Symmetric multisets of permutations

Abstract

The following long-standing problem in combinatorics was first posed in 1993 by Gessel and Reutenauer [5]. For which multisubsets B of the symmetric group Sn is the quasisymmetric functionQ(B)=∑π∈BFDes(π),n a symmetric function? Here Des(π) is the descent set of π and FDes(π),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 from [2]. Two other corollaries are also given. The first is a new and short proof that conjugacy classes are symmetric sets, a well known result first proved by Gessel and Reutenauer [5]. In our second corollary we give 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 in [2].