# Sums of Partitions and Stirling's formula

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?

 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 $|\mathbb{Y}_N|$.

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Presumably one would have to find the largest term and how nearby terms deviate from this maximum. Results of this nature are due to Kerov, e.g., references [57][61][64] of math.mit.edu/~rstan/papers/ids.pdf. I don't know whether Kerov's work can lead to a proof of Stirling's formula. If so, it will be a much more difficult proof than the standard ones. –  Richard Stanley Jul 18 '11 at 3:15
Yes, it would certainly be more difficult, though it might help with the following. I'm interested in replacing the exponent "2" with an arbitrary integer a, which recovers the partition function at a=0. Computing the asymptotics in n of f(n,a) could hopefully interpolate between Stirling's formula and Hardy-Ramanujan's formula. However, it doesn't seem that such a function f(n,a) would yield a harmonic function / coherent system of probability measures in the sense of Kerov's asymptotic representation theory (the case a=2 for the Plancherel measure seems special in that regard). –  Alexander Moll Jul 18 '11 at 4:38
Replacing the exponent "2" with a complex variable results in what's known as the "Witten zeta function" or "representation zeta function" of the symmetric group $S_N$. This function has most combinatorial meaning at negative integer values for "2". See arxiv.org/abs/1102.4353 –  John Wiltshire-Gordon Jul 18 '11 at 5:00
It might be of interest to look into the work of Berele-Regev concerned with the asymptotics of Young tableaux in (k,l)-hooks (I think they deal with a question related to yours). Link: arxiv.org/PS_cache/arxiv/pdf/1007/1007.3833v1.pdf –  Vasu vineet Jul 18 '11 at 6:40
A related, though slightly different, question is addressed in a paper of A. Maroti (see MR2006609), where asymptotic estimates for the partition function are addressed via character theory of $S_n$. –  Geoff Robinson Jul 18 '11 at 7:13
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