# A link between hooks, contents and parts of a partition

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 $$\sum_{\lambda\vdash n}\sum_{i\geq1}\lambda_i^2=\sum_{\lambda\vdash n}\sum_{u\in\lambda}h_u.$$

Another approach is to notice that $$\sum_{u\in\lambda}h_u=\sum_{u}d_u$$ where $$d_u=i+j-1$$ for $$u=(i,j)$$.
Proof: The easiest way to see this is that both sides count the number of pairs $$\{(i_1,j_1),(i_2,j_2)\}\in\lambda$$ such that either $$i_1=i_2$$ and $$j_1\le j_2$$ or $$j_1=j_2$$ and $$i_1\le i_2$$.
Using this we have $$\sum_{u\in \lambda}(h_u+c_u)=\sum_{u\in \lambda} (c_u+d_u)=\sum_{(i,j)\in \lambda} (2j-1)=\sum_{i\geq 1}\sum_{j=1}^{\lambda_i}(2j-1)=\sum_{i\geq1} \lambda_i^2$$ which gives us the equality we wanted.
You can prove this by inductively adding boxes to the outside corners of $$\lambda$$. It is true for $$\lambda=\varnothing$$. And when you add box $$(i,j)$$ as an outside corner, you change the sum $$\sum_{i \geq 1} \lambda_i^2$$ only in the $$i$$-term where you increase it by $$2j-1$$; while the hooks increase by one for the $$i-1+j-1$$ boxes directly to the left or directly above $$(i,j)$$, and the new hook length of $$(i,j)$$ plus the new content of $$(i,j)$$ gives you a total increase to the RHS of $$(i-1+j-1)+(1+j-i)=2j-1$$.