フック長の公式

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.

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.

Connection to representation theory

The hook-length formula is of great importance in the representation theory of the symmetric group , where the number  is known to be equal to the dimension of the complex irreducible representation  associated to , and is frequently denoted by ,  or .

Connection to longest increasing subsequences

The hook length formula also has important applications to the analysis of longest increasing subsequences in random permutations. If  denotes a uniformly random permutation of order ,  denotes the maximal length of an increasing subsequence of , and  denotes the expected (average) value of , Anatoly Vershik and Sergei Kerov  and independently Benjamin F. Logan and Lawrence A. Shepp  showed that when  is large,  is approximately equal to . This answers a question originally posed by Stanislaw Ulam. The proof is based on translating the question via the Robinson–Schensted correspondence to a problem about the limiting shape of a random Young tableau chosen according to Plancherel measure. Since the definition of Plancherel measure involves the quantity , the hook length formula can then be used to perform an asymptotic analysis of the limit shape and thereby also answer the original question.
The ideas of Vershik–Kerov and Logan–Shepp were later refined by Jinho Baik, Percy Deift and Kurt Johansson, who were able to achieve a much more precise analysis of the limiting behavior of the maximal increasing subsequence length, proving an important result now known as the Baik–Deift–Johansson theorem. Their analysis again makes crucial use of the fact that  has a number of good formulas, although instead of the hook length formula it made use of one of the determinantal expressions.