#####
Séminaire Lotharingien de Combinatoire, B21h (1989).

[Formerly: Publ. I.R.M.A. Strasbourg, 1990, 413/S-21, p.
99-110.]

# Christophe Reutenauer

# Number of Permutations with Given Descent Set and Cycle Structure

**Abstract.**
The purpose of this paper is
to count permutations in *S*_{n}
with a given cycle structure
and a given
descent set. Our main result asserts that the number of
these permutations can be expressed as a scalar product of two
symmetric
functions, one associated with the cycle structure and the other with
the descent set. Both of these symmetric functions can be
interpreted as
characteristics of certain representations of the symmetric group.

The paper has been finally published as a joint paper with Ira Gessel
under the title "Counting permutations with given cycle structure and
descent set" in
*J. Combin. Theory Ser. A* **64** (1993), 189-215.