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

• The paper introduces explicit combinatorial injections proving inversion monotonicity for specific 1324-avoiding subclasses, notably {1324, 231}.
• It develops a recursive construction method to build larger inversion-monotone sets, ensuring stabilization of inversion count sequences.
• The study links limit sequences in these permutation classes to classical partition families, advancing upper bounds on the Stanley–Wilf limit.

Inversion Monotonicity in Subclasses of the 1324-Avoiders

Introduction and Context

This paper investigates the structure of permutation classes defined by pattern avoidance, specifically focusing on the distribution of the inversion statistic over such classes. Of particular interest is the class of permutations avoiding the pattern 1324, due to its notoriety regarding its enigmatic enumeration and asymptotic growth properties.

The main framework of the study is the inversion monotonicity conjecture introduced by Claesson, Jelínek, and Steingrímsson (2012), which posits that for any non-identity pattern $p$, the count $\av_n^k(p)$—the number of $p$-avoiding permutations of length $n$ with exactly $k$ inversions—is non-decreasing in $n$ for fixed $k$. This property, if established for the 1324 pattern, would imply improved upper bounds for its Stanley–Wilf limit, which governs the exponential rate of growth of the $n$th term of $|Av_n(1324)|$ as $n \to \infty$.

The paper achieves several breakthroughs by establishing inversion monotonicity in explicit nontrivial subclasses wherein 1324 is combined with other patterns, formalizing the combinatorial mechanisms involved, and characterizing the pattern bases for which limit sequences—where the distribution $\av_n^k(p)$0 stabilizes for large $\av_n^k(p)$1—exist.

Main Contributions

Explicit Inversion-Monotone Subclasses

The authors present the first nontrivial examples of pattern sets $\av_n^k(p)$2 for which $\av_n^k(p)$3 is inversion-monotone, focusing on certain collections containing 1324. They provide explicit (and technically intricate) injective constructions that demonstrate inversion monotonicity for:

• $\av_n^k(p)$4
• $\av_n^k(p)$5

The proof for $\av_n^k(p)$6 utilizes a sophisticated mapping which handles the complex decomposition of $\av_n^k(p)$7-avoiding permutations and manages the inversion statistic through shift and insertion operators, ensuring the construction is indeed an injection that preserves the number of inversions. For the second collection, inversion monotonicity follows from a general recursive base-building construction; given an inversion-monotone base $\av_n^k(p)$8, certain symmetrically defined pattern extensions remain inversion monotone.

Systematic Construction of Larger Inversion-Monotone Sets

The paper introduces a general methodology for generating increasingly large (and structurally complex) inversion-monotone bases by inductively extending existing ones via the insertion of leading, trailing, maximal, or minimal entries in all possible ways. This combinatorial lifting is nontrivial since enlargement can disrupt, or, as shown, sometimes preserve monotonicity.

Characterization of Limit Sequences

An essential property associated with inversion monotonicity is the existence of limit sequences. For a pattern set $\av_n^k(p)$9, a limit sequence is a sequence $p$0 such that $p$1 stabilizes as $p$2 becomes large relative to $p$3. The paper generalizes prior results by proving that $p$4 has a limit sequence if and only if $p$5 contains a pattern with at most one inversion. This bridges the enumerative theory of permutation patterns with partition-theoretic phenomena.

For pattern pairs $p$6 with $p$7, the paper determines the limit sequence in all cases, identifying the underlying combinatorics (e.g., integer partitions, overpartitions, sandpile model partitions, convex penny arrangements, and more). The combinatorial interpretations are established rigorously and, where possible, explicit generating functions are derived.

Extension and Refinement of the "Half-Monotonicity" Paradigm

The authors leverage and extend recent work on "almost decomposable" $p$8-avoiding permutations. They define and analyze the notion of $p$9-compatibility—a technical condition ensuring that the injective inflation mapping preserves avoidance of additional patterns. Using a combination of injection priority arguments and structural decompositions, they provide necessary and sufficient conditions (in most cases) for a pattern $n$0 to be $n$1-compatible.

For particular patterns ($n$2), half-monotonicity is shown: $n$3 for all $n$4. Moreover, for $n$5 the mapping allows explicit enumeration formulae for all sufficiently large $n$6.

Strong Results and Claims

• Inversion monotonicity of $n$7 and $n$8: These are the first known nontrivial inversion-monotone pairs containing 1324, and explicit combinatorial injections are provided.
• Characterization of when a set $n$9 has a limit sequence: Precisely when $k$0 contains a pattern $k$1 with at most one inversion.
• Complete determination of limit sequences for all pairs $k$2 with $k$3. These span diverse partition-theoretic families, indicating deep enumerative structure.
• Broad family results for half-monotonicity: For many patterns $k$4, the pair $k$5 is monotone once $k$6 is large compared to $k$7.
• Enumeration of $k$8 for large $k$9: Utilizing secondary limit sequences and explicit generating functions.

Implications and Theoretical Developments

The paper's results establish that inversion monotonicity, previously only conjectured and known in trivial settings, in fact holds in larger, explicitly defined classes. The intricate mappings devised and the recursive constructions provided offer robust combinatorial tools with relevance well beyond the immediate context. These techniques give pathways towards incremental progress on the broader 1324 inversion monotonicity conjecture and related open problems in permutation pattern enumeration.

The characterization of limit sequences in terms of minimal-inversion patterns within $n$0 tightly connects the analytic aspects of permutation classes with their algebraic structure. The detailed correspondence between pattern-avoidance in indecomposable permutations and prominent partition families elucidates a much deeper connection between permutation combinatorics and classical partition theory (including the sand pile model and convex penny arrangements).

The paper’s approach to secondary (and tertiary) "limit-like" sequences in the finite differences $n$1 for various bases inspires further investigation in asymptotic stability phenomena for finer classes, potentially fostering new probabilistic models or limit theorems.

Future Directions

Several concrete open problems are posed:

• Establishing inversion monotonicity for additional nontrivial pairs $n$2.
• Determining limit sequences for other pattern bases, especially those not containing small-inversion patterns.
• Classifying Wilf-equivalence classes vis-à-vis their limit sequence behavior.
• Understanding the experimental phenomenon of higher-order limit sequences (secondary, tertiary,…) in the context of permutation classes.

These directions are likely to influence future work in enumerative combinatorics, with possible applications to related statistical physics models (due to the sandpile model connection), algebraic combinatorics, and symmetric function theory.

Conclusion

This paper provides the first explicit proofs of inversion monotonicity for nontrivial collections containing 1324, introduces systematic mechanisms for base extension, thoroughly characterizes limit sequence existence, and delineates new structural connections between pattern avoidance and partition theory. The technical breadth and variety of enumerative, structural, and algorithmic methods developed will be directly relevant to ongoing research in the combinatorics of permutation patterns, and may offer valuable tools for tackling longstanding open problems such as the Stanley–Wilf limit for 1324 and its subclasses.