The Expansion of a Chord Diagram and the Genocchi Numbers

Tomoki Nakamigawa

Abstract


A chord diagram E is a set of chords of a circle such that no pair of chords has a common endvertex. Let v1, v2, …, v2n be a sequence of vertices arranged in clockwise order along a circumference. A chord diagram {v1vn + 1, v2vn + 2, …, vnv2n} is called an n-crossing and a chord diagram {v1v2, v3v4, …, v2n − 1v2n} is called an n-necklace. 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}. 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.


Keywords


Chord diagram, chord expansion, Genocchi number, Seidel triangle.

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ISSN: 1855-3974

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