The University of Michigan Combinatorics Seminar
The structure of order ideals in the Bruhat order for the symmetric group is elucidated via permutation patterns. We characterize the permutations with boolean principal order ideals and show that they form an order ideal which is a simplicial poset. We generalize this characterization to describe a larger class of permutations whose principal order ideals are related to boolean posets. It is determined when the set of permutations avoiding a particular set of patterns is an order ideal, and the rank generating functions of these ideals are computed. Finally, we will discuss the Bruhat order in types B and D, and characterize the elements with boolean principal order ideals, as well as enumerate them by length.