Enumeration of linear chord diagrams

A linear chord diagram canonically determines a fatgraph and hence has an associated genus $g$. We compute the natural generating function ${\bf C}g(z)=\sum{n\geq 0} {\bf c}_g(n)z^n$forthenumber${\bf c}_g(n)$ 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 ${\bf C}_g(z)=z^{2g}R_g(z) (1-4z)^{{1\over 2}-3g}$,where$R_g(z)$isapolynomialofdegreeatmost$g-1$withintegralcoefficientssatisfying$R_g({1\over 4})\neq 0$and$R_g(0) = {\bf c}_g(2g)\neq 0.$Inparticular,${\bf C}_g(z)$isalgebraicover$\mathbb C(z)$, 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.