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 $$p_0(n,m)=\begin{cases} (n-1)!,&\text{$n>0, m=0$}\\ 0,&\text{$n>0, m>0$} \end{cases}$$ so the closed form $$p_{k}(n,m)=k!\sum\limits_{s=0}^{k-1}\binom{n+\left\lfloor\frac{k+s}{2}\right\rfloor-1}{n+s-1}\binom{m+\left\lfloor\frac{k+s-1}{2}\right\rfloor}{m+s}\frac{(n+s-1)!(m+s)!}{s!}$$ But how can one derive it from recurrence?