The Excedance Set of a Permutation

Abstract

The excedance set of a permutation π=π1π2···πn is the set of indices i for which πi>i. We give a formula for the number of permutations with a given excedance set and recursive formulas satisfied by these numbers. We prove log-concavity of certain sequences of these numbers and we show that the most common excedance set among permutations in the symmetric group Sn is {1,2,…,⌊n/2⌋}. We also relate certain excedance set numbers to Stirling numbers of the second kind, and others to the Genocchi numbers.