The excedance set of a permutation

The excedance set of a permutation pi = pi(1) pi(2) ... pi(n) is the set of indices i for which pi(i) > i. We give a formula for the number of permuta tions with a given excedance set and recursive formulas satisfied by these numbers. We prove log-concavity of certain sequences of these numbers and w e show that the most common excedance: set among permutations in the symmet ric group F-n is {1, 2,..., [n/2]}. We also relate certain excedance. set n umbers to Stirling numbers of the second kind, and others to the Genocchi n umbers. (C) 2000 Academic Press.