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 ${\mathfrak{S}}_n$ is the quasisymmetric function$Q(B)=\sum _{\pi \in B}{F}_{\mathrm{Des}(\pi ),n}$ a symmetric function? Here $\mathrm{Des}(\pi )$ is the descent set of π and $F_{\mathrm{Des}(\pi ),n}$ 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 [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].

Gessel和Reutenauer于1993年首次提出了组合学中的以下长期存在的问题[5]。对称组的哪个多子集B${\mathfrak{小号}}_{ñ}$ 是拟对称函数$问（乙）=\sum _{\pi \in 乙}{F}_{\mathrm{德斯}（\pi ），ñ}$对称函数？这里$\mathrm{德斯}（\pi ）$正在下降设置的π和$F_{\mathrm{德斯}（\pi ），ñ}$是Gessel拟对称函数向量空间的基本基础。本文的目的是提供这些多集的有用描述。使用这种表征，我们从[2]中证明了伊莱兹德和罗希曼的猜想。还给出了另外两个推论。第一个是新的简短证明，共轭类是对称集，这是由Gessel和Reutenauer [5]首次证明的众所周知的结果。在我们的第二个推论中，我们给出统一的解释，即对称多集的左乘和右乘通过逆J类进行对称。右乘的情况最早由Elizalde和Roichman在[2]中证明。