An elementary proof of Bevan's theorem on the growth of grid classes of permutations

TL;DR

This paper provides an elementary, self-contained proof of Bevan's theorem, showing that the growth rate of monotone grid classes of permutations equals the square of the spectral radius of an associated bipartite graph, using basic mathematical tools.

Contribution

It offers a simplified, accessible proof of a key result in permutation class growth rates, generalizing Bevan's theorem with elementary methods.

Findings

Growth rate equals the square of the spectral radius of the bipartite graph

Elementary proof uses Stirling's Formula, Lagrange multipliers, and SVD

Generalizes Bevan's theorem to broader classes

Abstract

Bevan established that the growth rate of a monotone grid class of permutations is equal to the square of the spectral radius of a related bipartite graph. We give an elementary and self-contained proof of a generalization of this result using only Stirling's Formula, the method of Lagrange multipliers, and the singular value decomposition of matrices.