Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 160 / 1 / d / i / Solution

In a Young-diagram hook, the cell $(i,j)$ has ${\lambda }_i-j$ cells to its right and ${\lambda }_j^{\prime }-i$ below it. Summing the arm lengths row by row and the leg lengths column by column givesAdding the content $c_{i,j}=j-i$ turns the summand on the right into $2j-1$. Since ${\sum }_{j=1}^m(2j-1)=m^2$, summing each row proves