フック長の公式
(1) | λ | {\displaystyle |\lambda |} 個のセルのなかから一様ランダムにひとつのセルを選び、そこからランダムウォークを始める。
(2) 現在の ( i , j ) {\displaystyle (i,j)} セルの次のセルは、一様ランダムにフック H λ ( i , j ) ∖ { ( i , j ) } {\displaystyle H_{\lambda }(i,j)\setminus \{(i,j)\}} から選ぶ。
(3) 一つのコーナーに到達するまでこれを続け、そのコーナーセルを c {\displaystyle {\textbf {c}}} とする。
命題: λに属するすべてのコーナーセル ( a , b ) {\displaystyle (a,b)} に対して、次が成り立つ。
P ( c = ( a , b ) ) = e μ e λ , {\displaystyle \mathbb {P} \left({\textbf {c}}=(a,b)\right)={\frac {e_{\mu }}{e_{\lambda }}},}
ここで、 μ = λ ∖ { ( a , b ) } {\displaystyle \mu =\lambda \setminus \{(a,b)\}} .
この命題を得れば、すべての c = ( a , b ) {\displaystyle {\textbf {c}}=(a,b)} に関して和をとることによって、 ∑ μ ↑ λ e μ e λ = 1 {\displaystyle \sum _{\mu \uparrow \lambda }{\frac {e_{\mu }}{e_{\lambda }}}=1} を得る。
The hook-length formula is of great importance in the representation theory of the symmetric group S n {\displaystyle S_{n}} , where the number d λ {\displaystyle d_{\lambda }} is known to be equal to the dimension of the complex irreducible representation V λ {\displaystyle V_{\lambda }} associated to λ {\displaystyle \lambda } , and is frequently denoted by dim  V λ {\displaystyle \dim V_{\lambda }} , dim  λ {\displaystyle \dim \lambda } or f λ {\displaystyle f^{\lambda }} .
There is a generalization of this formula for skew shapes,
s λ / μ ( 1 , t , t 2 , ⋯ ) = ∑ S ∈ E ( λ / μ ) ∏ ( i , j ) ∈ λ ∖ S t λ j ′ − i 1 − t h ( i , j ) {\displaystyle s_{\lambda /\mu }(1,t,t^{2},\cdots )=\sum _{S\in E(\lambda /\mu )}\prod _{(i,j)\in \lambda \setminus S}{\frac {t^{\lambda _{j}'-i}}{1-t^{h(i,j)}}}}
where the sum is taken over excited diagrams of shape λ {\displaystyle \lambda } and boxes distributed according to μ {\displaystyle \mu } .
S. Franzblau and D. Zeilberger in 1982. D. Zeilberger also converted the Greene–Nijenhuis–Wilf hook walk proof into a bijective proof in 1984. A simpler direct bijective proof was announced by Igor Pak and Alexander V. Stoyanovskii in 1992, and its complete proof was presented by the pair and Jean-Christophe Novelli in 1997.
Meanwhile, the hook-length formula has been generalized in several ways. R. M. Thrall found the analogue to the hook-length formula for shifted Young Tableaux in 1952. B. E. Sagan gave a shifted hook walk proof for the hook-length formula for shifted Young tableaux in 1980. B. E. Sagan and Y. N. Yeh proved the hook-length formula for binary trees using the hook walk in 1989.
Shepp showed that when n {\displaystyle n} is large, l n {\displaystyle \ell _{n}} is approximately equal to 2 n {\displaystyle 2{\sqrt {n}}} . This answers a question originally posed by Stanislaw Ulam. The proof is based on translating the question via the Robinson–Schensted correspondence to a problem about the limiting shape of a random Young tableau chosen according to Plancherel measure. Since the definition of Plancherel measure involves the quantity d λ {\displaystyle d_{\lambda }} , the hook length formula can then be used to perform an asymptotic analysis of the limit shape and thereby also answer the original question.
{\displaystyle \sum _{w\in S_{n}}\operatorname {sgn}(w){\frac {n!}{(l_{1}-w(1)+1)!(l_{2}-w(2)+1)!\cdots (l_{k}-w(k)+1)!}}}
which can be written as
n ! l 1 ! l 2 ! ⋯ l k ! ∑ w ∈ S n sgn  ( w ) [ ( l 1 ) ( l 1 − 1 ) ⋯ ( l 1 − w ( 1 ) + 2 ) ] [ ( l 2 ) ( l 2 − 1 ) ⋯ ( l 2 − w ( 2 ) + 2 ) ] [ ( l k ) ( l k − 1 ) ⋯ ( l k − w ( k ) + 2 ) ] {\displaystyle {\frac {n!}{l_{1}!l_{2}!\cdots l_{k}!}}\sum _{w\in S_{n}}\operatorname {sgn}(w)\left[(l_{1})(l_{1}-1)\cdots (l_{1}-w(1)+2)\right]\left[(l_{2})(l_{2}-1)\cdots (l_{2}-w(2)+2)\right]\left[(l_{k})(l_{k}-1)\cdots (l_{k}-w(k)+2)\right]}
The latter sum is equal to the following determinant
det [ 1 l 1 l 1 ( l 1 − 1 ) ...