Inversion monotonicity in subclasses of the 1324-avoiders

TL;DR

This paper investigates inversion monotonicity in pattern-avoiding permutations, proving new examples, characterizing limit sequences, and connecting these properties to integer partitions and permutation classes.

Contribution

It provides the first nontrivial examples of inversion-monotone sets, characterizes pattern sets with limit sequences, and extends results on almost decomposable permutations.

Findings

Proved that certain pattern collections are inversion monotone via explicit injections.

Characterized pattern sets with limit sequences and determined these sequences for pairs involving 1324.

Connected inversion monotonicity to integer partitions and permutation enumeration.

Abstract

A collection $B$ of patterns is called inversion monotone if $\mathrm{av}_n^k(B)$, the number of $B$-avoiding permutations of length $n$ with $k$ inversions, is weakly increasing in $n$ for any fixed $k$. In 2012, Claesson, Jel\'inek and Steingr\'imsson posed the inversion monotonicity conjecture, which states that the pattern $1324$ is inversion monotone and implies a new upper bound for its Stanley--Wilf limit. We prove that the collections $\{1324, 231\}$ and $\{1324, 2314, 3214, 4213\}$ are inversion monotone via explicit injections. The latter follows from a general procedure for constructing inversion-monotone sets. Our results constitute the first known nontrivial examples of inversion-monotone sets. A key feature of the inversion monotonicity conjecture is that $1324$ has a limit sequence: $\mathrm{av}_n^k(1324)$ is constant in $n$ when $n$ is large. We characterize the sets of patterns that have limit sequences, and determine the limit sequences of all pairs $\{1324, p\}$, where $p$ is a pattern of length four. Connections to various families of integer partitions arise. Finally, we expand on work by Linusson and Verkama (2025) on almost decomposable permutations to determine a broad family of sets containing $1324$ that are inversion monotone under the assumption $n \geq \frac{k+7}{2}$. The method yields an enumeration of $\mathrm{av}_n^k(1324, 1342)$ when $n \geq \frac{k+7}{2}$.