On symmetric pattern avoidance sets

Abstract

For a set of permutations $S\subseteq S_n$, consider the quasisymmetric generating function $$Q(S): = \sum_{w\in S}F_{n, \mathrm{Des}(w)},$$ where $\mathrm{Des}(w) := \{i\mid w(i)> w(i+1)\}$ is the descent set of $w$ and $F_{n, \mathrm{Des}(w)}$ is Gessel's fundamental quasisymmetric function. A set of permutations is said to be symmetric (respectively, Schur-positive) if its quasisymmetric generating function is symmetric (respectively, Schur-positive). Given a set $\Pi$ of permutations, let $S_n(\Pi)$ denote the set of permutations in $S_n$ that avoid all patterns in $\Pi.$ A set $\Pi$ is said to be symmetrically avoided (respectively, Schur-positively avoided) if $S_n(\Pi)$ is symmetric (respectively, Schur-positive) for all $n.$ Marmor proved in 2025 that for $n\ge 5$, a symmetric set $S\subseteq S_n$ has size at least $n-1$ unless $S\subseteq \{12\cdots n, n\cdots 21\}$ and asked for a general classification of the possible sizes of symmetric sets not containing the monotone elements $12\cdots n $ and $n\cdots 21$. We give a complete answer to this question for $n\ge 52.$ We also give a classification of symmetric sets of size at most $n-1$, thereby showing that they are actually Schur-positive, resolving a conjecture of Marmor. Finally, we give a classification of symmetrically avoided sets of size at most $n-1$, thereby showing that they are actually Schur-positively avoided.