Enumeration of $m$-endomorphisms

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.