  The ground state degeneracy of an $SU(N)_k$ topological phase with $n$
quasiparticle excitations is relevant quantity for quantum computation,
condensed matter physics, and knot theory. It is an open question to find a
closed formula for this degeneracy for any $N > 2$. Here we present the problem
in an explicit combinatorial way and analyze the case N=3. While not finding a
complete closed-form solution, we obtain generating functions and solve some
special cases.
