I have a proof that given a partition $\lambda=(\lambda_1,\dots,\lambda_l)$ then the number of semi-standard Young tableaux (SSYT) of shape $\lambda$ with entries in $1,2,\dots, n$ is given by
$$\frac{1}{1!2!\cdots (n-1)!} \prod_{1\leq i\lt j\leq n} (\lambda_i-i)-(\lambda_j-j).$$ (We define $\lambda_j:=0$ if $j \gt l.$)
The product is also recognized as a Vandermonde determinant.
There are plenty of product formulas (over boxes in the tableau) and determinant formulas (but not in Vandermonde form, as far as I can tell) for the number of such SSYTs, but I have not seen a this particular one in the literature or in any article I've come across.
Is this formula known? Is this formula of any interest?