Let \({{\mathfrak {S}}}_n\) be the nth symmetric group. Given a set of permutations \(\Pi \), we denote by \({{\mathfrak {S}}}_n(\Pi )\) the set of permutations in \({{\mathfrak {S}}}_n\) which avoid \(\Pi \) in the sense of pattern avoidance. Consider the generating function \(Q_n(\Pi )=\sum _\sigma F_{{{\,\mathrm{Des}\,}}\sigma }\) where the sum is over all \(\sigma \in {{\mathfrak {S}}}_n(\Pi )\) and \(F_{{{\,\mathrm{Des}\,}}\sigma }\) is the fundamental quasisymmetric function corresponding to the descent set of \(\sigma \). Hamaker, Pawlowski, and Sagan introduced \(Q_n(\Pi )\) and studied its properties, in particular, finding criteria for when this quasisymmetric function is symmetric or even Schur nonnegative for all \(n\ge 0\). The purpose of this paper is to continue their investigation by answering some of their questions, proving one of their conjectures, as well as considering other natural questions about \(Q_n(\Pi )\). In particular, we look at \(\Pi \) of small cardinality, superstandard hooks, partial shuffles, Knuth classes, and a stability property.

中文翻译：

---

再谈模式回避和拟对称函数

令\（{{\ mathfrak {S}}} _ n \）为第n个对称组。给定一组的排列\（\裨\），我们用\（{{\ mathfrak {S}}} _ N（\ PI）\）在该组的排列\（{{\ mathfrak {S}}} _Ñ \）在避免模式的意义上避免了\（\ Pi \）。考虑生成函数\（Q_n（\ Pi）= \ sum _ \ sigma F _ {{{\，\ mathrm {Des} \，}} \ sigma} \）其中总和在所有\（\ sigma \ in { {\ mathfrak {S}}} _ n（\ Pi）\）和\（F _ {{{\\ mathrm {Des} \，}} \ sigma} \）是与\的下降集相对应的基本拟对称函数。 （\ sigma \）。Hamaker，Pawlowski和Sagan进行了介绍\（Q_n（\ Pi）\）并研究了它的性质，尤其是为所有\（n \ ge 0 \）拟准对称函数是对称的甚至是Schur非负的。本文的目的是通过回答他们的一些问题，证明他们的一个猜想以及考虑关于\（Q_n（\ Pi）\）的其他自然问题来继续他们的研究。特别地，我们看一下小基数，超标准钩子，部分混洗，Knuth类和稳定性属性的\（\ Pi \）。