Hooklength

Hooklength refers to a concept in combinatorics associated with Young diagrams (Ferrers diagrams). In a partition λ = (λ1 ≥ λ2 ≥ … ≥ λk > 0) with n = sum λi boxes, the hook length h(i,j) of the cell in row i and column j is the number of boxes to the right in the same row plus the number of boxes below in the same column, plus one for the cell itself. Equivalently, if λ′ denotes the conjugate partition, h(i,j) = λi + λ′j − i − j + 1.

The hook-length formula gives the number of standard Young tableaux of shape λ: f^λ = n! / ∏_{(i,j)∈λ} h(i,j). A standard Young tableau is a filling of the diagram with the numbers 1 through n that increases along rows and columns. The formula shows that the product of hook lengths over all boxes divides n!, yielding an integer.

Applications of the hook-length formula include the representation theory of the symmetric group, where f^λ is the dimension of the irreducible representation corresponding to λ, as well as connections to symmetric functions, Schur functions, and various counting problems in combinatorics.

Generalizations exist to skew diagrams, shifted shapes, and other poset-ordered diagrams, with corresponding hook-length expressions. The concept also appears in algorithmic contexts where objects are counted with hook-length-like weights, though the standard formula is most tightly associated with Young diagrams and standard tableaux.