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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 ..."
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Cited by 8 (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
CROSSINGS OF SIGNED PERMUTATIONS AND qEULERIAN NUMBERS OF TYPE B
"... Abstract. In this paper we want to study combinatorics of the type B permutations and in particular the join statistics crossings, excedances and the number of negative entries. We generalize most of the results known for type A (i.e. zero negative entries) and use a mix of enumerative, algebraic a ..."
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Abstract. In this paper we want to study combinatorics of the type B permutations and in particular the join statistics crossings, excedances and the number of negative entries. We generalize most of the results known for type A (i.e. zero negative entries) and use a mix of enumerative, algebraic and bijective techniques. This work has been motivated by permutation tableaux of type B introduced by Lam and Williams, and natural statistics that can be read on these tableaux. We mostly use (pignose) diagrams and labelled Motzkin paths for the combinatorial interpretations of our results. 1.