Skip to main content
SpringerLink
Log in

Parity Alternating Permutations and Signed Eulerian Numbers

  • Published:
Annals of Combinatorics Aims and scope Submit manuscript

Abstract

This paper introduces subgroups of the symmetric group and studies their combinatorial properties. Their elements are called parity alternating, because they are permutations with even and odd entries alternately. The objective of this paper is twofold. The first is to derive several properties of such permutations by subdividing them into even and odd permutations. The second is to discuss their combinatorial properties; among others, relationships between those permutations and signed Eulerian numbers. Divisibility properties by prime powers are also deduced for signed Eulerian numbers and several related numbers.

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.

Similar content being viewed by others

References

  1. Désarménien J., Foata D.: The signed Eulerian numbers. Discrete Math. 99(1-3), 49–58 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  2. Foata D., Schützenberger M.-P.: Théorie Géométrique des Polynômes Eulériens, Lecture Notes in Mathematics, Vol. 138. Springer-Verlag, Berlin (1970)

  3. Graham R.L., Knuth D.E., Patashnik O.: Concrete Mathematics. Addison-Wesley Publishing Co., Reading, MA (1989)

    MATH  Google Scholar 

  4. Kerber A.: Algebraic Combinatorics via Finite Group Actions. Bibliographisches Institut, Mannheim (1991)

    MATH  Google Scholar 

  5. Knuth D.E.: The Art of Computer Programming, Vol. 3. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills (1973)

    Google Scholar 

  6. Lesieur L., Nicolas J.-L.: On the Eulerian numbers M n = max1≤k≤n A(n,k). European J. Combin. 13(5), 379–399 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  7. Loday J.-L.: Opérations sur l’homologie cyclique des algèbres commutatives. Invent. Math. 96(1), 205–230 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  8. Mantaci R.: Binomial coefficients and anti-exceedances of even permutations: a combinatorial proof. J. Combin. Theory Ser. A 63(2), 330–337 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  9. Regev A., Roichman Y.: Permutation statistics on the alternating group. Adv. Appl.Math. 33(4), 676–709 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Tanimoto S.: An operator on permutations and its application to Eulerian numbers. European J. Combin. 22(4), 569–576 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  11. Tanimoto S.: A study of Eulerian numbers by means of an operator on permutations. European J. Combin. 24(1), 33–43 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  12. Tanimoto, S.: A study of Eulerian numbers for permutations in the alternating group. Integers 6, #A31 (2006)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Kochi Joshi University, Kochi, 780-8515, Japan

    Shinji Tanimoto

Authors
  1. Shinji Tanimoto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shinji Tanimoto.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tanimoto, S. Parity Alternating Permutations and Signed Eulerian Numbers. Ann. Comb. 14, 355–366 (2010). https://doi.org/10.1007/s00026-010-0064-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00026-010-0064-3

AMS Subject Classification

Keywords

Search

Navigation