Below is the outline of a proof idea I have for the Hook Content Formula. I'm wondering whether anyone is aware of whether this technique has been used before, and if so, if they could give me a reference. It seems like it should have been done somewhere already, judging by its simplicity, but I'm having trouble finding it. Moreover, has anyone seen identity** appear somewhere else?

The Hook Content Formula gives the number of SSYT on a Young diagram of shape $\lambda$ with entries between $1$ and $N$. The idea of the proof is to use induction on $N$. To do this, we introduce a function: $C(N,n_0,\ldots,n_d)$, defined on $(d+2)$--tuples of integers, which returns the number of SSYT of shape $(n_0,\ldots,n_d)'$ using integers from $1$ to $N$ if $n_0\geq \cdots\geq n_d$, and returns $0$ otherwise. It is clear from the definition of $C$ that if each $n_i\geq 1$: \begin{eqnarray*} C(N,n_0,\ldots,n_d)=\sum_{(j_0,\ldots,j_d): j_i\in \{0,1\}} C(N-1,n_0-j_0,\ldots,n_d-j_d). \end{eqnarray*}

We claim that if $n_0-0\geq n_1-1 \geq \cdots\geq n_d-d$, then the function:

\begin{eqnarray} F(N,n_0,\ldots,n_d)=\prod_{i=0}^d\frac{(N+i)!}{(N+i-n_i)!}\times \frac{V(m_0,\ldots,m_d)}{m_0!\cdots m_d!},* \end{eqnarray}

(where $V$ refers to the Vandermonde polynomial, and $m_i=n_i+d-i$ for each $i$) coincides with $C(N,n_0,\ldots,n_d)$. The claim holds for $N=0$, so suppose it holds for $N-1$. We consider an arbitrary $(d+1)$--tuple of nonnegative integers, $(n_0,\ldots,n_d)$ satisfying $n_0-0\geq n_1-1 \geq \cdots\geq n_d-d$. In the case that $n_0\geq \cdots\geq n_d$ does not hold, both $F$ and $C$ vanish. So assume that $n_0\geq \cdots\geq n_d$. Moreover, we may assume WLOG, that $n_d \neq 0$. Hence the claim will follow by the inductive hypothesis if we can show that: \begin{eqnarray*} F(N,n_0,\ldots,n_d)=\sum_{(j_0,\ldots,j_d): j_i \in \{0,1\}} F(N-1,n_0-j_0,\ldots,n_d-j_d). \end{eqnarray*}

To do this we introduce and prove the following identity: \begin{eqnarray*} && X(X-t)(X-2t)\cdots(X-nt)\times V(x_0,\ldots,x_n)\\ &=&\sum_{(j_0,\ldots,j_n): j_i \in \{0,1\}}\Bigg(\bigg[ \prod_{i=0}^n (X-x_i)^{1-j_i}x_i^{j_i}\bigg] V(x_0-j_0t,\ldots,x_n-j_nt)\Bigg), ** \end{eqnarray*} from which the above follows, and our claim is complete.

In particular then, when $n_0\geq \cdots\geq n_d$, $F(N,n_0,\ldots,n_d)$ gives the number of SSYT of shape $(n_0,\ldots,n_d)'$ using integers from $1$ to $N$. Finally we show that this implies the Hook Content Formula: the left hand factor in * gives the product of the hook contents, the right hand factor gives the reciprocal of the product of the hook lengths. The former is easy to see, the latter takes a bit of work.