The multiset construction $\mathrm{MSET}(\mathcal{A})$ and the powerset construction $\mathrm{PSET}(\mathcal{A})$

Let $\mathcal{A}$ be a combinatorial class.

An object of $\mathrm{MSET}(\mathcal{A})$ is a finitely supported multiplicity function

$m:\mathcal{A}\to \mathbb{N},$

whose value $m(a)$ records how many copies of $a$ occur. Its size is

$\mathrm{\mid }m\mathrm{\mid }:={\sum }_{a\in \mathcal{A}}m(a)\mathrm{\mid }a\mathrm{\mid },$

which is a finite sum because the support of $m$ is finite.

An object of $\mathrm{PSET}(\mathcal{A})$ is such a multiplicity function with values only in $\{0,1\}$, so it records an ordinary finite subset of $\mathcal{A}$. Its size is given by the same formula.

If $\mathcal{A}$ has a size-zero object, then $\mathrm{MSET}(\mathcal{A})$ may fail to be a combinatorial class because that object can be repeated arbitrarily often without changing total size. The powerset construction has no such failure: its multiplicities are only $0$ and $1$, and the size-zero level of $\mathcal{A}$ is finite. Its generating function would, however, acquire the extra factor $2^{a_0}$. The product formulas below use the uniform no-size-zero hypothesis and therefore start at positive sizes.