AN ELEMENTARY PROOF of BEVAN'S THEOREM on the GROWTH of GRID CLASSES of PERMUTATIONS

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. Our proof relies on showing that the maximum over the space of n × n matrices with non-negative entries summing to one of a certain function of those entries, parametrized by the entries of another matrix Λ of non-negative real numbers, is equal to the square of the largest singular value of Λ and that the maximizing point can be expressed as a Hadamard product of Λ with the tensor product of singular vectors for its greatest singular value.