  A standard seaweed subalgebra of $A_{n-1}=\mathfrak{sl}(n)$ may be
parametrized by a pair of compositions of the positive integer $n$. For all $n$
and certain $k(n)$, we provide closed-form formulas and the generating
functions for $C(n,k)$ -- the number of parametrizing pairs which yield a
seaweed subalgebra of $\mathfrak{sl}(n)$ of index $k$. Our analysis sets the
framework for addressing similar questions in the other classical families.
