Papers › Sub-Fibonacci behavior in numerical semigroup enumeration
Sub-Fibonacci behavior in numerical semigroup enumeration
Daniel G. Zhu
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In 2013, Zhai proved that most numerical semigroups of a given genus have depth at most $3$ and that the number n_g of numerical semigroups of a genus g is asymptotic to Sφᵍ, where S is some positive constant and φ≈1.61803 is the golden ratio. In this paper, we prove exponential upper and lower bounds on the factors that cause n_g to deviate from a perfect exponential, including the number of semigroups with depth at least $4$. Among other applications, these results imply the sharpest known asymptotic bounds on n_g and shed light on a conjecture by Bras-Amor\'os (2008) that n_g ≥n_(g-1) + n_(g-2). Our main tools are the use of Kunz coordinates, introduced by Kunz (1987), and a result by Zhao (2011) bounding weighted graph homomorphisms.
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