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COUNTING NUMERICAL SEMIGROUPS WITH SHORT GENERATING FUNCTIONS

Published:2011-11 Issue:07 Volume:21 Page:1217-1235
ISSN:0218-1967
Container-title:International Journal of Algebra and Computation
language:en
Short-container-title:Int. J. Algebra Comput.

Author:

BLANCO VÍCTOR¹,GARCÍA-SÁNCHEZ PEDRO A.¹,PUERTO JUSTO²

Affiliation:

1. Departamento de Álgebra, Universidad de Granada, 18071 Granada, Spain

2. Departamento de Estadística e Investigación Operativa, Universidad de Sevilla, 41012 Sevilla, Spain

Abstract

This paper presents a new methodology to compute the number of numerical semigroups of given genus or Frobenius number. We apply generating function tools to the bounded polyhedron that classifies the semigroups with given genus (or Frobenius number) and multiplicity. First, we give theoretical results about the polynomial-time complexity of counting these semigroups. We also illustrate the methodology analyzing the cases of multiplicity 3 and 4 where some formulas for the number of numerical semigroups for any genus and Frobenius number are obtained.

Publisher

World Scientific Pub Co Pte Lt

Subject

General Mathematics

Link

https://www.worldscientific.com/doi/pdf/10.1142/S0218196711006911

Reference27 articles.

1. A Polynomial Time Algorithm for Counting Integral Points in Polyhedra When the Dimension is Fixed

2. Short rational generating functions for lattice point problems

3. Fibonacci-like behavior of the number of numerical semigroups of a given genus

4. Bounds on the number of numerical semigroups of a given genus

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