UCLA

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 indecomposable, non-almost-decomposable residuals at fixed defect. Contracting maximal increasing consecutive runs reduces residuality to a quadratic equation on a finite family of skeletons. For every fixed defect, the eventual residual count is quadratic in n. We determine the leading coefficient uniformly by a generating function, prove inversion monotonicity through k <= 2n+6, and obtain regions whose width grows with n. Complete catalogue and rational-sum certificates for defects 11, 12, and 13 are supplied in the archival supplement. The unrestricted Claesson--Jelinek--Steingrimsson conjecture, the full residual polynomials, and the sharp global base-length bound remain open.