Symmetric multisets of permutations

TL;DR

This paper characterizes multisets of permutations in the symmetric group for which the associated quasisymmetric functions are symmetric, proving a conjecture and providing new insights into the symmetry properties of conjugacy classes and multiplication actions.

Contribution

It offers a new characterization of multisets with symmetric quasisymmetric functions and proves a conjecture, enhancing understanding of symmetry in permutation multisets.

Findings

Characterization of multisets with symmetric quasisymmetric functions

Proof of Elizalde and Roichman's conjecture

New proof that conjugacy classes are symmetric sets

Abstract

The following long-standing problem in combinatorics was first posed in 1993 by Gessel and Reutenauer. For which multisubsets $B$ of the symmetric group $\fS_n$ is the quasisymmetric function $$Q(B) = \sum_{\pi \in B}F_{\Des(\pi), n}$$ a symmetric function? Here $\Des(\pi)$ is the descent set of $\pi$ and $F_{\Des(\pi), n}$ is Gessel's fundamental basis for the vector space of quasisymmetric functions. The purpose of this paper is to provide a useful characterization of these multisets. Using this characterization we prove a conjecture of Elizalde and Roichman. Two other corollaries are also given. The first is a short new proof that conjugacy classes are symmetric sets, a well known result first proved by Gessel and Reutenauer. Our second corollary is a unified explanation that both left and right multiplication of symmetric multisets, by inverse $J$-classes, is symmetric. The case of right multiplication was first proved by Elizalde and Roichman.