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GENERATING FUNCTIONS FOR THE QUOTIENTS OF NUMERICAL SEMIGROUPS

Part of: Diophantine equations Semigroups Combinatorics

Published online by Cambridge University Press:  29 February 2024

FEIHU LIU Open the ORCID record for FEIHU LIU [Opens in a new window]
FEIHU LIU*
Affiliation:
School of Mathematical Sciences, Capital Normal University, Beijing 100048, PR China
*
e-mail: [email protected]
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Abstract

We propose generating functions, $\textrm {RGF}_p(x)$, for the quotients of numerical semigroups which are related to the Sylvester denumerant. Using MacMahon’s partition analysis, we can obtain $\textrm {RGF}_p(x)$ by extracting the constant term of a rational function. We use $\textrm {RGF}_p(x)$ to give a system of generators for the quotient of the numerical semigroup $\langle a_1,a_2,a_3\rangle $ by p for a small positive integer p, and we characterise the generators of ${\langle A\rangle }/{p}$ for a general numerical semigroup A and any positive integer p.

Keywords

quotient of a numerical semigroupgeneratorSylvester denumerantMacMahon’s partition analysis

MSC classification

Primary: 11D07: The Frobenius problem
Secondary: 05A15: Exact enumeration problems, generating functions 11D04: Linear equations 20M13: Arithmetic theory of monoids
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 110 , Issue 3 , December 2024 , pp. 427 - 438
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

This work is partially supported by the National Natural Science Foundation of China (Grant No. 12071311).

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