Enumerating 1324-avoiders with few inversions

We enumerate the numbers Avₙᵏ(1324) of 1324-avoiding n-permutations with exactly k inversions for all k and n ≥ (k+7)/2. The result depends on a structural characterization of such permutations in terms of a new notion of almost-decomposability. In particular, our enumeration verifies half of a conjecture of Claesson, Jel\'inek and Steingr\'imsson, according to which Avₙᵏ(1324) ≤ Avₙ₊₁ᵏ(1324) for all n and k. Proving also the other half would improve the best known upper bound for the exponential growth rate of the number of $1324$-avoiders from $13.5$ to approximately $13.002$.