フック長の公式
It would be violent to plug in 1, but the right hand side is a continuous function inside the unit disk and a polynomial is continuous everywhere so at least we can say
P ( 1 ) = lim x → 1 ∏ k = 1 n ( 1 − x k ) ∏ ( i , j ) ∈ λ ( 1 − x h ( i , j ) ) . {\displaystyle P(1)=\lim _{x\to 1}{\frac {\prod _{k=1}^{n}(1-x^{k})}{\prod _{(i,j)\in \lambda }(1-x^{h_{(i,j)}})}}.}
Finally, applying L'Hopital's rule n {\displaystyle n} times yields the hook length formula
P ( 1 ) = n ! ∏ ( i , j ) ∈ λ h ( i , j ) . {\displaystyle P(1)={\frac {n!}{\prod _{(i,j)\in \lambda }h_{(i,j)}}}.}