Asymptotic enumeration of 1324-avoiding permutations with few inversions

Superseded by version 2.0. The combined manuscript, Inversion monotonicity and asymptotic enumeration of 1324-avoiding permutations with few inversions, is available at 10.5281/zenodo.22846587. It incorporates the monotonicity and asymptotic-enumeration results in the original version series. Let A(n,k) count the 1324-avoiding permutations of length n with k inversions, and let a(k) be the number of ordered pairs of integer partitions of total size k. We prove A(n,k)/a(k) = 1 + O(log log n / log n), uniformly for 0 ≤ k ≤ π2n2 / [12(log n − (1/2)log log n + 2)2], as n tends to infinity. The same error bounds the proportion of indecomposable permutations. All logarithms are natural. For each fixed 0 < c ≤ π2/12, the relative error in the range k ≤ c n2/log2n is Oc(n2−π/√(3c)/log n). The proof uses the partition description of decomposable avoiders, an inversion-preserving injection into excluded partition pairs, and a two-entry deletion recurrence. A refined bound on preimages of decomposable remainders gives the logarithmic enlargement. This is a companion paper to Inversion monotonicity for 1324-avoiding permutations in a near-quadratic range, version 1.1 (10.5281/zenodo.22764164). The present paper establishes asymptotic enumeration; monotonicity in the enlarged range remains open. The archive contains the PDF, LaTeX source, a portable Python program for supplementary exact checks, recorded output, and instructions. The mathematical proofs are given in the manuscript. Large language models assisted with the mathematical arguments, drafting, code development, and computational checks.