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?



New formula for counting semi-standard Young tableaux?
So, to answer this question, the formula is known. It is also (now) on the wikipedia page about Schur polynomials, so the next person will find it easily.


