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

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.

enumeration, free semigroup endomorphisms, semigroup
Mathematical Subject Classification 2010
Primary: 05A99
Secondary: 20M15
Received: 6 February 2015
Revised: 14 July 2015
Accepted: 20 July 2015
Published: 3 June 2016

Communicated by Vadim Ponomarenko
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