Journal of Algebraic Combinatorics

, Volume 24, Issue 3, pp 263–284 | Cite as

Reduced decompositions and permutation patterns



Billey, Jockusch, and Stanley characterized 321-avoiding permutations by a property of their reduced decompositions. This paper generalizes that result with a detailed study of permutations via their reduced decompositions and the notion of pattern containment. These techniques are used to prove a new characterization of vexillary permutations in terms of their principal dual order ideals in a particular poset. Additionally, the combined frameworks yield several new results about the commutation classes of a permutation. In particular, these describe structural aspects of the corresponding graph of the classes and the zonotopal tilings of a polygon defined by Elnitsky that is associated with the permutation.


Reduced decomposition Permutation pattern Vexillary permutation Zonotopal tiling Freely braided permutation 


  1. 1.
    S.C. Billey, W. Jockusch, and R.P. Stanley, “Some combinatorial properties of Schubert polynomials,” J. Alg. Combin. 2 (1993), 345–374.CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    A. Björner, “private communication,” April 2005.Google Scholar
  3. 3.
    A. Björner and F. Brenti, Combinatorics of Coxeter Groups, Graduate Texts in Mathematics 231, Springer, New York, 2005.Google Scholar
  4. 4.
    S. Elnitsky, “Rhombic tilings of polygons and classes of reduced words in Coxeter groups,” J. Combin. Theory, Ser. A 77 (1997), 193–221.CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    R.M. Green and J. Losonczy, “Freely braided elements in Coxeter groups,” Ann. Comb. 6 (2002), 337–348.MathSciNetMATHGoogle Scholar
  6. 6.
    R.M. Green and J. Losonczy, “Freely braided elements in Coxeter groups, II,” Adv. in Appl. Math. 33 (2004), 26–39.Google Scholar
  7. 7.
    L.M. Kelly and R. Rottenberg, “Simple points in pseudoline arrangements,” Pacific J. Math. 40 (1972), 617–622.MathSciNetMATHGoogle Scholar
  8. 8.
    A. Lascoux and M.P. Schützenberger, “Polynômes de Schubert,” C. R. Acad. Sci. Paris, Série I 294 (1982), 447–450.MATHGoogle Scholar
  9. 9.
    I.G. Macdonald, Notes on Schubert Polynomials, Laboratoire de combinatoire et d’;informatique mathématique (LACIM), Université du Québec à Montréal, Montreal, 1991.Google Scholar
  10. 10.
    T. Mansour, The enumeration of permutations whose posets have a maximal element, preprint.Google Scholar
  11. 11.
    T. Mansour, On an open problem of Green and Losonczy: Exact enumeration of freely braided permutations, Discrete Math. Theor. Comput. Sci. 6 (2004), 461–470.MathSciNetMATHGoogle Scholar
  12. 12.
    D.V. Pasechnik and B. Shapiro, “In search of higher permutahedra,” preprint.Google Scholar
  13. 13.
    V. Reiner, “Note on the expected number of Yang-Baxter moves applicable to reduced decompositions,” Europ. J. Combin. 26 (2005), 1019–1021.CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    B. Shapiro, M. Shapiro, and A. Vainshtein, “Connected components in the intersection of two open opposite Schubert cells in SL n /B,” Intern. Math. Res. Notices, no. 10 (1997), 469–493.Google Scholar
  15. 15.
    R. Simion and F.W. Schmidt, “Restricted permutations,” European J. Combin. 6 (1985), 383-406.MathSciNetMATHGoogle Scholar
  16. 16.
    R.P. Stanley, Enumerative Combinatorics, vol. 2, Cambridge Studies in Advanced Mathematics, no. 62, Cambridge University Press, Cambridge, 1999.Google Scholar
  17. 17.
    R.P. Stanley, “On the number of reduced decompositions of elements of Coxeter groups,” European J. Combin. 5 (1984), 359–372.MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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