Define $\delta_N$ as
$\delta_N = \big|\{(n_x,n_y)|n_x+n_y+1=N\}\big|$
where $n_x,n_y \in \mathbb{N}$ and $N \in \mathbb{N}^+$. I want to prove that $\delta_N = N$ for some choice of fixed $N$. The result of this problem is obvious to me, but I don't know how to formalize my intuition for it.
$n_x$ can be any value from $0$ to $N-1$, of which there are $N$. For each of these $n_y$ is fixed, so there are $N$ ordered pairs in your set.