Bijective Enumeration and Sign-Imbalance for Permutation Depth and Excedances

TL;DR

This paper introduces a simplified bijection linking permutations to 3-colored Motzkin paths, enabling analysis of permutation statistics like depth and excedances, and reveals parity and sign-imbalance properties.

Contribution

It generalizes previous results by providing a new bijection and involution that analyze permutation statistics and their sign-imbalances with respect to depth and excedance.

Findings

Equal number of permutations with even and odd depth when n is even

Difference in counts by tangent number when n is odd

Sign-imbalance results for permutations and derangements

Abstract

We present a simplified variant of Biane's bijection between permutations and 3-colored Motzkin paths with weight that keeps track of the inversion number, excedance number and a statistic so-called depth of a permutation. This generalizes a result by Guay-Paquet and Petersen about a continued fraction of the generating function for depth on the permutations of n elements. In terms of weighted Motzkin path, we establish an involution on the permutations that reverses the parities of depth and excedance numbers simultaneously, which proves that the numbers of permutations with even and odd depth (excedance numbers, respectively) are equal if n is even and differ by the tangent number if n is odd. Moreover, we present some interesting sign-imbalance results on permutations and derangements, refined with respect to depth and excedance numbers.