3
$\begingroup$

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 $n$?

|cite|improve this question
$\endgroup$
  • 2
    $\begingroup$ Isn't question 2 just: mathoverflow.net/questions/20711/… $\endgroup$ – Kevin Casto Mar 15 '17 at 17:02
  • $\begingroup$ @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. $\endgroup$ – user78249 Mar 15 '17 at 20:07
  • $\begingroup$ 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. $\endgroup$ – Sam Hopkins Mar 16 '17 at 15:17
  • $\begingroup$ I am almost sure it is not in EC2. $\endgroup$ – T. Amdeberhan Mar 16 '17 at 15:22
2
$\begingroup$

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

|cite|improve this answer
$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.