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The generating function for partitions where there are at most $r-1$ parts of each size is $$ (1+x+x^2+\dots+x^{r-1})(1+x^2+x^4+\dots+x^{2(r-1)})(1+x^3+\dots+x^{3(r-1)})\cdots $$ When expanding this out, the choice of the summand $x^{kj}$ in the factor $1+x^k+\dots+x^{k(r-1)}$ corresponds to having $j$ parts of size $k$ in the partition. We can write this as ...


As mentioned by Lord Shark the Unknown, it is easy to prove this using the fact that uniformly converging analytic functions converge to analytic functions. We have uniform convergence of an infinite product easily here on any disk of radius $r<1$ since $$\left|\log\left[\frac1{1-z^m}\right]\right|=|\log(1-z^m)|\le|z|^m+\frac{|z|^{2m}}{|1-z^m|^2}\le r^m+...

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