Excedance Numbers for Permutations in Complex Reflection Groups

The classical Weyl groups appear as special cases: for r = s = 1 we have the symmetric group G1,1,n = Sn, for r = 2s = 2 we have the hyperoctahedral group G2,1,n = Bn, and for r = s = 2 we have the group of even-signed permutations G2,2,n = Dn. We say that a permutation π ∈ Gr,s,n is an involution if π 2 = 1. More generally, we define G r,s,n = {σ ∈ Gr,s,n|σ m = 1}. Recently, Bagno, Garber and Mansour [2] studied an excedance number on the complex reflection groups (see [4]) and computed the number of involutions having specific numbers of fixed points and excedances. In this note, we consider the similar problems on the set G r,s,n. This paper is organized as follows. In Section 2, we recall some properties of Gr,s,n and define some parameters on Gr,n = Gr,1,n and hence also on Gr,s,n. In Section 3 we present our main results and compute the corresponding recurrences together with explicit formulas.