Let a partition $P$ of some polynomial consist of the multiplicities of every root of that polynomial. We analyze the function $\overline{w}$ of a partition $P$, which counts the number of polynomials with coefficients modulo a prime q and with root multiplicities given by $P$ or a degeneration of $P$. We present a class of partitions with formula $\overline{w}$ that is very simple, namely a power of $q$, and wonder what other cases yield the same formula.