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)≤ a(n+1,k) for k≤2n−7. We study the obstruction beyond that line: the residuals ℛδ,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 |ℛδ,n|=Aδn2+Bδn+Cδ. Our central uniform result determines the quadratic coefficient at every defect: with P(q)=∏j≥1(1−qj)−1, ∑δ≥0Aδqδ=4q3(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)≤ a(n+1,k) for every n≥1 and k≤2n+6, and of regions whose width grows with n: for n≥216,218,220 the defect may be as large as ⌊√n/4⌋, ⌊√n/3⌋, ⌊√n/2⌋, respectively. More generally, every fixed c<√2log(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.
Related
- Inversion-descent enumerators of underline321-avoiding permutations
- Avoiding patterns with three distinct letters in Canon permutations
- Descent Tops and Pinnacles on 2143- and 3421-Avoiding Permutations
- Shuffle automata and the growth of 1324-avoiding permutations
- The Distribution of Double Deficiencies in Pattern-Avoiding Permutations
BibTeX
@misc{meng2026residual,
title = {{Residual structure and growing inversion-monotonicity regions for 1324-avoiding permutations}},
author = {Lingsen Meng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06717},
eprint = {2609.06717},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06717},
}
RIS
TY - UNPB
TI - Residual structure and growing inversion-monotonicity regions for 1324-avoiding permutations
AU - Lingsen Meng
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.06717
UR - https://arxiv.org/abs/2609.06717
AB - Let a(n,k) be the number of $1324$-avoiding permutations of length n with k inversions. Linusson and Verkama proved a(n,k)≤ a(n+1,k) for k≤2n−7. We study the obstruction beyond that line: the residuals 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 |ℛ_δ,n|=A_δn^2+B_δn+C_δ. Our central uniform result determines the quadratic coefficient at every defect: with P(q)=∏_j≥1(1−q^j)^−1, ∑_δ≥0A_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)≤ a(n+1,k) for every n≥1 and k≤2n+6, and of regions whose width grows with n: for n≥2^16,2^18,2^20 the defect may be as large as ⌊√n/4⌋, ⌊√n/3⌋, ⌊√n/2⌋, respectively. More generally, every fixed c<√2log(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.
ER -