The direct multiplicity product and the published multiset proof give the same Euler product

The published item Integer partitions have generating function ${\prod }_{n\ge 1}(1-x^n{)}^{-1}$ proves

${\sum }_{n\ge 0}p(n)x^n={\prod }_{m\ge 1}(1-x^m{)}^{-1}$

by treating a partition as a multiset of one abstract atom of each positive size and then invoking If $\mathcal{A}$ has no size-zero objects then $\mathrm{MSET}(\mathcal{A})$ has generating function ${\prod }_{n\ge 1}(1-x^n{)}^{-a_n}$.

The same identity also has a direct coefficientwise reading. A partition is equally a multiplicity sequence $(c_1,c_2,\dots )$ with only finitely many nonzero entries in each fixed total degree, and the coefficient of $x^n$ in

${\prod }_{m\ge 1}(1+x^m+x^{2m}+\cdots )$

depends only on the finitely many multiplicity choices satisfying ${\sum }_{m\ge 1}m{c}_m=n$. The summability hypothesis of Summable formal families may be regrouped and rearranged, distribute over multiplication, and have well-defined locally finite products is exactly what legitimizes turning that degreewise finite counting argument into a formal infinite product. So the "direct" product and the published multiset product are not two different series: they encode the same multiplicity data in two equivalent ways.