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：序列所有排列的结构可以表示为一个排列多面体，这是一个图，其中顶点为排列，若一个顶点的排列中交换相邻元素可得到另一个顶点的排列，则这两个顶点相连。已有假说认为语言中的词序能够最小化排列多面体中的交换距离：给定一个源顺序，在排列多面体中距离更近的词序应成本更低，因此更可能出现。在此，我们阐述如何测量词序变异相对于交换距离最小化的最优性程度。通过展示跨语言手势至少达到77%的最优性，我们证明了这一新型数学框架的效力。跨语言手势多次达到最优性，这几乎不可能是偶然所致。我们为研究通信系统中词序或手势顺序相对于交换距离最小化的最优性奠定了理论基础。最后，我们将二次分配问题引入语言研究，作为多个优化问题的统一框架，并据此提出一个统一包括交换距离最小化在内的多种语言学原则的最优分配普适原则。