On the degeneracy of SU(3)k topological phases

The ground state degeneracy of an SU(N)k topological phase with n quasiparticle excitations is a 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.