Inversion monotonicity and asymptotic enumeration of 1324-avoiding permutations with few inversions

Abstract

Let A(n,k) count the 1324-avoiding permutations of length n with k inversions, let D(n,k) and I(n,k) count the decomposable and indecomposable ones, and put a(k) = sumj=0k p(j)p(k-j), where p is the partition function. We prove A(n,k) ≤ A(n+1,k) for 0 ≤ k ≤ floor(n2 / [20 log2(n+1)]), for every n ≥ 1. We characterize the range in which the partition approximation holds and almost every avoider is decomposable. For k tending to infinity and bk = √(3k)/π, the following are equivalent: D(n,k)/a(k) tends to 1; jointly A(n,k)/a(k) tends to 1 and I(n,k)/a(k) tends to 0; and n/bk − 2 log bk tends to infinity. This yields uniform convergence up to π2n2 / [12(log n − log log n + ω(n))2] for every ω(n) tending to infinity, without a regularity assumption. For every fixed η < 1 and real σ, set Kn = floor(π2n2 / [12(log n − η log log n + σ)2]). The maximum over k ≤ Kn of the joint error |A(n,k)/a(k) − 1| + I(n,k)/a(k) has order (log log n)/(log n)2(1−η), with a matching lower bound at k = Kn. The lower bound concerns the sum of the two errors. All logarithms are natural. The monotonicity proof uses the Linusson–Verkama injection, a finite classification and explicit partition estimates. The asymptotic results have independent proofs using partition pairs and a weighted deletion recurrence. The weighted Schröder recurrence also proves inversion monotonicity for Av(1324,1342) and its three symmetry-equivalent classes for all lengths and inversion numbers, answering Problem 8.1 of Claesson, Linusson, Ulfarsson and Verkama. The enlarged asymptotic range does not assert monotonicity there; the full conjecture remains open. Version 3.0 extends the asymptotic results of version 2.0 with the sharp joint criterion and matching error estimates. The full-class monotonicity range and two-pattern theorem are unchanged. Earlier versions remain available in the same Zenodo series. The upload contains the PDF and a self-contained source archive with LaTeX, proof notes, verification programs, recorded output, citation metadata and a SHA-256 manifest. A new exhaustive encoding test covers 28,091,496 residual permutations through length 12, with no collisions at fixed length and inversion number. The command python -B verify_release.py repeats the Python finite checks and verifies the stored C++ inputs and summaries; instructions for rerunning the exhaustive C++ test are included. Large language models assisted with mathematical arguments, drafting, code development and computational checks.