Inversion monotonicity in subclasses of the 1324-avoiders

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 $n \geq \frac{k+7}{2}$.