TL;DR
This paper derives a formula to count equivalence classes of m-endomorphisms on free semigroups, revealing new integer sequences and advancing combinatorial understanding of semigroup endomorphisms.
Contribution
It specializes de Bruijn's argument to produce a formula for counting equivalence classes of m-endomorphisms, a novel combinatorial result.
Findings
Derived a formula for the number of equivalence classes
Identified several new integer sequences
Enhanced understanding of m-endomorphism combinatorics
Abstract
An m-endomorphism of a free semigroup is an endomorphism that sends every generator to a word of length at most 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.
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