The following long-standing problem in combinatorics was first posed in 1993 by Gessel and Reutenauer. For which multisubsets \(B\) of the symmetric group \(S_n\) 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 from 2015. 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. 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.
Published: J. Combin. Theory Ser. A, 2020.