Séminaire Lotharingien de Combinatoire, B64b (2010), 17 pp.
Generalized Dumont-Foata Polynomials and Alternative Tableaux
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. 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 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.
Received: May 21, 2010.
Accepted: October 20, 2010.
Final Version: November 18, 2010.
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