フック長の公式
The hook-length h λ ( i , j ) {\displaystyle h_{\lambda }(i,j)} is the number of cells in the hook H λ ( i , j ) {\displaystyle H_{\lambda }(i,j)} .
Then the hook-length formula expresses the number of standard Young tableaux of shape λ {\displaystyle \lambda } , sometimes denoted by d λ {\displaystyle d_{\lambda }} , as
d λ = n ! ∏ h λ ( i , j ) , {\displaystyle d_{\lambda }={\frac {n!}{\prod h_{\lambda }(i,j)}},}
where the product is over all cells ( i , j ) {\displaystyle (i,j)} of λ {\displaystyle \lambda } .
Let λ = ( λ 1 , ... , λ m ) {\displaystyle \lambda =(\lambda _{1},\ldots ,\lambda _{m})} be a partition of n {\displaystyle n} . It is customary to interpret λ {\displaystyle \lambda } graphically as a Young diagram, namely a left-justified array of square cells with m {\displaystyle m} rows and λ i {\displaystyle \lambda _{i}} cells in the i {\displaystyle i} th row for each 1 ≤ i ≤ m {\displaystyle 1\leq i\leq m} . A standard Young tableau of shape λ {\displaystyle \lambda } is a Young diagram of shape λ {\displaystyle \lambda } in which each of the n {\displaystyle n} cells contains a distinct integer between 1 and n {\displaystyle n} (i.
All that remains is tracking that coefficient with a mixture of cleverness and brute force: Multiplying
Δ ( x ) = ∑ w ∈ S n sgn  ( w ) x 1 w ( 1 ) − 1 x 2 w ( 2 ) − 1 ⋯ x k w ( k ) − 1 {\displaystyle \Delta (x)=\sum _{w\in S_{n}}\operatorname {sgn}(w)x_{1}^{w(1)-1}x_{2}^{w(2)-1}\cdots x_{k}^{w(k)-1}}
and
p 1 ( n ) = ( x 1 + x 2 + ⋯ + x k ) n = ∑ n ! d 1 ! d 2 ! ⋯ d k ! x 1 d 1 x 2 d 2 ⋯ x k d k {\displaystyle p_{1^{(n)}}=(x_{1}+x_{2}+\cdots +x_{k})^{n}=\sum {\frac {n!}{d_{1}!d_{2}!\cdots d_{k}!}}x_{1}^{d_{1}}x_{2}^{d_{2}}\cdots x_{k}^{d_{k}}}
we conclude that the coefficient that we are looking for is
∑ w ∈ S n sgn  ( w ) n ! ( l 1 − w ( 1 ) + 1 ) ! ( l 2 − w ( 2 ) + 1 ) ! ⋯ ( l k − w ( k ) + 1 ) !
数学の組合せ論において、フック長の公式(フックちょうのこうしき、英語: Hook length formula)とは、与えられたヤング図形の形をした標準盤を数える公式である。表現論や、確率論、アルゴリズム解析などの多種多様な分野に応用があり、最長増加部分列問題(英語版)などが例としてあげられる。