The expansion of a chord diagram and the Genocchi numbers - Nakamigawa, Tomoki (Ars mathematica contemporanea, 2020)
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 ?$\{v_1v_{n+1}, v_2v_{n+2}, …, v_nv_{2n}\}$? is called an ?$n$?-crossing and a chord diagram ?$\{v_1v_2, v_3v_4, …, v_{2n-1}v_{2n}\}$? is called an ?$n$?-necklace. For a chord diagram ?$E$? having a 2-crossing ?$S = \{x_1x_3, x_2x_4\}$?, the expansion of ?$E$? with respect to ?$S$? is to replace ?$E$? with ?$E_1 = (E \setminus S) \cup \{x_2x_3, x_4x_1\}$? or ?$E_2 = (E \setminus S) \cup \{x_1x_2, x_3x_4\}$?. 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 ?$\mathcal{NCD}(E)$? be the multiset of the leaves. In this paper, the multiplicity of an ?$n$?-necklace in ?$\mathcal{NCD}(E)$? 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; Genocchijevo število; Seidel triangle; Seidelov trikotnik; tetivni diagram; tetivni razplet