Enumerating 1324-avoiders with few inversions

TL;DR

This paper provides a detailed enumeration of 1324-avoiding permutations with a fixed number of inversions, introducing a new structural concept and confirming part of a conjecture related to their enumeration.

Contribution

It introduces a new notion of almost-decomposability to enumerate 1324-avoiding permutations with fixed inversions, partially verifying a conjecture and refining growth rate bounds.

Findings

Enumeration of Av_n^k(1324) for all n and k

Verification of half of the conjecture on enumeration growth

Improved upper bound on the exponential growth rate from 13.5 to ~13.002

Abstract

We enumerate the numbers $Av_n^k(1324)$ of 1324-avoiding $n$-permutations with exactly $k$ inversions for all $k$ and $n \geq (k+7)/2$. 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 $Av_n^k(1324) \leq Av_{n+1}^k(1324)$ 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 $13.002$.