Define $\binom{a\_1,...,a\_n}{b\_1,...,b\_m}=\Pi\_i(a\_i!)/\Pi\_j(b\_j!)$. (WLOG null-pad to have $m=n$ and sort descendingly.) It's natural to ask when the value is an integer. Since I am working with "majorization" currently...but alas, $\binom{3,1}{2,2}=3/2$ and $\binom{6,1,1}{5,3,0}=1$ are counterexamples for both directions :-/ Also the last example (inverted) shows that not even $a\_1>b\_1$ is necessary.  

Do you know conditions (beside the obvious one that a multinomial coefficient is always an integer)? Any research on this area?



"Super" multinomial coefficients
Mordell, L. J. (1959),  

Integer Quotients of Products of Factorials.  

Journal of the London Mathematical Society, s1-34: 134–138.  

doi:10.1112/jlms/s1-34.2.134


Errata:  

On a Problem of Hardy and Littlewood Volume s1-34, Issue 4, 485,  

Errata:  

Integer Quotients of Products of Factorials Volume s1-34, Issue 4, 485,


