Is there a closed form expression for the number of terms in a monomial symmetric polynomial in a given number of variables for a particular partition of exponents, in terms of which/how many exponents are distinct?
I feel the answer should be straightforward, and it's probably just a contrived statement of a more elementary number theoretic question, but I'm just drawing a blank at the moment. I at first thought that the answer might be
for $N$ variables and $k$ distinct exponents, which works for all $N=2$ and $N=3$ cases, but predicts 4 terms for a partition $\alpha=(1,1,0,0)$, while there are in fact 6:
$$m_\alpha(a,b,c,d) = ab+ac+ad+bc+bd+cd$$
Am I being an idiot?