In
a Young-diagram hook, the
cell (i,j) has
λi−j cells to its right and
λj′−i below it. Summing the arm
lengths row by row and the leg
lengths column by column gives
∑(i,j)∈λhi,j=∑(i,j)∈λ((j−1)+(i−1)+1)=∑(i,j)∈λ(i+j−1).
Adding the content
ci,j=j−i turns the summand on the right into
2j−1. Since
∑j=1m(2j−1)=m2, summing each row proves
∑(i,j)∈λ(hi,j+ci,j)=∑iλi2.
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