Séminaire Lotharingien de Combinatoire, B13a (1986), 19 pp.
[Formerly: Publ. I.R.M.A. Strasbourg, 1986, 316/S-13, p. 27-44.]

Dominique Dumont

Pics de cycles et dérivées partielles

Abstract. We introduce four sequences of polynomials in three variables that present strong analogies. Each of those sequences is defined by a recurrence involving partial derivatives, each of them can also be defined in a counting permutation context involving even and odd cycle peaks. Furthermore, when one variable vanishes, the ordinary generating functions for the four sequences of polynomials in two variables thereby derived have continued fraction expansions, that appear to be extensions of the continued fraction expansions for the Euler and Genocchi numbers. Finally, when two variables are identified, the polynomials thereby obtained have ordinary generating functions that can be expressed as series of rational fractions having the same form. The proofs derived here are purely combinatorial. This rises the problem of getting an analytical proof. No exponential generating function is known for those four sequences, except for the first one that is an elliptic function.

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