Solution
= 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> are
$$
\begin{array}{rrrr}7&4&2&1\\4&1\\2\\1\end{array},
$$
with product $448$. The <hook-length formula> gives $f^\lambda=8!/448=90$. Therefore
$$
\boxed{\text{class size}=90^2=8100.}
$$