Stirling's formula $$N! \sim \sqrt{2 \pi}\ N^{N+ \frac{1}{2}} e^{-N}$$ follows easily from Laplace's method in light of the famous integral representation $$N! = \int_0^{\infty} e^{-z} z^N dz.$$ Basic representation theory of the symmetric group $S(N)$ gives the remarkable finite sum $$N! = \sum_{\lambda \in \mathbb{Y}_N} (\dim \lambda)^2$$ over all partitions of $N$, where $\dim \lambda$ counts the number of paths from $\emptyset$ to $\lambda$ in Young's lattice $\mathbb{Y}$.

- Is it possible to derive Stirling's formula directly from this finite sum?

[Edit] I'd like to thank everyone for the very interesting comments below. For those looking at this thread for the first time, I was hoping that an answer to the question above might help us calculate the asymptotics of $$f(N,a) = \sum_{\lambda \in \mathbb{Y}_N} (\dim \lambda)^{a}.$$ Notice that the case $a=0$ is the partition function $|\mathbb{Y}_N|$.