Symmetric multisets of permutations

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.