Growth rates of geometric grid classes of permutations

TL;DR

This paper establishes a mathematical framework linking geometric grid classes of permutations to spectral graph theory, characterizing their growth rates through roots of matching polynomials and exploring structural influences.

Contribution

It introduces a novel connection between geometric grid classes and graph spectral properties, providing explicit formulas for growth rates and analyzing structural effects.

Findings

Growth rates are squares of the largest roots of matching polynomials.

Characterization of growth rates via spectral radii of trees.

Edge subdivision impacts the roots of matching polynomials.

Abstract

Geometric grid classes of permutations have proven to be key in investigations of classical permutation pattern classes. By considering the representation of gridded permutations as words in a trace monoid, we prove that every geometric grid class has a growth rate which is given by the square of the largest root of the matching polynomial of a related graph. As a consequence, we characterise the set of growth rates of geometric grid classes in terms of the spectral radii of trees, explore the influence of "cycle parity" on the growth rate, compare the growth rates of geometric grid classes against those of the corresponding monotone grid classes, and present new results concerning the effect of edge subdivision on the largest root of the matching polynomial.