Expansions of a Chord Diagram and Alternating Permutations

A chord diagram is a set of chords of a circle such that no pair of chords has a common endvertex. A chord diagram E with n chords is called an n-crossing if all chords of E are mutually crossing. A chord diagram E is called nonintersecting if E contains no 2-crossing. For a chord diagram E having a 2-crossing S = {x1x3, x2x4}, the expansion of E with respect to S is to replace E with E1 = (E\S)∪{x2x3, x4x1} or E2 = (E\S)∪{x1x2, x3x4}. It is shown that there is a one-to-one correspondence between the multiset of all nonintersecting chord diagrams generated from an ncrossing with a finite sequence of expansions and the set of alternating permutations of order n + 1.