Counting Multiderangements by Excedances

We consider the enumeration of multiderangements of a multiset n = {11 , 22 , . . . ,mm} by the number of excedances. We prove several properties, including the invariance under permutations of {n1, n2, . . . , nm}, the symmetry, recurrence relations, the real-rootedness, and a combinatorial expansion, of the generating function dn(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.