Transformation à la Foata for special kinds of descents and excedances

Abstract

A pure excedance in a permutation $\pi ={\pi }_1{\pi }_2\dots {\pi }_n$ is a position $i<{\pi }_i$ such that there is no $j<i$ with $i\le {\pi }_j<{\pi }_i$. 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.

12 pages, 2 figures