A structural characterisation of Av(1324) and new bounds on its growth rate

TL;DR

This paper improves bounds on the exponential growth rate of 1324-avoiding permutations by providing a new structural characterization and analyzing related subclasses, leading to tighter bounds.

Contribution

It introduces a novel exact structural characterization of 1324-avoiders and derives new bounds on their growth rate using detailed combinatorial analysis.

Findings

Lower bound of 10.271 on growth rate

Upper bound of 13.5 on growth rate

Structural characterization as a subclass of an infinite staircase grid class

Abstract

We establish an improved lower bound of 10.271 for the exponential growth rate of the class of permutations avoiding the pattern 1324, and an improved upper bound of 13.5. These results depend on a new exact structural characterisation of 1324-avoiders as a subclass of an infinite staircase grid class, together with precise asymptotics of a small domino subclass whose enumeration we relate to West-two-stack-sortable permutations and planar maps. The bounds are established by carefully combining copies of the dominoes in particular ways consistent with the structural characterisation. The lower bound depends on concentration results concerning the substructure of a typical domino, the determination of exactly when dominoes can be combined in the fewest distinct ways, and technical analysis of the resulting generating function.