Distribution of genus among numerical semigroups with fixed Frobenius number

A numerical semigroup is a sub-monoid of the natural numbers under addition that has a finite complement. The size of its complement is called the genus and the largest number in the complement is called its Frobenius number. We consider the set of numerical semigroups with a fixed Frobenius number...

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Bibliographic Details
Published in:Semigroup forum 2022-06, Vol.104 (3), p.704-723
Main Author: Singhal, Deepesh
Format: Article
Language:English
Subjects:
Algebra
Complement
Mathematics
Mathematics and Statistics
Monoids
Number theory
Power series
Research Article
Semigroups
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Summary:A numerical semigroup is a sub-monoid of the natural numbers under addition that has a finite complement. The size of its complement is called the genus and the largest number in the complement is called its Frobenius number. We consider the set of numerical semigroups with a fixed Frobenius number f and analyse their genus. We find the asymptotic distribution of genus in this set of numerical semigroups and show that it is a product of a Gaussian and a power series. We show that almost all numerical semigroups with Frobenius number f have genus close to 3 f 4 . We denote the number of numerical semigroups of Frobenius number f by N ( f ). While N ( f ) is not monotonic we prove that N ( f ) < N ( f + 2 ) for every f .
ISSN:0037-1912
1432-2137
DOI:10.1007/s00233-022-10282-6