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?