On compositions associated to seasweed subalgebras of sl(n)

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.