We enumerate the numbers $\mathrm{av}_n^k(1324)$ 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, Jelínek and Steingrímsson, according to which $\mathrm{av}_n^k(1324) \leq \mathrm{av}_{n+1}^k(1324)$ 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 $13.002$.
@article{10_37236_13387,
author = {Svante Linusson and Emil Verkama},
title = {Enumerating 1324-avoiders with few inversions},
journal = {The electronic journal of combinatorics},
year = {2025},
volume = {32},
number = {3},
doi = {10.37236/13387},
zbl = {8097672},
url = {http://geodesic.mathdoc.fr/articles/10.37236/13387/}
}
TY - JOUR
AU - Svante Linusson
AU - Emil Verkama
TI - Enumerating 1324-avoiders with few inversions
JO - The electronic journal of combinatorics
PY - 2025
VL - 32
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/13387/
DO - 10.37236/13387
ID - 10_37236_13387
ER -
%0 Journal Article
%A Svante Linusson
%A Emil Verkama
%T Enumerating 1324-avoiders with few inversions
%J The electronic journal of combinatorics
%D 2025
%V 32
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/13387/
%R 10.37236/13387
%F 10_37236_13387
Svante Linusson; Emil Verkama. Enumerating 1324-avoiders with few inversions. The electronic journal of combinatorics, Tome 32 (2025) no. 3. doi: 10.37236/13387
