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Inversion monotonicity in subclasses of the 1324-avoiders

Anders Claesson, Svante Linusson, Henning Ulfarsson, Emil Verkama

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, Jelínek and Steingrímsson 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 $n \geq \frac{k+7}{2}$.

Inversion monotonicity in subclasses of the 1324-avoiders

Abstract

A collection of patterns is called inversion monotone if , the number of -avoiding permutations of length with inversions, is weakly increasing in for any fixed . In 2012, Claesson, Jelínek and Steingrímsson posed the inversion monotonicity conjecture, which states that the pattern is inversion monotone and implies a new upper bound for its Stanley--Wilf limit. We prove that the collections and 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 has a limit sequence: is constant in when is large. We characterize the sets of patterns that have limit sequences, and determine the limit sequences of all pairs , where 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 that are inversion monotone under the assumption . The method yields an enumeration of when .

Paper Structure

This paper contains 32 sections, 39 theorems, 129 equations, 10 figures, 26 tables.

Table of Contents

  1. Introduction
  2. Inversion-monotone patterns
  3. Limit sequences
  4. Half-monotone collections
  5. Outline
  6. Preliminaries
  7. Inversions and decomposability
  8. Limit sequences
  9. 132-avoiders and partitions
  10. {1324, 231} is inversion monotone
  11. Building inversion-monotone sets
  12. Limit sequences of sets containing 1324
  13. On limit sequences
  14. Representatives and overview
  15. Non-inversion-monotone pairs
  16. ...and 17 more sections

Key Result

Theorem 1

The collections $\{1324, 231\}$ and $\{1324, 2314, 3214, 4213\}$ are inversion monotone. $\blacktriangleleft$$\blacktriangleleft$

Figures (10)

  • Figure 1: The direct sum and skew sum of $\sigma = 21$ and $\tau = 231$.
  • Figure 2: The structure of a $\{213, 231\}$-avoider (left). The different ways to insert a point in the lower arm (middle) and in the upper arm (right), with the increase in inversions indicated.
  • Figure 3: A schematic of the injection $\mathop{\mathrm{Av}}\nolimits_n^k(1324, 231) \to \mathop{\mathrm{Av}}\nolimits_{n+1}^k(1324, 231)$.
  • Figure 4: A $\{1324, 231\}$-avoiding permutation $\pi$ (left) and its image $f(\pi)$ (right).
  • Figure 5: The structure of an almost decomposable permutation in $\mathop{\mathrm{Av}}\nolimits(1324,1342)$ whose image under $f$ contains $1342$.
  • ...and 5 more figures

Theorems & Definitions (76)

  • Conjecture 1.1: Conjecture 20 in claesson_upper_2012
  • Theorem
  • Proposition
  • Theorem
  • Theorem
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Lemma 2.5
  • ...and 66 more