Vol. 9, No. 3, 2016

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Enumeration of $m$-endomorphisms

Louis Rubin and Brian Rushton

Vol. 9 (2016), No. 3, 423–435
Abstract

An $m$-endomorphism on a free semigroup is an endomorphism that sends every generator to a word of length $\le m$. 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.

Keywords
enumeration, free semigroup endomorphisms, semigroup
Primary: 05A99
Secondary: 20M15
Milestones
Received: 6 February 2015
Revised: 14 July 2015
Accepted: 20 July 2015
Published: 3 June 2016

Communicated by Vadim Ponomarenko
Authors
 Louis Rubin Department of Mathematics and Computer Science St. Louis University 220 North Grand Boulevard St. Louis, MO 63103 United States Brian Rushton Department of Mathematics Brigham Young University 268 TMCB Provo, UT 84602 United States