Residual structure and growing inversion-monotonicity regions for 1324-avoiding permutations

Abstract

Let $a(n,k)$ be the number of $1324$-avoiding permutations of length $n$ with $k$ inversions. Linusson and Verkama proved $a(n,k)\le a(n+1,k)$ for $k\le2n-7$. We study the obstruction beyond that line: the residuals $\mathcal R_{δ,n}$, namely the indecomposable, non-almost-decomposable avoiders at defect $δ=k-2n+7$. Contracting maximal increasing consecutive runs reduces residuality to a quadratic equation on a finite family of skeletons. It follows that, for every fixed $δ$, the eventual count has the form $|\mathcal R_{δ,n}|=A_δn^2+B_δn+C_δ$. Our central uniform result determines the quadratic coefficient at every defect: with $P(q)=\prod_{j\ge1}(1-q^j)^{-1}$, $\sum_{δ\ge0}A_δq^δ=4q^3(1+q)P(q)^2/(1-q)^2$. This is a formula for the leading coefficient of the residual count, not for the full count. The same structural estimates give computer-assisted proofs of $a(n,k)\le a(n+1,k)$ for every $n\ge1$ and $k\le2n+6$, and of regions whose width grows with $n$: for $n\ge2^{16},2^{18},2^{20}$ the defect may be as large as $\lfloor\sqrt n/4\rfloor$, $\lfloor\sqrt n/3\rfloor$, $\lfloor\sqrt n/2\rfloor$, respectively. More generally, every fixed $c<\sqrt2\log(5)/\log(68)$ is admissible for all sufficiently large $n$. The three added fixed defects $11,12,13$ use complete catalogue and rational-sum certificates supplied in the accompanying archival supplement. The leading-coefficient theorem is obtained from a complete finite classification of marked rank-three cores and all-parameter extension lemmas. We also determine the exact rank-three stabilization onset, while keeping it separate from the still unknown onset of the complete residual count. The unrestricted Claesson--Jel'inek--Steingr'imsson conjecture, the full residual polynomials, and the sharp global base-length bound remain open.