# $f^{\lambda}$: asymptotics and analytic continuations

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 $$g_3(n)$$?

Question 2. Is there an analytic continuation of $$g_3(n)$$, as for $$g_2(n)$$, in the variable $$n$$?

• Isn't question 2 just: mathoverflow.net/questions/20711/… Mar 15, 2017 at 17:02
• @KevinCasto Perhaps he means there may be some number theoretic property of $g_3$ that he desires in the continuation. Otherwise the solution is pretty straight forward.
– user78249
Mar 15, 2017 at 20:07
• I have the vague memory that $\sum_{\lambda \vdash n} f_{\lambda}^{3}$ should count the number of triples? of permutations that satisfy some condition like multiplying to the identity, but I looked a little bit and cannot seem to find this in EC2. Mar 16, 2017 at 15:17
• I am almost sure it is not in EC2. Mar 16, 2017 at 15:22

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