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Séminaire Lotharingien de Combinatoire, B59d (2008), 15 pp.

# Chak-On Chow

# Counting Multiderangements by Excedances

**Abstract.**
We consider in this work the enumeration of multiderangements of a
multiset
**n**={1^{n1},2^{n2},...,*m*^{nm}} by the
number of excedances. We prove several properties, including the
invariance under permutations of
{*n*_{1},*n*_{2},...,*n*_{m}}, the
symmetry, the recurrence relation, the real-rootedness, and a
combinatorial expansion, of the generating function
*d*_{n}(*x*)
of multiderangements by excedances, thus generalizing the
corresponding results for the classical derangements. By a further
extension, the generating function for multipermutations by
numbers of excedances and fixed points is also given.

Received: October 8, 2007.
Revised: April 28, 2008; May 13, 2008.
Accepted: May 18, 2008.
Final Version: May 18, 2008.

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