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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.

Let λ = {λ₁,λ₂,…,λ_m } be a partition of n. That is, λ_k for k=1,2,…,m are positive integers such that n = λ₁ + λ₂ + … + λ_m and λ₁ ≥ λ₂ ≥ … ≥ λ_m. Then the Young diagram of λ is an array of cells with indices (i,j) where 1 ≤ i ≤ m and 1 ≤ j ≤ λ_i. The Young diagram 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 with an integer between 1 and n in each cell without repetition such that each row and each column form increasing sequences. For each cell (i,j) of λ, the hook H_ij is the set of all cells to (a,b) such that a=i and b≥j or a≥i and b=j. The hook-length h_ij is defined as |H_ij|, the number of cells in the hook H_ij.