TITLE: SYT and contents of a partition
QUESTION [7 upvotes]: 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 $t\rightarrow\infty$.

REPLY [6 votes]: 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.