Enumerating 1324-avoiders with few inversions

2024/08/27 by Linusson, Svante, Verkama, Emil
#05A05 #05A15 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2408.15075

Abstract

We enumerate the numbers Av_n^k(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, Jelínek and Steingrímsson, according to which Av_n^k(1324) ≤ 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.

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