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Article Dans Une Revue European Journal of Combinatorics Année : 2010

A q-enumeration of alternating permutations

Résumé

A classical result of Euler states that the tangent numbers are an alternating sum of Eulerian numbers. A dual result of Roselle states that the secant numbers can be obtained by a signed enumeration of derangements. We show that both identities can be refined with the following statistics: the number of crossings in permutations and derangements, and the number of patterns 31-2 in alternating permutations. Using previous results of Corteel, Rubey, Prellberg, and the author, we derive closed formulas for both q-tangent and q-secant numbers. There are two different methods for obtaining these formulas: one with permutation tableaux and one with weighted Motzkin paths (Laguerre histories).

Dates et versions

hal-03618495 , version 1 (24-03-2022)

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Matthieu Josuat-Vergès. A q-enumeration of alternating permutations. European Journal of Combinatorics, 2010, 31 (7), pp.1892-1906. ⟨10.1016/j.ejc.2010.01.008⟩. ⟨hal-03618495⟩
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