フック長の公式
The formula for the number of Young tableau of a given shape was originally derived from the Frobenius determinant formula in connection to representation theory. If the shape of a Young diagram is given by the row lengths n 1 , ... , n m {\displaystyle n_{1},\dots ,n_{m}} , then the number of tableau with that shape is given by
f ( n 1 , n 2 , ... , n m ) = n ! Δ ( n m , n m − 1 + 1 , ... , n 1 + m − 1 ) n m ! ( n m − 1 + 1 ) ! ⋯ ( n 1 + m − 1 ) ! {\displaystyle f(n_{1},n_{2},\ldots ,n_{m})={\frac {n!\,\Delta (n_{m},n_{m-1}+1,\ldots ,n_{1}+m-1)}{n_{m}!(n_{m-1}+1)!\cdots (n_{1}+m-1)!}}}
Hook lengths can also be used to give a product representation to the generating function for the number of reverse plane partitions of a given shape.
∏ i = 0 k − 2 ( l 1 − i ) 1 l 2 l 2 ( l 2 − 1 ) ... ∏ i = 0 k − 2 ( l 2 − i ) ⋮ ⋮ ⋮ ⋱ ⋮ 1 l k l k ( l k − 1 ) ... ∏ i = 0 k − 2 ( l k − i ) ] {\displaystyle \det \left[{\begin{matrix}1&l_{1}&l_{1}(l_{1}-1)&\dots &\prod _{i=0}^{k-2}(l_{1}-i)\\1&l_{2}&l_{2}(l_{2}-1)&\dots &\prod _{i=0}^{k-2}(l_{2}-i)\\\vdots &\vdots &\vdots &\ddots &\vdots \\1&l_{k}&l_{k}(l_{k}-1)&\dots &\prod _{i=0}^{k-2}(l_{k}-i)\end{matrix}}\right]}
which column reduces to the Vandermonde determinant, and we obtain the formula
d λ = n ! l 1 ! l 2 ! ⋯ l k ! ∏ i < j ( l i − l j ) {\displaystyle d_{\lambda }={\frac {n!}{l_{1}!l_{2}!\cdots l_{k}!}}\prod _{i<j}(l_{i}-l_{j})}