Skip to main content
SpringerLink
Log in

A Generalization of Euler Numbers to Finite Coxeter Groups

  • Published:
Annals of Combinatorics Aims and scope Submit manuscript

Abstract

It is known that Euler numbers, defined as the Taylor coefficients of the tangent and secant functions, count alternating permutations in the symmetric group. Springer defined a generalization of these numbers for each finite Coxeter group by considering the largest descent class, and computed the value in each case of the classification. We consider here another generalization of Euler numbers for finite Coxeter groups, building on Stanley’s result about the number of orbits of maximal chains of set partitions. We present a method to compute these integers and obtain the value in each case of the classification.

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. André D.: Développement de sec x and tg x. C. R. Math. Acad. Sci. Paris 88, 965–979 (1879)

    MATH  Google Scholar 

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

  3. Armstrong D., Reiner V., Rhoades B.: Parking spaces. Adv. Math. 269, 647–706 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  4. Arnol’d V.I.: The calculus of snakes and the combinatorics of Bernoulli, Euler and Springer numbers for Coxeter groups. Russian Math. Surveys 47(1), 1–51 (1992)

    Article  MathSciNet  Google Scholar 

  5. Björner, A., Brenti, F.: Combinatorics of Coxeter Groups. Graduate Texts in Math., Vol. 231. Springer, New York (2005)

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

    Book  MATH  Google Scholar 

  7. Josuat-Vergès, M.: Refined enumeration of noncrossing chains and hook formulas. In preparation

  8. Josuat-Vergès M., Novelli J.-C., Thibon J.-Y.: The algebraic combinatorics of snakes. J. Combin. Theory Ser. A 119(8), 1613–1638 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  10. Saito K.: Principal Γ-cone for a tree. Adv. Math. 212(2), 645–668 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  11. Springer T.A.: Remarks on a combinatorial problem. Nieuw Arch. Wisk. 3(19), 30–36 (1971)

    MathSciNet  Google Scholar 

  12. Stanley R.P.: Some aspects of groups acting on finite posets. J. Combin. Theory Ser. A 32(2), 132–161 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  13. 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., Vol. 531, pp. 165–196. Amer. Math. Soc., Providence, RI (2010)

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. A Generalization of Euler Numbers to Finite Coxeter Groups. Ann. Comb. 19, 325–336 (2015). https://doi.org/10.1007/s00026-015-0267-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00026-015-0267-8

Mathematics Subject Classification

Keywords

Search

Navigation