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Formulae for Askey–Wilson moments and enumeration of staircase tableaux
, 2010
"... 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 [11, 12]. This gives us a direct combinatorial formula for these moments, which is related to, but more elegant th ..."
Abstract

Cited by 7 (2 self)
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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 [11, 12]. This gives us a direct combinatorial formula for these moments, which is related to, but more elegant than the formula given in [11]. 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.
GENERALIZED DUMONTFOATA POLYNOMIALS AND ALTERNATIVE TABLEAUX
"... Dumont and Foata introduced in 1976 a threevariable symmetric refinement of Genocchi numbers, which satisfies a simple recurrence relation. A sixvariable generalization with many similar properties was later considered by Dumont. They generalize a lot of known integer sequences, and their ordina ..."
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Dumont and Foata introduced in 1976 a threevariable symmetric refinement of Genocchi numbers, which satisfies a simple recurrence relation. A sixvariable generalization with many similar properties was later considered by Dumont. They generalize a lot of known integer sequences, and their ordinary generating function can be expanded as a Jacobi continued fraction. We give here a new combinatorial interpretation of the sixvariable polynomials in terms of the alternative tableaux introduced by Viennot. A powerful tool to enumerate alternative tableaux is the socalled “matrix Ansatz”, and using this we show that our combinatorial interpretation naturally leads to a new proof of the continued fraction expansion.
Symmetric distribution of crossings and nestings in permutations of type B
"... This note contains two results on the distribution of crossing numbers and nesting numbers in permutations of type B. More precisely, we prove a Bnanalogue of the symmetric distribution of crossings and nestings of permutations due to Corteel [Adv. Appl. Math. 38(2)(2007), 149–163] as well as the s ..."
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This note contains two results on the distribution of crossing numbers and nesting numbers in permutations of type B. More precisely, we prove a Bnanalogue of the symmetric distribution of crossings and nestings of permutations due to Corteel [Adv. Appl. Math. 38(2)(2007), 149–163] as well as the symmetric distribution of kcrossings and knestings of permutations due to Burrill et al. [DMTCS proc. AN (2010), 593–600]. 1
Séminaire Lotharingien de Combinatoire 64 (2010), Article B64b GENERALIZED DUMONT–FOATA POLYNOMIALS AND ALTERNATIVE TABLEAUX
"... Abstract. Dumont and Foata introduced in 1976 a threevariable symmetric refinement of Genocchi numbers, which satisfies a simple recurrence relation. A sixvariable generalization with many similar properties was later considered by Dumont. It generalizes a lot of known integer sequences, and its o ..."
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Abstract. Dumont and Foata introduced in 1976 a threevariable symmetric refinement of Genocchi numbers, which satisfies a simple recurrence relation. A sixvariable generalization with many similar properties was later considered by Dumont. It generalizes a lot of known integer sequences, and its ordinary generating function can be expanded as a Jacobi continued fraction. We give here a new combinatorial interpretation of the sixvariable polynomials in terms of the alternative tableaux introduced by Viennot. A powerful tool to enumerate alternative tableaux is the socalled “matrix Ansatz, ” and using this we show that our combinatorial interpretation naturally leads to a new proof of the continued fraction expansion. 1.