Permutations selon leurs pics, creux, doubles montées et double descentes, nombres d'euler et nombres de Genocchi

Abstract

We study permutations whose type is given, the type being the sets of the values of the peaks, throughs, doubles rises and double falls. We show that the type of a permutation on n letters is caracterized by a map γ[n]→[n]; the number of possible types is the Catalan number; the number of permutations whose type is associated with γ is the product γ(1)γ(2)·γ(n). This result is a corollary of an explicit bijection between permutations and pairs (γ, ƒ) where ƒ is a map dominated by γ. Specifying this bijection tG various classes of permutations provides enumerative formulas for classical numbers, e.g. Euler and Genocchi numbers. It has been proved recently that each enumerative formula of this work is equivalent to a continued fraction expansion of a generating serie.