Enumeration Problems on the Expansion of a Chord Diagram

A chord diagram is a set of chords of a circle such that no pair of chords has a common endvertex. A pair of chords is called a crossing if the two chords intersect. A chord diagram E is called nonintersecting if E contains no crossing. For a chord diagram E having a 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}. A chord diagram E = E1 ∪E2 is called complete bipartite of type (m,n), denoted by Cm,n, if (1) both E1 and E2 are nonintersecting, (2) for every pair e1 ∈ E1 and e2 ∈ E2, e1 and e2 are crossing, and (3) |E1| = m, |E2| = n. Let fm,n be the cardinality of the multiset of all nonintersecting chord diagrams generated from Cm,n with a finite sequence of expansions. In this paper, it is shown ∑ m,n fm,n(x m/m!)(yn/n!) is 1/(coshx cosh y − (sinhx+ sinh y)).