Boolean elements in the Bruhat order on twisted involutions

Delong Meng
2012 Involve. A Journal of Mathematics  
We prove that a permutation in the Bruhat order on twisted involutions is Boolean if and only if it avoids the following patterns: 4321, and 36712845. This result answers a question proposed by Hultman and Vorwerk. Our technique provides an application of the pictorial representation of the Bruhat order given by Incitti. We first define some requisite terms in the problem statement (see [Björner and Brenti 2005] for background reading). Definition. Let denote the number of inversions of w. The
more » ... ruhat order of the symmetric group, denoted by Br (S n ), is a partial order on S n defined as follows: w covers w if and only if l(w) = l(w ) + 1 and w is obtained from w by a transposition of w (i) and w ( j) for some 1 ≤ i, j ≤ n. MSC2010: 05E15. 339 340 DELONG MENG Definition. The Bruhat order on twist involutions, denoted by Tw(S n ), is the poset on twisted involutions defined by u ≤ v in Tw(S n ) if and only if u ≤ v in Br (S n ). Let Q be a poset. The principal order ideal of w ∈ Q, denoted by P Q (w) (or P(w) when the context is clear), is the subposet of Q induced by the set of elements less than or equal to w. The Boolean poset B k is the poset on the subsets of {1, 2, . . . , k} partially ordered by inclusion. A twisted involution w is said to be Boolean if its principal order ideal in Tw(S n ) is isomorphic to a Boolean poset.
doi:10.2140/involve.2012.5.339 fatcat:p2dqw7nd2jbohc6jy74d6rpdsq