Involve: A Journal of Mathematics

  • Involve
  • Volume 9, Number 3 (2016), 423-435.

Enumeration of $m$-endomorphisms

Louis Rubin and Brian Rushton

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Abstract

An m-endomorphism on a free semigroup is an endomorphism that sends every generator to a word of length m. Two m-endomorphisms are combinatorially equivalent if they are conjugate under an automorphism of the semigroup. In this paper, we specialize an argument of N. G. de Bruijn to produce a formula for the number of combinatorial equivalence classes of m-endomorphisms on a rank-n semigroup. From this formula, we derive several little-known integer sequences.

Article information

Source
Involve, Volume 9, Number 3 (2016), 423-435.

Dates
Received: 6 February 2015
Revised: 14 July 2015
Accepted: 20 July 2015
First available in Project Euclid: 22 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.involve/1511371023

Digital Object Identifier
doi:10.2140/involve.2016.9.423

Mathematical Reviews number (MathSciNet)
MR3509336

Zentralblatt MATH identifier
1338.05013

Subjects
Primary: 05A99: None of the above, but in this section
Secondary: 20M15: Mappings of semigroups

Keywords
enumeration free semigroup endomorphisms semigroup

Citation

Rubin, Louis; Rushton, Brian. Enumeration of $m$-endomorphisms. Involve 9 (2016), no. 3, 423--435. doi:10.2140/involve.2016.9.423. https://projecteuclid.org/euclid.involve/1511371023


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References

  • N. G. de Bruijn, “Pólya's theory of counting”, pp. 144–184 in Applied combinatorial mathematics, edited by E. F. Bechenbach, Wiley, New York, 1964.
  • N. G. de Bruijn, “Enumeration of mapping patterns”, J. Combinatorial Theory Ser. A 12:1 (1972), 14–20.
  • D. S. Dummit and R. M. Foote, Abstract algebra, 3rd ed., Wiley, Hoboken, NJ, 2004.
  • D. S. Malik, J. N. Mordeson, and M. Sen, Fundamentals of abstract algebra, McGraw-Hill, New York, 1997.
  • OEIS, “The on-line encyclopedia of integer sequences”, 1996, hook http://oeis.org \posturlhook.