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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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