## Involve: A Journal of Mathematics

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

### Enumeration of $m$-endomorphisms

#### 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
Revised: 14 July 2015
Accepted: 20 July 2015
First available in Project Euclid: 22 November 2017

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

#### 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

#### 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.