Boolean elements in the Bruhat order

We show that a Weyl group element is boolean if and only if it avoids a set of Billey-Postnikov patterns, which we describe explicitly. Our proof is based on analysis of inversion sets, and it is in large part type-uniform. We also introduce the notion of linear pattern avoidance, and show that boolean elements are characterized by avoiding just \ (3\) linear patterns in types \ (A₂\), \ (A₃\), and \ (D₄\), respectively. We also consider the more general case of \ (k\) -boolean Weyl group elements. We say that a Weyl group element \ (w\) is \ (k\) -boolean if every reduced expression for \ (w\) contains at most \ (k\) copies of each generator. We show that the \ (2\) -boolean elements of the symmetric group are characterized by avoiding the patterns \ (3421, 4312, 4321, \) and \ (456123\), and obtain their generating function. Mathematics Subject Classifications: 05A05, 20F55Keywords: Boolean permutations, Bruhat orders, Billey-Postnikov patterns, Weyl groups