Skip to main content
SpringerLink
Log in

Refined Enumeration of Noncrossing Chains and Hook Formulas

  • Published:
Annals of Combinatorics Aims and scope Submit manuscript

Abstract

In the combinatorics of finite Coxeter groups, there is a simple formula giving the number of maximal chains of noncrossing partitions. It is a reinterpretation of a result by Deligne which is due to Chapoton, and the goal of this article is to refine the formula. First, we prove a one-parameter generalization, by considering the enumeration of noncrossing chains where we put a weight on some relations. Second, we consider an equivalence relation on noncrossing chains coming from the natural action of the group on set partitions, and we show that each equivalence class has a simple generating function. Using this we recover Postnikov’s hook length formula in type A and obtain a variant in type B.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Armstrong, D.: Generalized noncrossing partitions and combinatorics of Coxeter groups. Mem. Amer. Math. Soc. 202(949) (2009)

  2. Bessis D.: The dual braid monoid. Ann. Scient. Éc. Norm. Sup. 36(5), 647–683 (2003)

    MATH  MathSciNet  Google Scholar 

  3. Brady T., Watt C.: A partial order on the orthogonal group. Comm. Algebra 30(8), 3749–3754 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  4. Chapoton, F.: Enumerative properties of generalized associahedra. Sém. Lothar. Combin. 51, Art. B51b (2004)

  5. Deligne, P.: Letter to E. Looijenga, 9/3/1974. Available online at: http://homepage.univie.ac.at/christian.stump/Deligne_Looijenga_Letter_09-03-1974.pdf

  6. Du R.R.X., Liu F.: (k, m)-Catalan numbers and hook length polynomials for plane trees. European J. Combin. 28(4), 1312–1321 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  7. Foata D., Strehl V.: Rearrangements of the symmetric group and enumerative properties of the tangent and secant numbers. Math. Z. 137, 257–264 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  8. Fomin S., Reading N.: Generalized cluster complexes and Coxeter combinatorics. Int. Math. Res. Not. 2005(44), 2709–2757 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Humphreys J.E.: Reflection Groups and Coxeter Groups. Cambridge University Press, Cambridge (1990)

    Book  MATH  Google Scholar 

  10. Josuat-Vergès, M.: A generalization of Euler numbers to finite Coxeter group. Ann. Combin. (to appear)

  11. Postnikov A.: Permutohedra, associahedra, and beyond. Int. Math. Res. Not. 2009(6), 1026–1106 (2009)

    MATH  MathSciNet  Google Scholar 

  12. Purtill M.: André permutations, lexicographic shellability and the cd-index of a convex polytope. Trans. Amer. Math. Soc. 338(1), 77–104 (1993)

    MATH  MathSciNet  Google Scholar 

  13. Reading N.: Clusters, Coxeter-sortable elements and noncrossing partitions. Trans. Amer. Math. Soc. 359(12), 5931–5958 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Reading N.: Chains in the noncrossing partition lattice. SIAM J. Discrete Math. 22(3), 875–886 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. Stanley, R.P.: A survey of alternating permutations. In: Brualdi, R.A., Hedayat, S., Kharaghani, H., Khosrovshahi, G.B., Shahriari, S. (eds.) Combinatorics and Graphs. Contemp. Math. 531, pp. 165–196. American Mathematical Society, Providence, RI (2010)

  16. Stein, W.A. et al.: Sage Mathematics Software (Version 5.4). The Sage Development Team, http://www.sagemath.org (2013)

  17. Steinberg R.: Finite reflection groups. Trans. Amer. Math. Soc. 91, 493–504 (1959)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CNRS and Institut Gaspard Monge, Université Paris-Est Marne-la-Vallée, 5 Boulevard Descartes, Champs-sur-Marne, 77454, Marne-la-Vallée cedex 2, France

    Matthieu Josuat-Vergès

Authors
  1. Matthieu Josuat-Vergès
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matthieu Josuat-Vergès.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Josuat-Vergès, M. Refined Enumeration of Noncrossing Chains and Hook Formulas. Ann. Comb. 19, 443–460 (2015). https://doi.org/10.1007/s00026-015-0278-5

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00026-015-0278-5

Mathematics Subject Classification

Keywords

Search

Navigation