Result: GENERATING FUNCTIONS FOR THE QUOTIENTS OF NUMERICAL SEMIGROUPS: Generating functions for the quotients of numerical semigroups
Title:
GENERATING FUNCTIONS FOR THE QUOTIENTS OF NUMERICAL SEMIGROUPS: Generating functions for the quotients of numerical semigroups
Authors:
Source:
Bulletin of the Australian Mathematical Society. 110:427-438
Publication Status:
Preprint
Publisher Information:
Cambridge University Press (CUP), 2024.
Publication Year:
2024
Subject Terms:
quotient of a numerical semigroup, Exact enumeration problems, generating functions, 0102 computer and information sciences, systems of generators, 01 natural sciences, Arithmetic theory of semigroups, The Frobenius problem, FOS: Mathematics, Linear Diophantine equations, Sylvester denumerant, Mathematics - Combinatorics, Combinatorics (math.CO), 0101 mathematics, MacMahon's partition analysis
Document Type:
Academic journal
Article
File Description:
application/xml
Language:
English
ISSN:
1755-1633
0004-9727
0004-9727
DOI:
10.1017/s0004972724000054
DOI:
10.48550/arxiv.2312.10889
Access URL:
Rights:
Cambridge Core User Agreement
CC BY
CC BY
Accession Number:
edsair.doi.dedup.....cf77fa6db57f7f5a31b9ec2a74deabd2
Database:
OpenAIRE
Further Information
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.