Transformation à la Foata for special kinds of descents and excedances

Jean-Luc Baril, Sergey Kirgizov

Published 2021-01-06Version 1

A pure excedance in a permutation $\pi=\pi_1\pi_2\ldots \pi_n$ is a position $i<\pi_i$, $1\leq i\leq n-1$, so that there is no $j<i$ such that $i\leq \pi_j<\pi_i$. We present a one-to-one correspondence on the symmetric group that transports pure excedances to descents of special kind. As a byproduct, we prove that the popularity of pure excedances equals those of pure descents on permutations, while their distributions are different.