$${P}_{m}(k,S)=\sum _{\sum _{s\in S}{t}_{s}=m,\sum _{s\in S}s{t}_{s}=k}\prod _{s\in S-0}\frac{k!}{{t}_{s}!(s!{)}^{{t}_{s}}}$$

and generating function considering parameter $k$ is

$$\sum _{k=0}^{\mathrm{\infty}}{P}_{m}(k,S)\frac{{x}^{k}}{k!}=\sum _{{t}_{0}=0}^{m-1}\frac{1}{(m-{t}_{0})!}{\left(\sum _{s\in S-0}\frac{{x}^{s}}{s!}\right)}^{m-{t}_{0}}$$

Can someone find the generating function with two variables considering parameters $m$ and $k$?