Cyclomatic numbers and permutations

We show that the inversion graph of a permutation governs several apparently different aspects of the permutation: the gap between its Coxeter and reflection lengths, the number of repeated letters in its reduced words, and its cycle structure, and it also bounds the number of $321$ and $3412$ patterns. The mechanism is that a reduced word orders the edges of the inversion graph, with the edges from first-occurrence letters forming a spanning forest and the edges from repeated letters accounting for the rest. The case where the inversion graph is a forest unifies several classical characterizations of these permutations, due to Edelman, Tenner, and Petersen and Tenner. We also give a new proof that every connected acyclic inversion graph is a caterpillar.