Definitions and statement 
Let  be a partition of . It is customary to interpret  graphically as a Young diagram, namely a left-justified array of square cells with  rows and  cells in the th row for each . A standard Young tableau of shape  is a Young diagram of shape  in which each of the  cells contains a distinct integer between 1 and  (i.e., no repetition), such that each row and each column form increasing sequences. For each cell of the Young diagram in coordinates  (that is, the cell in the th row and th column), the hook  is the set of cells  such that  and  or  and . The hook-length  is the number of cells in the hook .

Then the hook-length formula expresses the number of standard Young tableaux of shape , sometimes denoted by , as

 

where the product is over all cells  of .

Example 

The figure on the right shows hook-lengths for all cells in the Young diagram  of the partition 

9 = 4 + 3 + 1 + 1. Then the number of standard Young tableaux  for this Young diagram can be computed as

History 
There are other formulas for , but the hook-length formula is particularly simple and elegant. The hook-length formula was discovered in 1954 by J. S. Frame, G. de B. Robinson, and R. M. Thrall by improving a less convenient formula expressing  in terms of a determinant.  This earlier formula was deduced independently by G. Frobenius and A. Young in 1900 and 1902 respectively using algebraic methods.   P. A. MacMahon found an alternate proof for the Young–Frobenius formula in 1916 using difference methods. 

Despite the simplicity of the hook-length formula, the Frame–Robinson–Thrall proof is uninsightful and does not provide an intuitive argument as to why hooks appear in the formula. The search for a short, intuitive explanation befitting such a simple result gave rise to many alternate proofs for the hook-length formula.  A. P. Hillman and R. M. Grassl gave the first proof that illuminates the role of hooks in 1976 by proving a special case of the Stanley hook-content formula, which is known to imply the hook-length formula.  C. Greene, A. Nijenhuis, and H. S. Wilf found a probabilistic proof using the hook walk in which the hook lengths appear naturally in 1979.  J. B. Remmel adapted the original Frame–Robinson–Thrall proof into the first bijective proof for the hook-length formula in 1982.  A direct bijective proof was first discovered by D. S. Franzblau and D. Zeilberger in 1982.  D. Zeilberger also converted the Greene–Nijenhuis–Wilf hook walk proof into a bijective proof in 1984.  A simpler direct bijective proof was announced by Igor Pak and Alexander V. Stoyanovskii in 1992, and its complete proof was presented by the pair and Jean-Christophe Novelli in 1997.  

Meanwhile, the hook-length formula has been generalized in several ways. R. M. Thrall found the analogue to the hook-length formula for shifted Young Tableaux in 1952.  B. E. Sagan gave a shifted hook walk proof for the hook-length formula for shifted Young tableaux in 1980.  B. E. Sagan and Y. N. Yeh proved the hook-length formula for binary trees using the hook walk in 1989.

Proofs

Knuth's heuristic argument 
The hook-length formula can be understood intuitively using the following heuristic, but incorrect, argument suggested by D. E. Knuth. Given that each element of a tableau is the smallest in its hook and filling the tableau shape at random, the probability that cell  will contain the minimum element of the corresponding hook is the reciprocal of the hook length.  Multiplying these probabilities over all  and  gives the formula. This argument is fallacious since the events are not independent.

Knuth's argument is however correct for the enumeration of labellings on trees satisfying monotonicity properties analogous to those of a Young tableau. In this case, the 'hook' events in question are in fact independent events.

フックウォークを用いた確率論的証明法 
これは1979年に、  C. Greene, A. Nijenhuis と H. S. Wilf によって発見された、確率論的証明法である。