The expansion of a chord diagram and the Genocchi numbers

Abstract

A chord diagram E is a set of chords of a circle such that no pair of chords has a common endvertex. Let v 1 ,  v 2 , …,  v 2 n be a sequence of vertices arranged in clockwise order along a circumference. A chord diagram { v 1 v n  + 1 ,  v 2 v n  + 2 , …,  v n v 2 n } is called an n -crossing and a chord diagram { v 1 v 2 ,  v 3 v 4 , …,  v 2 n  − 1 v 2 n } is called an n -necklace. For a chord diagram E having a 2 -crossing S  = { x 1 x 3 ,  x 2 x 4 } , the expansion of E with respect to S is to replace E with E 1  = ( E  \  S ) ∪ { x 2 x 3 ,  x 4 x 1 } or E 2  = ( E  \  S ) ∪ { x 1 x 2 ,  x 3 x 4 } . Beginning from a given chord diagram E as the root, by iterating chord expansions in both ways, we have a binary tree whose all leaves are nonintersecting chord diagrams. Let NCD ( E ) be the multiset of the leaves. In this paper, the multiplicity of an n -necklace in NCD ( E ) is studied. Among other results, it is shown that the multiplicity of an n -necklace generated from an n -crossing equals the Genocchi number when n is odd and the median Genocchi number when n is even.