I'd like to try to develop some formal maths for listing the degeneracies of spinless fermion states in a harmonic oscillator. For those who don't know much quantum physics, I'm essentially trying to count the number distinct k-tuples whose entries sum to some number n (up to commutativity, ie. (123) = (213) = (312) = (321)), as well as adding the restriction that no two numbers in this k-tuple can be repeated.
The post in the link above helped me with the case of bosons (the same deal, but no repetition restrictions). I'm hoping someone could help me out, as I started to develop a flawed formalism and I'm too motivated to stop now. If you test the case j=3, k=3, you obtain 2 ways to write the j-tuple: (210) and (012), but these are just the same in my terms. Thanks a lot!