Skip to main content
SpringerLink
Log in

Number of cyclically irreducible words in the alphabet of a free group of finite rank

  • Published:
Cybernetics and Systems Analysis Aims and scope

Abstract

It is shown that a formula that was independently obtained earlier for the number of cyclically irreducible words of length n in a symmetric alphabet of a finitely generated free group of rank k and the Whitney formula for a chromatic polynomial of a simple nonself-intersecting cycle of length n with a variable λ are mutually deducible from one another when λ = 2k. The necessary bijections differ for even and odd values of n.

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. L. M. Koganov, “Number of cyclically irreducible words in the alphabet of a free group,” in: O. B. Lupanov (ed.), Abstracts of the XIIIth Intern. Conf. on Problems of Theor. Cybernetics, Izd-vo Tsentra Prikl. Issled., Mekh.-Mat. Fak. MGU, Moscow (2002), p. 85.

    Google Scholar 

  2. L. M. Koganov, “Two deductions of an explicit closed expression for the number of cyclically irreducible words of fixed length in a free group,” in: O. B. Lupanov (ed.), Proc. VIIIth Intern. Seminar “ Discrete Mathematics and Its Applications,” Izd-vo Mekh.-Mat. Fak. MGU, Moscow (2004), pp. 203–204.

    Google Scholar 

  3. L. M. Koganov, “A question of R. I. Grigorchuk and problems connected with it,” in: Proc. VIIth Intern. Seminar “Discrete Mathematics and Its Applications,” Vol. 3, Izd-vo Mekh.-Mat. Fak. MGU, Moscow (2001), pp. 363–364.

    Google Scholar 

  4. I. Rivin, “Growth in free groups (and other stories),” in: arXiv: math. CO/9911076_v 2; (1999), pp. 1–31.

  5. I. Rivin, “Some properties of the conjugacy class growth function,” Contemporary Mathematics, 360, 113–117 (2004).

    MathSciNet  Google Scholar 

  6. H. Whitney, “The colorings of graphs,” Ann. Math. (2), 33, 688–718 (1932).

    Article  MathSciNet  Google Scholar 

  7. H. Whitney, “A logical expansion in mathematics,” Bull. Amer. Math. Soc., 38, 572–579 (1932).

    Article  MATH  MathSciNet  Google Scholar 

  8. R. C. Read, “An introduction to chromatic polynomials,” J. Combinat. Theory, 4, No. 1, 52–71 (1968).

    MathSciNet  Google Scholar 

  9. M. O. Asanov, V. A. Baranskii, and V. V. Rasin, Discrete Mathematics: Graphs, Matroids, and Algorithms [in Russian], Regular and Chaotic Dynamics (R&C Dynamics), Moscow-Izhevsk (2001).

    Google Scholar 

  10. M. Bocher, Introduction to Higher Algebra [Russian translation], GTTI (1933).

  11. A. D. Mednykh, “Determination of the number of nonequivalent coverings over a compact Riemann surface,” in: Dokl. Akad. Nauk SSSR, 239, No. 2, 269–271 (1978).

    MathSciNet  Google Scholar 

  12. A. D. Mednykh, “Branched coverings of Riemann surfaces whose branch orders coincide with the multiplicity,” Comm. in Algebra, 18, No. 5, 1517–1533 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  13. I. M. Vinogradov, Foundations of Number Theory [in Russian], ONTI, Moscow-Leningrad (1936).

    Google Scholar 

  14. Yu. V. Matiyasevich, “One representation of a chromatic polynomial,” in: Proc. Sobolev Institute of Mathematics SB RAS, Methods of Discrete Analysis in Control Systems Theory, No. 31, 61–70, Novosibirsk (1977).

  15. R. P. Stanley, Enumerative Combinatorics [Russian translation], Vol. 1, Mir, Moscow (1990).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Scientific Center of Nonlinear Wave Mechanics and Technology, Russian Academy of Science, Moscow, Russia

    L. M. Koganov

Authors
  1. L. M. Koganov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

To the memory of William T. Tutte (05.14.1917–05.02.2002)

__________

Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 39–48, July–August 2007.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Koganov, L.M. Number of cyclically irreducible words in the alphabet of a free group of finite rank. Cybern Syst Anal 43, 499–506 (2007). https://doi.org/10.1007/s10559-007-0076-0

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10559-007-0076-0

Keywords

Search

Navigation