Annals of Combinatorics

, Volume 20, Issue 3, pp 453–485 | Cite as

A Generating Tree Approach to k-Nonnesting Partitions and Permutations

  • Sophie Burrill
  • Sergi Elizalde
  • Marni Mishna
  • Lily Yen


We describe a generating tree approach to the enumeration and exhaustive generation of k-nonnesting set partitions and permutations. Unlike previous work in the literature which uses the connections of these objects to Young tableaux and restricted lattice walks, our approach deals directly with partition and permutation diagrams. We provide explicit functional equations for the generating functions, with k as a parameter. Key to the solution is a superset of diagrams that permit semi-arcs. Many of the resulting counting sequences also count other well-known objects, such as Baxter permutations, and Young tableaux of bounded height.

Mathematics Subject Classification

05A15 05A18 


enumeration generating tree partition permutation nonnesting 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Banderier C., Bousquet-Mélou M., Denise A., Flajolet P., Gardy D., Gouyou-Beauchamps D.: Generating functions for generating trees. Discrete Math. 246(1-3), 29–55 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Barcucci E., Del Lungo A., Pergola E., Pinzani R.: ECO: a methodology for the enumeration of combinatorial objects. J. Differ. Equations Appl. 5(4-5), 435–490 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bonichon, N., Bousquet-Mélou, M., Fusy, É.: Baxter permutations and plane bipolar orientations. Sém. Lothar. Combin. 61A, Art. B61Ah (2009/2011)Google Scholar
  4. 4.
    Bousquet-Mélou M.: Counting permutations with no long monotone subsequence via generating trees and the kernel method. J. Algebraic Combin. 33(4), 571–608 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bousquet-Mélou, M., Xin, G.: On partitions avoiding 3-crossings. Sém. Lothar. Combin. 54, Art. B54e (2005/07)Google Scholar
  6. 6.
    Bouvel M., Guibert O.: Refined enumeration of permutations sorted with two stacks and a D 8-symmetry. Ann. Combin. 18(2), 199–232 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Boyce W.M.: Baxter permutations and functional composition. Houston J. Math. 7(2), 175–189 (1981)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Burrill, S.R.: A generating tree approach to k-nonnesting arc diagrams. PhD thesis, Simon Fraser University, Burnaby (2014)Google Scholar
  9. 9.
    Burrill, S., Melczer, S., Mishna, M.: A Baxter class of a different kind, and other bijective results using tableau sequences ending with a row shape. In: Proceedings of the 27th International Conference on Formal Power Series and Algebraic Combinatorics, (FPSAC’15), Discrete Math. Theor. Comput. Sci. Proc., AS, pp. 369–380. Assoc. Discrete Math. Theor. Comput. Sci., Nancy, (2015)Google Scholar
  10. 10.
    Burrill, S., Mishna, M., Post, J.: On k-crossings and k-nestings of permutations. In: Proceedings of 22nd International Conference on Formal Power Series and Algebraic Combinatorics, Discrete Math. Theor. Comput. Sci. Proc., AS, pp. 593–600. Assoc. Discrete Math. Theor. Comput. Sci., Nancy (2010)Google Scholar
  11. 11.
    Burrill, S., Yen, L.: Constructing Skolem sequences via generating trees. (2013) arXiv:1301.6424
  12. 12.
    Chen W.Y.C., Deng E.Y.P., Du R.R.X., Stanley R.P., Yan C.H.: Crossings and nestings of matchings and partitions. Trans. Amer. Math. Soc. 359(4), 1555–1575 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chen W.Y.C., Han H.S.W., Reidys C.M.: Random k-noncrossing RNA structures. Proc. Natl. Acad. Sci. USA 106(52), 22061–22066 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Chen W.Y.C., Qin J., Reidys C.M.: Crossings and nestings in tangled diagrams. Electron. J. Combin. 15(1), #R86 (2008)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Corteel S.: Crossings and alignments of permutations. Adv. Appl. Math. 38(2), 149–163 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    de Mier, A.: On the symmetry of the distribution of k-crossings and k-nestings in graphs. Electron. J. Combin. 13(1), #N21 (2006)Google Scholar
  17. 17.
    de Mier A.: k-noncrossing and k-nonnesting graphs and fillings of Ferrers diagrams. Combinatorica 27(6), 699–720 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Dershowitz N., Zaks S.: Ordered trees and noncrossing partitions. Discrete Math. 62(2), 215–218 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Elizalde S.: The X-class and almost-increasing permutations. Ann. Combin. 15(1), 51–68 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Felsner S., Fusy É., Noy M., Orden D.: Bijections for Baxter families and related objects. J. Combin. Theory Ser. A 118(3), 993–1020 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Flajolet P., Sedgewick R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)CrossRefzbMATHGoogle Scholar
  22. 22.
    Jin E.Y., Qin J., Reidys C.M.: Combinatorics of RNA structures with pseudoknots. Bull. Math. Biol. 70(1), 45–67 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kim J.S.: New interpretations for noncrossing partitions of classical types. J. Combin. Theory Ser. A 118(4), 1168–1189 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Krattenthaler, C.: Bijections between oscillating tableaux and (semi)standard tableaux via growth diagrams. arXiv:1412.5646 (2014)
  25. 25.
    Ma G., Reidys C.M.: Canonical RNA pseudoknot structures. J. Comput. Biol. 15(10), 1257–1273 (2008)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Marberg, E.: Crossings and nestings in colored set partitions. Electron. J. Combin. 20(4), #P6 (2013)Google Scholar
  27. 27.
    Mishna, M., Yen, L.: Set partitions with no m-nesting. In: Kotsireas, I.S., Zima, E.V. (edsAdvances in Combinatorics, pp. 249–258. Springer, Heidelberg (2013)Google Scholar
  28. 28.
    Noonan J., Zeilberger D.: The enumeration of permutations with a prescribed number of “forbidden” patterns. Adv. Appl. Math. 17(4), 381–407 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    OEIS Foundation Inc.: The On-Line Encyclopedia of Integer Sequences. Published electronically at (2011)
  30. 30.
    Saule C., Régnier M., Steyaert J.-M., Denise A.: Counting RNA pseudoknotted structures. J. Comput. Biol. 18(10), 1339–1351 (2011)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Simion R.: Noncrossing partitions. Discrete Math. 217(1-3), 367–409 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    West J.: Generating trees and the Catalan and Schrö.der numbers. Discrete Math. 146(1-3), 247–262 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Yen, L.: Crossings and nestings for arc-coloured permutations. In: Proceedings of 25th International Conference on Formal Power Series and Algebraic Combinatorics, Discrete Math. Theor. Comput. Sci. Proc., AS, pp. 743–754. Assoc. Discrete Math. Theor. Comput. Sci., Nancy (2013)Google Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  • Sophie Burrill
    • 1
  • Sergi Elizalde
    • 2
  • Marni Mishna
    • 1
  • Lily Yen
    • 1
    • 3
  1. 1.Department of MathematicsSimon Fraser UniversityBurnabyCanada
  2. 2.Department of MathematicsDartmouth CollegeHanoverUSA
  3. 3.Department of Mathematics and StatisticsCapilano UniversityNorth VancouverCanada

Personalised recommendations