalan numbers, Fibonacci numbers, Eulerian numbers, the number ofpermutations, and the number of matchings.

Formulae for Askey-Wilson moments and enumeration of staircase tableaux