Enumeration of linear chord diagrams

Abstract

A linear chord diagram canonically determines a fatgraph and hence has an associated genus $g$. We compute the natural generating function ${\mathbf{C}}_g(z)={\sum }_{n\ge 0}{\mathbf{c}}_g(n)z^n$ for the number ${\mathbf{c}}_g(n)$ of linear chord diagrams of fixed genus $g\ge 1$ with a given number $n\ge 0$ of chords and find the remarkably simple formula ${\mathbf{C}}_g(z)=z^{2g}{R}_g(z)(1-4z{)}^{\frac{1}{2}-3g}$, where $R_g(z)$ is a polynomial of degree at most $g-1$ with integral coefficients satisfying $R_g(\frac{1}{4})\ne 0$ and $R_g(0)={\mathbf{c}}_g(2g)\ne 0.$ In particular, ${\mathbf{C}}_g(z)$ is algebraic over $\mathbb{C}(z)$, which generalizes the corresponding classical fact for the generating function ${\mathbf{C}}_0(z)$ of the Catalan numbers. As a corollary, we also calculate a related generating function germaine to the enumeration of knotted RNA secondary structures, which is again found to be algebraic.

Cited by in corpus (1)

• Linear chord diagrams on two intervals