Recall the following:
Multinomial Theorem. For all finite sets $X$, we have:
$$\left(\sum_{x \in X} x\right)^n = \sum_{a}[a](X)^a$$
where $a$ ranges over the set of partitions of $n$ into $|X|$-many parts (with each part possibly $0$), $[a]$ is the relevant multinomial coefficient, and $(X)^a$ is the relevant monomial symmetric polynomial.
For example:
$$\left(x+y+z\right)^3 = [3,0,0](x,y,z)^{3,0,0}+[2,1,0](x,y,z)^{2,1,0}+[1,1,1](x,y,z)^{1,1,1}$$
$$= (x^3+y^3+z^3)+3(x^2y+x^2z+y^2z)+6xyz$$
Anyway, I don't particularly like this business of leaving out the domain of quantification for $a$ and just putting it in words.
Question. It would be good if there were a moderately standard notation for the set of all ways of partitioning a natural number $a$ into $b$ parts, so the multinomial theorem can be stated properly. Is there?