### 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*_{2n} 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*_{2n}} is called an *n*-crossing and a chord diagram {*v*_{1}*v*_{2}, *v*_{3}*v*_{4}, …, *v*_{2n − 1}*v*_{2n}} 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.

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MANUSCRIPTISSN: 1855-3974

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