#### 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