Growth rates of permutation grid classes, tours on graphs, and the spectral radius

TL;DR

This paper establishes a precise relationship between the growth rates of permutation grid classes and the spectral radius of associated graphs, using spectral graph theory and tour enumeration to characterize growth behaviors.

Contribution

It proves that the exponential growth rate of grid classes equals the square of the spectral radius of their row-column graphs, linking permutation classes to spectral graph theory.

Findings

Growth rate equals spectral radius squared

Characterization of slowly growing grid classes

Existence of grid classes with arbitrary large growth rates

Abstract

Monotone grid classes of permutations have proven very effective in helping to determine structural and enumerative properties of classical permutation pattern classes. Associated with grid class $\mathrm{Grid}(M)$ is a graph, $G(M)$, known as its "row-column" graph. We prove that the exponential growth rate of $\mathrm{Grid}(M)$ is equal to the square of the spectral radius of $G(M)$. Consequently, we utilize spectral graph theoretic results to characterise all slowly growing grid classes and to show that for every $\gamma\geqslant2+\sqrt{5}$ there is a grid class with growth rate arbitrarily close to $\gamma$. To prove our main result, we establish bounds on the size of certain families of tours on graphs. In the process, we prove that the family of tours of even length on a connected graph grows at the same rate as the family of "balanced" tours on the graph (in which the number of times an edge is traversed in one direction is the same as the number of times it is traversed in the other direction).