Let $\mathbb{Y}\_n$ denote the set of all partitions of $n\in\mathbb{N}$ and $\mathbb{Y}$ Young's lattice of all partitions. The partition function $g\_0(n)=\sum\_{\lambda\in\mathbb{Y}\_n}1$ has an asymptotic formula proven by Hardy and Ramanujan. This sequence is A000041 in the OEIS.


Given a partition $\lambda$, let $\dim\lambda$ be the number of paths in $\mathbb{y}$ starting at $\emptyset$ and ending at $\lambda$, also known as the number of standard Young tableau of shape $\lambda$. The number
of Young tableau of size $n$ is given by $g\_1(n)=\sum\_{\lambda\in\mathbb{Y}\_n}\dim\lambda$ which has known asymptotics. Incidentally, $g\_1(n)$ counts the number of involutions in $\mathfrak{S}\_n$. This is sequence A000085 in the OEIS.


Basic representation theory of $\mathfrak{S}\_n$ and Stirling's approximation give
$$g\_2(n)=\sum\_{\lambda\in\mathbb{Y}\_n}(\dim\lambda)^2=n!\sim \sqrt{2\pi n}\,n^ne^{-n}.$$
A special feature of $g\_2(n)$ is: it allows an analytic continuation via the Euler's Gamma function $\Gamma(n)$.


Define $g\_k(n)=\sum\_{\lambda\in\mathbb{Y}\_n}(\dim\lambda)^k$. Now, I would like to inquire:



> 
> **Question 1.** What is the growth rate of $g\_3(n)$?
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> 
> **Question 2.** Is there an analytic continuation of $g\_3(n)$, as for $g\_2(n)$, in the variable $n$?
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> 
> 



$f^{\lambda}$: asymptotics and analytic continuations
Consider the Plancherel measure on partitions: the probability of $\lambda$ equals $\dim^2 \lambda/n!$. Then $g\_3(n)/n!$ is an expectation of $\dim \lambda$. It grows like $\sqrt{n!} e^{-c\sqrt{n}}$ for some $c$, see [Vershik and Kerov](https://link.springer.com/article/10.1007%2FBF01086021)


