Let $\lambda$ be an integer partition: $\lambda=(\lambda_1\geq\lambda_2\geq\dots\geq0)$. Denote its conjugate partition by $\lambda'$. For example, if $\lambda=(4,3,1)$ then $\lambda'=(3,2,2,1)$.

Recall also the notation for the content of a cell $u=(i,j)$ in a partition is $c_u=j−i$.

I have made the following observation which seems interesting enough to ask here.

QUESTION.Is this true? It might even be known. Is it? Any reference? $$\sum_{i\geq1}\lambda_i^2=\sum_{u\in\lambda}(h_u+c_u).$$

**REMARK 1.** The above identity implies
$$\sum_{i\geq1}(\lambda_i^2+(\lambda_i')^2)=2\sum_{u\in\lambda}h_u.$$

**REMARK 2.** It also implies that
$$\sum_{\lambda\vdash n}\sum_{i\geq1}\lambda_i^2=\sum_{\lambda\vdash n}\sum_{u\in\lambda}h_u.$$