Permutations avoiding 1324 and patterns in {\L}ukasiewicz paths

TL;DR

This paper investigates the permutation class avoiding 1324, establishing a new lower bound on its growth rate by analyzing substructures and pattern distributions in related combinatorial objects.

Contribution

It introduces a novel approach to bounding the growth rate of Av(1324) using asymptotic analysis of substructures and pattern distributions in Lukasiewicz paths.

Findings

Growth rate of Av(1324) exceeds 9.81

Substructure distributions in Hasse graphs are Gaussian in the limit

Pattern occurrences in Lukasiewicz paths are concentrated and normally distributed

Abstract

The class Av(1324), of permutations avoiding the pattern 1324, is one of the simplest sets of combinatorial objects to define that has, thus far, failed to reveal its enumerative secrets. By considering certain large subsets of the class, which consist of permutations with a particularly regular structure, we prove that the growth rate of the class exceeds 9.81. This improves on a previous lower bound of 9.47. Central to our proof is an examination of the asymptotic distributions of certain substructures in the Hasse graphs of the permutations. In this context, we consider occurrences of patterns in {\L}ukasiewicz paths and prove that in the limit they exhibit a concentrated Gaussian distribution.