TITLE: Why this polynomial represents this figure?
QUESTION [6 upvotes]: I'm trying to understand why this figure is represented by this polynomial expression:

My goal is to prove directly why cartesian product of natural numbers is equinumerous to the natural numbers. I've already checked to the first elements of $(m,n)$ and indeed this polynomial seems to represent this figure.
I'm having problems to see why this figure is represented by this polynomial expression.
Thanks

REPLY [4 votes]: The $k^{th}$ downward diagonal runs from $(k,0)$ to $(0,k)$ and traverses $k+1$ points. So the indices of the points on the vertical axis are $0,1,3,6\dots=0,0+1,0+1+2,0+1+2+3\cdots$, i.e. the triangular numbers $\frac12k(k+1)$.
To get to a point $(m,n)$, you move $m$ steps along the diagonal $k=m+n$, so that the index of this point is $\frac12k(k+1)+m=\frac12(n+m)(n+m+1)+m$.