フック長の公式
See Schur function. Since the vector ( l 1 , l 2 , ⋯ , l k ) {\displaystyle (l_{1},l_{2},\cdots ,l_{k})} is different for each partition, this means that the coefficient of x 1 l 1 ⋯ x k l k {\displaystyle x_{1}^{l_{1}}\cdots x_{k}^{l_{k}}} in Δ ( x ) p 1 ( n ) {\displaystyle \Delta (x)p_{1^{(n)}}} , denoted [ Δ ( x ) p 1 ( n ) ] l 1 , ⋯ , l k {\displaystyle \left[\Delta (x)p_{1^{(n)}}\right]_{l_{1},\cdots ,l_{k}}} , is equal to f λ {\displaystyle f^{\lambda }} . This is known as the Frobenius Character Formula, which gives one of the earliest proofs.
The hook length formula also has important applications to the analysis of longest increasing subsequences in random permutations. If σ n {\displaystyle \sigma _{n}} denotes a uniformly random permutation of order n {\displaystyle n} , L ( σ n ) {\displaystyle L(\sigma _{n})} denotes the maximal length of an increasing subsequence of σ n {\displaystyle \sigma _{n}} , and l n {\displaystyle \ell _{n}} denotes the expected (average) value of L ( σ n ) {\displaystyle L(\sigma _{n})} , Anatoly Vershik and Sergei Kerov and independently Benjamin F. Logan and Lawrence A.