Countability of the Rationals: A Pictoral Diagram

Countability of the Rationals: A Pictoral Diagram

I am trying to reproduce the diagram below:

I'm wondering what I should use to generate this. I can make the matrix,
that's simple, but the arrows are what's hard... It would be interesting
to use this example, where the S_{ij}'s are circled nodes, and then some
nodes just aren't connected, and the ellipses and the S_i:'s are nodes,
but they wouldn't be circled. I'm looking here, and I think this would
work too, but I don't yet know how to interpret the (m-i-j) symbolism, or
how to connect arrows to nodes.



So far, I've got the following:

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{arrows}
\begin{document}
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=3cm,
thick,main
node/.style={circle,fill=blue!20,draw,font=\sffamily\Large\bfseries}]
\node[main node] (1) {$S_{11}$};
\node[main node] (2) [right of=1] {$S_{12}$};
\node[main node] (3) [below of=1] {$S_{21}$};
\node[main node] (4) [below of=3] {$S_{31}$};
\node[main node] (5) [below of=2] {$S_{22}$};
\node[main node] (6) [right of=2] {$S_{13}$};
\node[main node] (7) [right of=6] {$S_{14}$};
\node[main node] (8) [below of=6] {$S_{23}$};
\node[main node] (9) [below of=5] {$S_{32}$};
\node[main node] (10) [below of=4] {$S_{41}$};
\node[main node] (11) [right of=10] {$S_{42}$};
\node[main node] (12) [right of=11] {$S_{43}$};
\node[main node] (13) [right of=12] {$S_{44}$};
\node[main node] (14) [right of=9] {$S_{33}$};
\node[main node] (15) [right of=14] {$S_{34}$};
\node[main node] (16) [right of=8] {$S_{24}$};
\path[every node/.style={font=\sffamily\small}]
(1) edge node [right] {} (2)
(2) edge node [right] {} (3)
(3) edge node [right] {} (4)
(4) edge node [right] {} (5)
(5) edge node [right] {} (6)
(6) edge node [right] {} (7)
(7) edge node [right] {} (8)
(8) edge node [right] {} (9)
(9) edge node [right] {} (10);
\end{tikzpicture}
\end{document}