Pattern avoidance and the Bruhat order

The structure of order ideals in the Bruhat order for the symmetric group is elucidated via permutation patterns. 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.

[1]  Julian West,et al.  Generating trees and forbidden subsequences , 1996, Discret. Math..

[2]  Richard P. Stanley,et al.  f-vectors and h-vectors of simplicial posets , 1991 .

[3]  Jonas Sjöstrand,et al.  Bruhat intervals as rooks on skew Ferrers boards , 2006, J. Comb. Theory, Ser. A.

[4]  Axel Hultman,et al.  Bruhat intervals of length 4 in Weyl groups , 2003, J. Comb. Theory, Ser. A.

[5]  M. Dyer,et al.  On the "Bruhat graph" of a Coxeter system , 1991 .

[6]  Axel Hultman,et al.  Combinatorial complexes, Bruhat intervals and reflection distances , 2003 .

[7]  Francesco Brenti.  A combinatorial formula for Kazhdan-Lusztig polynomials , 1994 .

[8]  Charalambos A. Charalambides,et al.  Enumerative combinatorics , 2018, SIGA.

[9]  Bridget Eileen Tenner.  Reduced decompositions and permutation patterns , 2005 .

[10]  Richard P. Stanley,et al.  Generalized $H$-Vectors, Intersection Cohomology of Toric Varieties, and Related Results , 1987 .

[11]  N. J. A. Sloane,et al.  The On-Line Encyclopedia of Integer Sequences , 2003, Electron. J. Comb..

[12]  Rodica Simion,et al.  Restricted Permutations , 1985, Eur. J. Comb..

[13]  J. Jantzen.  Moduln mit einem höchsten Gewicht , 1979 .

[14]  C. Fan.  Schubert varieties and short braidedness , 1998 .

[15]  Serge Elnitsky,et al.  Rhombic Tilings of Polygons and Classes of Reduced Words in Coxeter Groups , 1997, J. Comb. Theory A.

[16]  Daya-Nand Verma,et al.  Möbius inversion for the Bruhat ordering on a Weyl group , 1971 .

[17]  V Lakshmibai,et al.  Criterion for smoothness of Schubert varieties in Sl(n)/B , 1990 .

[18]  A. Björner,et al.  Combinatorics of Coxeter Groups , 2005 .

[19]  R. Carter.  REFLECTION GROUPS AND COXETER GROUPS (Cambridge Studies in Advanced Mathematics 29) , 1991 .

[20]  Michelle L. Wachs,et al.  Bruhat Order of Coxeter Groups and Shellability , 1982 .

[21]  J. Humphreys.  Reflection groups and coxeter groups , 1990 .