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In combinatorial mathematics, the hook-length formula gives d_λ, the number of Young tableau of shape λ, as a function of the hook-lengths of λ. It has applications to representation theory and the problem of longest increasing subsequences.

Let λ = {λ₁,λ₂,…,λ_m} be a partition of n. That is, λ₁,λ₂,…,λ_m are positive integers such that n = λ₁ + λ₂ + … + λ_m and λ₁ ≥ λ₂ ≥ … ≥ λ_m. Then the Young diagram of shape λ, denoted Y(λ), is an array of cells with indices (i,j) where 1 ≤ i ≤ m and 1 ≤ j ≤ λ_i. 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 λ, the hook H_λ(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 h_λ(i,j) is defined as |H_λ(i,j)|, the number of cells in the hook H_λ(i,j).

Then the hook-length formula is

$d_{\lambda }={\frac {n!}{\prod _{(i,j)\in Y(\lambda )}\mathrm {h_{\lambda }} (i,j)}}.$

There are other formulas for d_λ, but the hook-length formula is particularly simple. Indeed, 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 determinant formula for d_λ. 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.

Since its discovery in 1954, many alternate proofs for the hook-length have been found. C. Greene, A. Nijenhuis, and H. S. Wilf found a probabilistic proof in which the hook lengths appear naturally in 1979. A direct bijective proof was first discovered by D. S. Franzblau and D. Zeilberger in 1982. A simpler bijective proof was announced in 1992 by Pak, I. M. and Stoyanovskii, A. V., and it was fully proven by Jean-Christophe Novelli, Igor Pak, and Alexander V. Stoyanovskii in 1997.

Stanley 1972 Hillman and Grassl provided in 1976 proof uses hooks in a natural way to derive a combinatorial correspondence involving plane partitions, from which the Frame-Robinson-Thrall formula can be derived by an asymptotic argument.

For the latter, the bijection without the complete proof was announced in 1992 by presented

References

Frame, J. S., Robinson, G. de B. and Thrall, R. M. (1954). The hook graphs of the symmetric group. Canad. J. Math. 6, 316–325.

Hillman, A. P. and Grassl, R. M. (1976). Reverse plane partitions and tableau hook numbers. J. Combin. Theory Ser A 21, 216–221.

Stanley, R. (1972). Ordered structures and partitions.Memoirs Amer. Soc. 119.

Greene, C., Nijenhuis, A. and Wilf, H. S. (1979) A probabilistic proof of a formula for the number of Young tableaux of a given shape. Adv. in Math. 31, 104–109.

Franzblau, D. S. and Zeilberger, D. (1982). A bijective proof of the hook-length formula. J. Algorithms 3, 317–343.

Novelli, J.-C., Pak, I. M. and Stoyanovskii, A. V. (1997). A direct bijective proof of the hook-length formula. Discrete Mathematics and Theoretical Computer Science 1, 1997, 53–67

P. A. MACMAHON, “Combinatory Analysis,” Cambridge Univ. Press, London/New York, 1916; reprinted by Chelsea, New York, 1960.

A. YOUNG, Quantitative substitutional analysis II, Proc. London Math. Sot., Ser. 1, 35 (1902), 361-397.

G. FROBENIUS, Uber die charaktere der symmetrischer gruppe, Preuss. &ad. Wk. sitz. (1900), 516-534.

4. D. E. I(NUTH, “The Art of Computer Programming,” Vol. 3, “Sorting and Searching,” Addison-Wesley, Reading, Mass., 1973. NUMBER OF YOUNG TABLEAUX OF A GIVEN SHAPE 109