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A Formula for the number of Young Tableaux associated with a given Young Diagram. In
each box, write the sum of one plus the number of boxes horizontally to the right and vertically below the box (the ``hook
length''). The number of tableaux is then 
divided by the product of all ``hook lengths''. The Combinatorica`NumberOfTableaux function in Mathematica
implements the hook length formula.
See also Young Diagram, Young Tableau
References
Jones, V. ``Hecke Algebra Representations of Braid Groups and Link Polynomials.'' Ann. Math. 126, 335-388, 1987.
Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica.
Reading, MA: Addison-Wesley, 1990.