  The structure of order ideals in the Bruhat order for the symmetric group is
elucidated via permutation patterns. A method for determining non-isomorphic
principal order ideals is described and applied for small lengths. The
permutations with boolean principal order ideals are characterized. These form
an order ideal which is a simplicial poset, and its rank generating function is
computed. Moreover, the permutations whose principal order ideals have a form
related to boolean posets are also completely described. 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,
the Bruhat order in types B and D is studied, and the elements with boolean
principal order ideals are characterized and enumerated by length.
