Two-sided permutation statistics via symmetric functions

TL;DR

This paper develops a symmetric function-based method to analyze joint distributions of descent-based permutation statistics and applies it to various statistics, revealing simple sum formulas and rederiving known generating functions.

Contribution

Introduces a general symmetric function approach for joint distributions of descent statistics, leading to new formulas and rederivations in permutation enumeration.

Findings

Joint distributions can be expressed as sums of products of individual distributions.

Re-derivation of Stanley's generating function for doubly alternating permutations.

Conjectures on real-rootedness and gamma-positivity of certain polynomials.

Abstract

Given a permutation statistic $\operatorname{st}$, define its inverse statistic $\operatorname{ist}$ by $\operatorname{ist}(\pi):=\operatorname{st}(\pi^{-1})$. We give a general approach, based on the theory of symmetric functions, for finding the joint distribution of $\operatorname{st}_{1}$ and $\operatorname{st}_{2}$ whenever $\operatorname{st}_{1}$ and $\operatorname{st}_{2}$ are descent statistics: permutation statistics that depend only on the descent composition. We apply this method to a number of descent statistics, including the descent number, the peak number, the left peak number, the number of up-down runs, and the major index. Perhaps surprisingly, in many cases the polynomial giving the joint distribution of $\operatorname{st}_{1}$ and $\operatorname{ist}_{2}$ can be expressed as a simple sum involving products of the polynomials giving the (individual) distributions of $\operatorname{st}_{1}$ and $\operatorname{st}_{2}$. Our work leads to a rederivation of Stanley's generating function for doubly alternating permutations, as well as several conjectures concerning real-rootedness and $\gamma$-positivity.