Bruhat graphs and pattern avoidance

Given a permutation (thought of as a sequence of integers), one can construct a graph (in the sense of vertices and edges) called the Bruhat graph.  I will give a characterization of permutations whose Bruhat graphs can be drawn without edges crossing in terms of the elementary notion of pattern avoidance.  The motivation for this theorem comes from theoretical questions about pattern avoidance further motivated by questions in representation theory.  Though the motivation involves deep mathematics, the new theorems will be accessible to most middle schoolers and were proven by Chris Conklin, an undergraduate student who worked with me in the summer of 2010.  Undergraduate students are encouraged to attend.