Inversion monotonicity for 1324-avoiding permutations in a near-quadratic range

Let A(n,k) denote the number of 1324-avoiding permutations of length n with k inversions. Claesson, Jelínek and Steingrímsson conjectured that A(n,k) ≤ A(n+1,k) for all n and k. We prove A(n,k) ≤ A(n+1,k) for 0 ≤ k ≤ floor(n2 / (20 log2(n+1))). The proof starts with the injection of Linusson and Verkama. The remaining permutations are encoded by a shorter permutation, a small pattern and at most seven integer coordinates, and explicit estimates for two-coloured partitions show that a family outside the image of the injection is large enough to accommodate them. The finite classification steps are computer-assisted. We also prove that the pair {1324, 1342} is inversion monotone for all n and k, answering Problem 8.1 of Claesson, Linusson, Ulfarsson and Verkama (arXiv:2604.01143). This follows from the q-Schröder recurrence of Barcucci, Del Lungo, Pergola and Pinzani for the inversion enumerator of the Schröder permutations, proved bijectively by Bandlow, Egge and Killpatrick. By symmetry the same holds for {1324, p} with p in {1423, 2314, 3124}. The complete conjecture for Av(1324) remains open. Changes from version 1.0: the section on the line k = 2n+7 has been withdrawn from this version; the two-pattern theorem has been added; the attribution of the q-Schröder recurrence has been made precise. The archive contains the manuscript and LaTeX source, verification programs and their output, proof notes, and a SHA-256 manifest; python verify_release.py reruns the finite checks.