This picture by Tilman Piesk shows the 14 Dyck words of length 8. A Dyck word is a balanced string of left and parentheses. In the picture, a left parenthesis is shown as upward-slanting line segment, and a right parenthesis as a downward-slanting one.
Suppose you have a set with 5 elements. There are 10 ways to choose a 2-element subset. Form a graph with these 10 choices as vertices, and with two vertices connected by an edge precisely when the corresponding subsets are disjoint. You get the graph shown here, called the Petersen graph.