Recursions for Excedance number in some permutations groups

TL;DR

This paper provides a direct recursive proof of the Eulerian distribution of excedance numbers in permutation groups, extends it to colored groups, and explores related combinatorial properties like Stirling numbers and symmetry of generating functions.

Contribution

It introduces a new recursive proof for excedance distributions, extends results to colored permutation groups, and investigates their combinatorial properties.

Findings

Recursive formulas for excedance numbers in permutation groups

Connection between excedance numbers and Stirling numbers of the second kind

Symmetry of the generating function for excedance numbers

Abstract

The excedance number for S_n is known to have an Eulerian distribution. Nevertheless, the classical proof uses descents rather than excedances. We present a direct recursive proof which seems to be folklore and extend it to the colored permutation groups G_r,n. The generalized recursion yields some interesting connection to Stirling numbers of the second kind. We also show some logconcavity result concerning a variant of the excedance number. Finally, we show that the generating function of the excedance number defined on G_r,n is symmetric.