フック長の公式
Specializing the schur functions to the variables 1 , t , t 2 , t 3 , ⋯ {\displaystyle 1,t,t^{2},t^{3},\cdots } there is the formula
s λ ( 1 , t , t 2 , ⋯ ) = t n ( λ ) ∏ ( i , j ) ∈ Y ( λ ) ( 1 − t h λ ( i , j ) ) {\displaystyle s_{\lambda }(1,t,t^{2},\cdots )={\frac {t^{n\left(\lambda \right)}}{\prod _{(i,j)\in Y(\lambda )}(1-t^{h_{\lambda }(i,j)})}}}
The number n ( λ ) {\displaystyle n(\lambda )} is defined as
n ( λ ) = ∑ i ( i − 1 ) λ i = ∑ i ( λ i ′ 2 ) {\displaystyle n(\lambda )=\sum _{i}(i-1)\lambda _{i}=\sum _{i}{\binom {\lambda _{i}'}{2}}}
where λ ′ {\displaystyle \lambda '} is the conjugate partition
Multiplying these probabilities over all i {\displaystyle i} and j {\displaystyle j} gives the formula. This argument is fallacious since the events are not independent.
Knuth's argument is however correct for the enumeration of labellings on trees satisfying monotonicity properties analogous to those of a Young tableau. In this case, the 'hook' events in question are in fact independent events.
μの形のヤングタブローの数が d μ {\displaystyle d_{\mu }} であり、それらμに関して和をとることでその式を得る。
ここで、 e φ = 1 {\displaystyle e_{\phi }=1} であることにも注意されたい。それゆえ、次を示せば十分で、
e λ = ∑ μ ↑ λ e μ , {\displaystyle e_{\lambda }=\sum _{\mu \uparrow \lambda }e_{\mu },}
その結果、 d λ = e λ {\displaystyle d_{\lambda }=e_{\lambda }} は帰納的に証明される。
上の式の和は、次のように式を書き換えることによって、確率の和と捉えることができる。
∑ μ ↑ λ e μ e λ = 1. {\displaystyle \sum _{\mu \uparrow \lambda }{\frac {e_{\mu }}{e_{\lambda }}}=1.}
われわれはこのようにして、 e μ e λ {\displaystyle {\frac {e_{\mu }}{e_{\lambda }}}} が、ヤング図形μの集合上の確率測度を定めている。(ここで μ ↑ λ {\displaystyle \mu \uparrow \lambda } )
これはフックウォークと呼ばれるランダムウォークを定義を用いた構成法によるものである。
フックウォークは、ヤング図形λの上で、ひとつのコーナーセルを選択する。
フックウォークは次のルールによって定義される。
There are other formulas for d λ {\displaystyle d_{\lambda }} , but the hook-length formula is particularly simple and elegant. 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 formula expressing d λ {\displaystyle d_{\lambda }} in terms of a determinant. 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.
Despite the simplicity of the hook-length formula, the Frame–Robinson–Thrall proof is uninsightful and does not provide an intuitive argument as to why hooks appear in the formula.
The Catalan numbers are ubiquitous in enumerative combinatorics. Not surprisingly, they are also part of this story:
C n = f ( n , n ) {\displaystyle C_{n}=f^{(n,n)}}
Lets briefly mention why. When doing a Dyck path we may go up (U) or down (D). So for any Dyck path of length n {\displaystyle n} consider the tableaux of shape ( n , n ) {\displaystyle (n,n)} such that the first row is given by the numbers i {\displaystyle i} such that the i {\displaystyle i} -th step was up and in the second row given by the positions in which it goes down.
By the above considerations
p 1 ( n ) = ∑ λ ⊢ n s λ f λ {\displaystyle p_{1^{(n)}}=\sum _{\lambda \vdash n}s_{\lambda }f^{\lambda }}
So that
Δ ( x ) p 1 ( n ) = ∑ λ ⊢ n Δ ( x ) s λ f λ {\displaystyle \Delta (x)p_{1^{(n)}}=\sum _{\lambda \vdash n}\Delta (x)s_{\lambda }f^{\lambda }}
where Δ ( x ) = ∏ i < j ( x i − x j ) {\displaystyle \Delta (x)=\prod _{i<j}(x_{i}-x_{j})} is the Vandermonde determinant.
For a given partition λ = ( λ 1 , λ 2 , ⋯ , λ k ) {\displaystyle \lambda =(\lambda _{1},\lambda _{2},\cdots ,\lambda _{k})} define l i = λ i + k − i {\displaystyle l_{i}=\lambda _{i}+k-i} for i = 1 , 2 , ⋯ , k {\displaystyle i=1,2,\cdots ,k} .