Let $\lambda=(\lambda_1,...,\lambda_n)$ be a partition. My goal is to prove the following formula $$\sum\limits_{x\in\Lambda}(h(x)^2-c(x)^2)=|\lambda|^2,$$ where for $x=(i,j)\in\Lambda:=\{(i,j)\in\mathbb{Z}^2 :\ 1\le j\le \lambda_i \}$ we define $$h(x)=\lambda_i+\lambda_j'-i-j+1,$$ $$c(x)=j-i,$$ $$|\lambda|=\lambda_1+...+\lambda_n,$$ and $\lambda_j'=\mathrm{card}\{j: \ \lambda_j\ge i\}$.

I did it by a straightforward calculation (using induction), but I found an information that it is possible to do it in a purely combinatorial way. I will be very grateful for any hint how to see this formula without tedious calculations.


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