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In combinatorial mathematics, the hook-length formula gives dλ, the number of Young tableau of shape λ, as a function of the hook-lengths of λ. It has applications in diverse areas such as probability, algebraic geometry, group theory, representation theory, and algorithm analysis; for example, the problem of longest increasing subsequences.

Let λ={λ1,λ2,…,λm} be a partition of n. That is, λ1,λ2,…,λm are positive integers such that n=λ1+λ2+…+λm and λ1≥λ2≥…≥λm. Then the Young diagram of shape λ, denoted Y(λ), is an array of cells with indices (i,j) where i=1,2,…,m and j=1,2,…,λi. The Young diagram of shape λ has n cells and has a natural bijection to a partition of n. A standard Young tableau of shape λ is a Young diagram of shape λ in which each of the n cells contains a distinct integer between 1 and n (i.e., no repetition), such that each row and each column form increasing sequences. For each cell (i,j) of Y(λ), the hook Hλ(i,j) is the set of all cells (a,b) such that a=i and b≥j or a≥i and b=j. The hook-length hλ(i,j) is |Hλ(i,j)|, the number of cells in the hook Hλ(i,j).

Then the hook-length formula is
 * $$ d_\lambda = \frac{n!}{\prod_{(i,j) \in Y(\lambda)} h_\lambda (i,j)} . $$

There are other formulas for dλ, but the hook-length formula is particularly simple. In fact, 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 for dλ 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 highly non-trivial 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.