Generalized Dumont-Foata polynomials and alternative tableaux

Details Generalized Dumont-Foata polynomials and alternative tableaux Generalized Dumont-Foata polynomials and alternative tableaux

2010-05-21

arxiv.org/abs/1005.4007v2

S\'eminaire Lotharingien de Combinatoire 64 (2010), Article B64b, 17pp

Mathematics

Combinatorics

17 pages

Scientific paper

Dumont and Foata introduced in 1976 a three-variable symmetric refinement of Genocchi numbers, which satisfies a simple recurrence relation. A six-variable 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 six-variable polynomials in terms of the alternative tableaux introduced by Viennot. A powerful tool to enumerate alternative tableaux is the so-called "matrix Ansatz", and using this we show that our combinatorial interpretation naturally leads to a new proof of the continued fraction expansion.

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