Journal of Algebraic Combinatorics

, Volume 22, Issue 4, pp 383–409 | Cite as

Decreasing Subsequences in Permutations and Wilf Equivalence for Involutions

  • Mireille Bousquet-MélouEmail author
  • Einar Steingrímsson


In a recent paper, Backelin, West and Xin describe a map φ* that recursively replaces all occurrences of the pattern k... 21 in a permutation σ by occurrences of the pattern (k−1)... 21 k. The resulting permutation φ*(σ) contains no decreasing subsequence of length k. We prove that, rather unexpectedly, the map φ* commutes with taking the inverse of a permutation.

In the BWX paper, the definition of φ* is actually extended to full rook placements on a Ferrers board (the permutations correspond to square boards), and the construction of the map φ* is the key step in proving the following result. Let T be a set of patterns starting with the prefix 12... k. Let T′ be the set of patterns obtained by replacing this prefix by k... 21 in every pattern of T. Then for all n, the number of permutations of the symmetric group \({\cal S}\) n that avoid T equals the number of permutations of \({\cal S}\) n that avoid T′.

Our commutation result, generalized to Ferrers boards, implies that the number of involutions of \({\cal S}\) n that avoid T is equal to the number of involutions of \({\cal S}\) n avoiding T′, as recently conjectured by Jaggard.


pattern avoiding permutations Wilf equivalence involutions decreasing subsequences prefix exchange 


  1. 1.
    E. Babson and E. Steingrímsson, “Generalized permutation patterns and a classification of the Mahonian statistics,” Sém. Lothar. Combin., 44:Art. B44b, (2000), 18 pp. (electronic).Google Scholar
  2. 2.
    E. Babson and J. West, “The permutations 123p4 ... pm and 321p4 ... pm are Wilf-equivalent,” Graphs Combin. 16(4) (2000), 373–380.MathSciNetGoogle Scholar
  3. 3.
    J. Backelin, J. West, and G. Xin, “Wilf-equivalence for singleton classes,” in Proceeedings of the 13th Conference on Formal Power Series and Algebraic Combinatorics, H. Barcelo and V. Welker (Eds.), Arizona State University, 2001, pp. 29–38 (To appear in Adv. in Appl. Math.).Google Scholar
  4. 4.
    E. Barcucci, A. Del Lungo, E. Pergola, and R. Pinzani, “Some permutations with forbidden subsequences and their inversion number,” Discrete Math. 234(1–3) (2001), 1–15.MathSciNetGoogle Scholar
  5. 5.
    M. Bóna, “Exact enumeration of 1342-avoiding permutations: A close link with labeled trees and planar maps,” J. Combin. Theory Ser. A 80(2) (1997), 257–272.CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    M. Bousquet-Mélou, “Four classes of pattern-avoiding permutations under one roof: Generating trees with two labels,” Electron. J. Combin. 9(2), (2003).Google Scholar
  7. 7.
    A. Claesson and T. Mansour, “Enumerating permutations avoiding a pair of Babson-Steingrí msson patterns,” Ars Combinatoria, to appear.Google Scholar
  8. 8.
    I. Gessel, “Symmetric functions and P-recursiveness,” J. Combin. Theory Ser. A 53(2) (1990), 257–285.CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    I. Gessel, J. Weinstein, and H.S. Wilf, “Lattice walks in Zd and permutations with no long ascending subsequences,” Electron. J. Combin. 5(1) (1998), 11.MathSciNetGoogle Scholar
  10. 10.
    D. Gouyou-Beauchamps, “Standard Young tableaux of height 4 and 5,” European J. Combin. 10(1) (1989), 69–82.zbMATHMathSciNetGoogle Scholar
  11. 11.
    O. Guibert, “Combinatoire des permutations à motifs exclus, en liaison avec mots, cartes planaires et tableaux de Young,” Ph.D. thesis, LaBRI, Université Bordeaux 1, 1995.Google Scholar
  12. 12.
    O. Guibert, E. Pergola, and R. Pinzani, “Vexillary involutions are enumerated by Motzkin numbers,” Ann. Comb. 5(2) (2001), 153–174.MathSciNetGoogle Scholar
  13. 13.
    G. Huet, “Confluent reductions: Abstract properties and applications to term rewriting systems,” J. Assoc. Comput. Mach. 27(4) (1980) 797–821.zbMATHMathSciNetGoogle Scholar
  14. 14.
    A.D. Jaggard, “Prefix exchanging and pattern avoidance for involutions,” Electron. J. Combin. 9(2)(2003), 16.MathSciNetGoogle Scholar
  15. 15.
    S. Kitaev and T. Mansour, “A survey of certain pattern problems,” Preprint, 2004.Google Scholar
  16. 16.
    C. Krattenthaler, “Permutations with restricted patterns and Dyck paths,” Adv. in Appl. Math. 27(2/3) (2001), 510–530.zbMATHMathSciNetGoogle Scholar
  17. 17.
    A. Regev, “Asymptotic values for degrees associated with strips of Young diagrams,” Adv. in Math. 41(2) (1981) 115–136.CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    C. Schensted, “Longest increasing and decreasing subsequences,” Canad. J. Math. 13 (1961), 179–191.zbMATHMathSciNetGoogle Scholar
  19. 19.
    R. Simion and F.W. Schmidt, “Restricted permutations,” European J. Combin. 6(4) (1985) 383–406.MathSciNetGoogle Scholar
  20. 20.
    Z.E. Stankova, “Forbidden subsequences,” Discrete Math. 132(1/3) (1994), 291–316.zbMATHMathSciNetGoogle Scholar
  21. 21.
    Z.E. Stankova, “Classification of forbidden subsequences of length 4,” European J. Combin. 17(5) (1996), 501–517.CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Z.E. Stankova and J. West, “A new class of Wilf-equivalent permutations,” J. Algebraic Combin. 15(3) (2002), 271–290.CrossRefMathSciNetGoogle Scholar
  23. 23.
    G. Viennot, “Une forme géométrique de la correspondance de Robinson-Schensted,” in Combinatoire et représentation du groupe symétrique (Actes Table Ronde CNRS, Univ. Louis-Pasteur, Strasbourg, 1976). Lecture Notes in Math., vol. 579. Springer, Berlin, 1977, pp. 29–58.Google Scholar
  24. 24.
    J. West, “Permutations with Forbidden Subsequences, and Stack-Sortable Permutations,” Ph.D. thesis, MIT, 1990.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  • Mireille Bousquet-Mélou
    • 1
    Email author
  • Einar Steingrímsson
    • 1
  1. 1.CNRSLaBRI, Université Bordeaux 1Talence CedexFrance

Personalised recommendations