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We have for $k>0$, $n>0$, $m\geqslant0$ $$p_k(n,m)=k(n-1)!\sum\limits_{s=0}^{n-1}\frac{p_{k-1}(s+1,m+1)+p_{k-1}(m+1,s)}{s!}$$ also $$p_0(n,m)=\begin{cases} (n-1)!,&\text{$n>0, m=0$}\\ 0,&...