Past exam of the Lie algebras and representation theory of symmetric groups course of the University of Oxford / 2007 / B / 6 / d / Solution

The Robinson-Schensted-Knuth correspondence is a bijection between permutations and ordered pairs of standard Young tableaux of the same shape. Hence the class with shape $\lambda$ has size $(f^{\lambda }{)}^2$. For $\lambda =(4,2,1,1)$, the hook lengths arewith product $448$. The hook-length formula gives $f^{\lambda }=8!/448=90$. Therefore