# SYT and contents of a partition

Let $$\lambda$$ be an integer partition, denote the number of Standard Young Tableaux of shape $$\lambda$$ by $$f_{\lambda}$$. This number is computed by the formula $$f_{\lambda}=\frac{n!}{\prod_{u\in\lambda}h_u}$$ where $$h_u$$ is a hook length. It is also well-known that $$\sum_{\lambda\vdash n}f_{\lambda}^2=n!\tag1$$ Recall the notation for the content of a cell $$u=(i,j)$$ in a partition is $$c_u=j-i$$.

Question. Is this true? $$\sum_{\lambda\vdash n}f_{\lambda}^2\prod_{u\in\lambda}(t+c_u)=n!\,t^n.$$

NOTE. An affirmative answer would imply (1): divide both sides by $$t^n$$ and take the limit $$t\rightarrow\infty$$.

• For any cell $u$ of $\lambda$, let $h_{\lambda, u}$ denote the hook length of $u$ in $\lambda$. Using the formula quoted in mathoverflow.net/a/263658 as Corollary 7.21.4, we can rewrite $\prod\limits_{u \in \lambda} \left(t+c_u\right) = s_{\lambda}\left(1^t\right) \prod\limits_{u \in \lambda} h_{\lambda, u} = s_{\lambda}\left(1^t\right) \dfrac{n!}{f_{\lambda}}$ (by the hook-length formula). Thus, the left hand side of your question simplifies to $n! \cdot \sum\limits_{\lambda \vdash n} f_{\lambda} s_{\lambda}\left(1^t\right)$. The rest ... – darij grinberg Oct 11 '18 at 20:49
• ... is an easy application of RSK: The symmetric function $\sum\limits_{\lambda \vdash n} f_{\lambda} s_{\lambda}$ is the generating function of all $n$-letter words over the alphabet $\left\{x_1,x_2,x_3,\ldots\right\}$ (with the $f_{\lambda}$ counting all possible Q-tableaux and the $s_{\lambda}$ being the generating function of all possible P-tableaux) and thus equals $\left(x_1+x_2+x_3+\cdots\right)^n$ (a fact that also follows by applying the Pieri rule $n$ times); now substitute $1^t$ for the variables and you're done. – darij grinberg Oct 11 '18 at 20:51

Notice that $$f_{\lambda}$$ is the number of standard Young tableaux of shape $$\lambda$$, whereas $$f_{\lambda}\frac{\prod_{u\in \lambda} (t+c_u)}{n!}$$ is the number of semistandard Young tableaux of shape $$\lambda$$ and content $$t$$. It was mentioned in a previous question that the identity $$\sum_{|\lambda|=n}\left|\text{SSYT}(\lambda)\right|\left|\text{SYT}(\lambda)\right|=t^n.$$ can either be proved bijectively from RSK (as Darij explains in the comments) or by appealing to Schur-Weyl duality (as David explains in the link), which gives an isomorphism of $$S_n\times GL(V)$$ representations $$\sum_{|\lambda|=n} Sp_{\lambda} \boxtimes S_{\lambda}(V) \cong V^{\otimes n}$$ where $$V$$ is a $$t$$ dimensional vector space, $$S_{\lambda}$$ is the Schur functor, and $$Sp_{\lambda}$$ the corresponding Specht module. Our identity follows by taking the dimension of both sides.