Upper bound of $S=\sum_{k=1}^{P}k!\binom{P}{k}\binom{Q}{k}$

Question

EDIT: How can I find a good upper bound to this quantity ?

$$S_{n,m}=\sum_{k=1}^{P}k!\binom{P}{k}\binom{Q}{k}$$

where $P=\min\{m,n\}$ et $Q=\max\{m,n\}$.

Answer 1

I don't know how this upper bound is good but this is one that you can use.

So here is the hint: In this paper, Asymptotic Enumeration Methods (1995), there is an upper bound as follow (which can be found on wikipedia too):

$$\left(\frac{n}{k}\right)^k \le {n \choose k} \le \frac{n^k}{k!} \le \left(\frac{n\cdot e}{k}\right)^k,\; \text{for}\; 1\leqslant k\leqslant n.$$

Now, suppose that $M>N$ and find the upper bound. Do the same for $N>M$.