  Consider all possible ways of attaching disjoint chords to two ordered and
oriented disjoint intervals so as to produce a connected graph. Taking the
intervals to lie in the real axis with the induced orientation and the chords
to lie in the upper half plane canonically determines a corresponding fatgraph
which has some associated genus $g\geq 0$, and we consider the natural
generating function ${\bf C}_g^{[2]}(z)=\sum_{n\geq 0} {\bf c}^{[2]}_g(n)z^n$
for the number ${\bf c}^{[2]}_g(n)$ of distinct such chord diagrams of fixed
genus $g\geq 0$ with a given number $n\geq 0$ of chords. We prove here the
surprising fact that ${\bf C}^{[2]}_g(z)=z^{2g+1} R_g^{[2]}(z)/(1-4z)^{3g+2} $
is a rational function, for $g\geq 0$, where the polynomial $R^{[2]}_g(z)$ with
degree at most $g$ has integer coefficients and satisfies $R_g^{[2]}({1\over
4})\neq 0$. Earlier work had already determined that the analogous generating
function ${\bf C}_g(z)=z^{2g}R_g(z)/(1-4z)^{3g-{1\over 2}}$ for chords attached
to a single interval is algebraic, for $g\geq 1$, where the polynomial $R_g(z)$
with degree at most $g-1$ has integer coefficients and satisfies $R_g(1/4)\neq
0$ in analogy to the generating function ${\bf C}_0(z)$ for the Catalan
numbers. The new results here on ${\bf C}_g^{[2]}(z)$ rely on this earlier
work, and indeed, we find that $R_g^{[2]}(z)=R_{g+1}(z) -z\sum_{g_1=1}^g
R_{g_1}(z) R_{g+1-g_1}(z)$, for $g\geq 1$.
