Transformation \`a la Foata for special kinds of descents and excedances

TL;DR

This paper introduces a bijection in permutations that maps pure excedances to a special kind of descents, revealing new insights into their distribution and popularity.

Contribution

It presents a novel one-to-one correspondence in permutations that connects pure excedances with a specific type of descents, advancing combinatorial understanding.

Findings

Pure excedances and pure descents have equal popularity in permutations.

The distribution of pure excedances differs from that of pure descents.

A bijection preserves certain permutation statistics.

Abstract

A pure excedance in a permutation $\pi=\pi_1\pi_2\ldots \pi_n$ is a position $i<\pi_i$ such that there is no $j<i$ with $i\leq \pi_j<\pi_i$. 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.