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.

I've finally found a place that gives this formula explicitly, without any need for piecing together or rewriting or taking limits using L'Hopital: Martin Aigner, A Course in Enumeration, Springer 2007, Section 8.5, formula (7) and the sentence that follows it.

I am wondering if anyone has ever given a bijective proof, though (after multiplying with the denominator).