On symmetric pattern avoidance sets

TL;DR

This work characterizes when sets of permutations have symmetric or Schur-positive quasisymmetric generating functions by translating the problem into harmonic set systems tied to descent structures. It proves a complete size-classification for symmetric sets without monotone elements when $n\ge 52$, and shows all symmetric sets of size at most $n-1$ are Schur-positive, resolving a conjecture of Marmor. It also classifies symmetrically avoided sets of size at most $n-1$, establishing they are Schur-positively avoided. A central tool is the Robinson–Schensted correspondence together with the novel framework of harmonic set systems, enabling precise structural descriptions (Knuth-closed classes, partial shuffles, and their complements) that govern symmetry and Schur-positivity in these permutation families.

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 $Π$ of permutations, let $S_n(Π)$ denote the set of permutations in $S_n$ that avoid all patterns in $Π.$ A set $Π$ is said to be symmetrically avoided (respectively, Schur-positively avoided) if $S_n(Π)$ 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.

Figures (4)

• Figure 1: A plot of $2413$
• Figure 2: Here, $k = 5$, $i = 2, j = 3$. Adding a point to one of the regions indicated with an x creates a permutation with descents at $2$ and $4$. The two grayed regions give the same permutations.
• Figure 3: Here, $k = 6, i = 3$. Left: Add a point in the $(2, 3)\times (4,5)$ region to obtain $Q(3,5) = 125346$. Middle: Add a point in the $(3, 4)\times (1,2)$ region to obtain $Q(3,2) = 134256$. Right: Add a point to $(2, 3)\times (3,4)$ or to $(3, 4)\times (2,3)$ to obtain $Q(3,3) = Q(3,4) = 124356$.
• Figure 4: Here, $k = 6.$ Left: adding a point in one of regions with letters creates a permutation in with descent set $\{4\}$. Regions with the same letter give the same permutation. Right: adding point in one of regions with letters creates a permutation with descent set $\{4\}.$

Theorems & Definitions (143)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Example 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6