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} 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,nfm,n(xm/m!)(yn/n!) is 1/(coshxcoshy−(sinhx+sinhy)).