69 research outputs found
Intervals and factors in the Bruhat order
In this paper we study those generic intervals in the Bruhat order of the
symmetric group that are isomorphic to the principal order ideal of a
permutation w, and consider when the minimum and maximum elements of those
intervals are related by a certain property of their reduced words. We show
that the property does not hold when w is a decomposable permutation, and that
the property always holds when w is the longest permutation.Comment: to appear in DMTC
Pattern Avoidance and the Bruhat Order
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.Comment: 18 pages, 7 figure
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