Numerical semigroups counted by maximum primitive are Möbius transforms of those counted by Frobenius number, and almost all with large enough maximum primitive satisfy Wilf's conjecture.
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Finite generation of the additive monoid N_0[alpha] is fully characterized for minimal polynomials of the form p(X) - c and implies that alpha must be a weak Perron number, with applications to cubic cases and rank-3 monoids.
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On counting numerical semigroups by maximum primitive and Wilf's conjecture
Numerical semigroups counted by maximum primitive are Möbius transforms of those counted by Frobenius number, and almost all with large enough maximum primitive satisfy Wilf's conjecture.
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Finite Generation in Polynomial Semirings
Finite generation of the additive monoid N_0[alpha] is fully characterized for minimal polynomials of the form p(X) - c and implies that alpha must be a weak Perron number, with applications to cubic cases and rank-3 monoids.