Let p be a permutation of {1,...,k} and w be a permutation of {1,...,n}. The permutation w contains a p-pattern if there exist letters in w which are in the same relative order as p. Otherwise, w avoids the pattern p.

The study of permutation patterns has blossomed in recent years, leading to a number of well known results and open problems. Recent work has uncovered significant links between patterns and both the reduced decompositions of a permutation and the structure of the Bruhat order. These connections yield a new definition of vexillary permutations, a description of permutations with Boolean order ideals, and a characterization of when the permutations avoiding a particular set of patterns is an order ideal. The boolean permutations themselves form a simplicial poset that is homotopy equivalent to a wedge of spheres. Additionally, results of Elnitsky about the rhombic tilings of 2n-gons are expanded, and lead to, for example, the fact that a particular class of 2n-gons can be tiled by convex centrally symmetric 2k-gons iff k is 2 or n.