AI 中文总结
研究排列的逆序图，证明其能支配排列的多方面特性，如科克斯特与反射长度差距等，阐述了简约表达式对逆序图边排序的机制，统一了相关经典特征，还给出连通无环逆序图是毛毛虫图的新证明。
AI 中文摘要
我们证明排列的逆序图支配着排列的几个看似不同的方面：它的科克斯特长度和反射长度之间的差距、其简约表达式中重复字母的数量及其循环结构，并且它还限制了321和3412模式的数量。其机制是简约表达式对逆序图的边进行排序，首次出现字母的边形成一个生成森林，重复字母的边构成其余部分。逆序图为森林的情况统一了由埃德尔曼、特纳以及彼得森和特纳给出的这些排列的几个经典特征。我们还给出了一个新的证明，即每个连通无环逆序图都是毛毛虫图。
英文摘要
We show that several apparently different aspects of a permutation are all tied to a single quantity, the cyclomatic number of its inversion graph. Every reduced word for the permutation orders the edges of the inversion graph one at a time, with the edges from first-occurrence letters forming a spanning forest and the edges from repeated letters accounting for the rest; the number of repeated letters is therefore the cyclomatic number. The excess of permutation cycles over sum components is also at most this quantity, and it follows that the gap between the Coxeter and reflection lengths is at least the cyclomatic number and at most twice it. When the inversion graph is a forest, these results unify classical characterizations of the boolean permutations due to Edelman, to Tenner, and to Petersen and Tenner. We also give a new proof that every connected acyclic inversion graph is a caterpillar.
Comments19 pages
