Pattern avoidance and the Bruhat order
Copyright policy )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.
18 pages, 7 figures
- Massachusetts Institute of Technology United States
Bruhat order, pattern, Theoretical Computer Science, Combinatorics of partially ordered sets, Reflection and Coxeter groups (group-theoretic aspects), 05A05, FOS: Mathematics, Discrete Mathematics and Combinatorics, Mathematics - Combinatorics, Order ideal, Pattern, order ideal, Interval, Coxeter group, Boolean poset, Combinatorial aspects of groups and algebras, 05E15; 06A07; 05A05, 05E15, Computational Theory and Mathematics, 06A07, Reduced decomposition, reduced decomposition, Combinatorics (math.CO), interval
Bruhat order, pattern, Theoretical Computer Science, Combinatorics of partially ordered sets, Reflection and Coxeter groups (group-theoretic aspects), 05A05, FOS: Mathematics, Discrete Mathematics and Combinatorics, Mathematics - Combinatorics, Order ideal, Pattern, order ideal, Interval, Coxeter group, Boolean poset, Combinatorial aspects of groups and algebras, 05E15; 06A07; 05A05, 05E15, Computational Theory and Mathematics, 06A07, Reduced decomposition, reduced decomposition, Combinatorics (math.CO), interval
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