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

Bevan, David; Brignall, Robert; Elvey Price, Andrew and Pantone, Jay (2020). A structural characterisation of Av(1324) and new bounds on its growth rate. European Journal of Combinatorics, 88, article no. 103115.

DOI: https://doi.org/10.1016/j.ejc.2020.103115

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 is related 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.