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 Γ of non-negative real numbers, is equal to the square of the largest singular value of Γ and that the maximizing point can be expressed as a Hadamard product of Γ with the tensor product of singular vectors for its greatest singular value.

中文翻译：

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贝文关于排列网格类增长的定理的基本证明

Bevan 建立了单调网格类排列的增长率等于相关二分图谱半径的平方。我们仅使用斯特林公式、拉格朗日乘子法和矩阵的奇异值分解给出了对该结果的推广的基本且自足的证明。我们的证明依赖于证明空间上的最大值n×n具有非负条目的矩阵求和为这些条目的某个函数之一，由另一个矩阵的条目参数化Γ的非负实数，等于最大奇异值的平方Γ并且最大化点可以表示为的 Hadamard 乘积Γ与奇异向量的张量积为其最大奇异值。