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 v₁, v₂, …, v_2n be a sequence of vertices arranged in clockwise order along a circumference. A chord diagram {v₁v_n + 1, v₂v_n + 2, …, v_nv_2n} is called an n-crossing and a chord diagram {v₁v₂, v₃v₄, …, v_2n − 1v_2n} 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.