The structure of all the permutations of a sequence can be represented as a permutohedron, a graph where vertices are permutations and two vertices are linked if a swap of adjacent elements in the permutation of one of the vertices produces the permutation of the other vertex. It has been hypothesized that word orders in languages minimize the swap distance in the permutohedron: given a source order, word orders that are closer in the permutohedron should be less costly and thus more likely. Here we explain how to measure the degree of optimality of word order variation with respect to swap distance minimization. We illustrate the power of our novel mathematical framework by showing that crosslinguistic gestures are at least $77\%$ optimal. It is unlikely that the multiple times where crosslinguistic gestures hit optimality are due to chance. We establish the theoretical foundations for research on the optimality of word or gesture order with respect to swap distance minimization in communication systems. Finally, we introduce the quadratic assignment problem (QAP) into language research as an umbrella for multiple optimization problems and, accordingly, postulate a general principle of optimal assignment that unifies various linguistic principles including swap distance minimization.

翻译：序列的所有排列结构可以表示为置换多面体，这是一个图，其中顶点代表排列，若一个顶点的排列中相邻元素交换后得到另一个顶点的排列，则这两个顶点相连。已有假设认为，语言中的词汇顺序在置换多面体中最小化交换距离：给定一个源顺序，在置换多面体中距离更近的词汇顺序成本更低，因此更可能被采用。本文解释了如何衡量词汇顺序变异相对于交换距离最小化的最优程度。我们通过展示跨语言手势至少具有$77\%$的最优性，证明了这一新颖数学框架的效力。跨语言手势多次达到最优性，这几乎不可能是偶然现象。我们为通信系统中词汇或手势顺序相对于交换距离最小化最优性的研究奠定了理论基础。最后，我们将二次分配问题（QAP）引入语言研究，作为多种优化问题的总框架，并据此提出了统一多种语言学原则（包括交换距离最小化）的通用最优分配原则。