  A linear chord diagram canonically determines a fatgraph and hence has an
associated genus $g$. We compute the natural generating function ${\bf
C}_g(z)=\sum_{n\geq 0} {\bf c}_g(n)z^n$ for the number ${\bf c}_g(n)$ of linear
chord diagrams of fixed genus $g\geq 1$ with a given number $n\geq 0$ of chords
and find the remarkably simple formula ${\bf C}_g(z)=z^{2g}R_g(z)
(1-4z)^{{1\over 2}-3g}$, where $R_g(z)$ is a polynomial of degree at most $g-1$
with integral coefficients satisfying $R_g({1\over 4})\neq 0$ and $R_g(0) =
{\bf c}_g(2g)\neq 0.$ In particular, ${\bf C}_g(z)$ is algebraic over $\mathbb
C(z)$, which generalizes the corresponding classical fact for the generating
function ${\bf C}_0(z)$ 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.
