Papers › Counting numerical semigroups by Frobenius number, multiplicity, and depth
Counting numerical semigroups by Frobenius number, multiplicity, and depth
Sean Li
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In 1990, Backelin showed that the number of numerical semigroups with Frobenius number f approaches Cᵢ ·2^(f/2) for constants C₀ and C₁ depending on the parity of f. In this paper, we generalize this result to semigroups of arbitrary depth by showing there are ⌊(q+1)²/4⌋^(f/(2q-2)+o(f)) semigroups with Frobenius number f and depth q. More generally, for fixed q ≥3, we show that, given (q-1)m < f < qm, the number of numerical semigroups with Frobenius number f and multiplicity m is(⌊((q+2)²)/4 ⌋^(α/2) ⌊((q+1)²)/4 ⌋^((1-α)/2))^(m + o(m)) where α= f/m - (q-1). Among other things, these results imply Backelin's result, strengthen bounds on Cᵢ, characterize the limiting distribution of multiplicity and genus with respect to Frobenius number, and resolve a recent conjecture of Singhal on the number of semigroups with fixed Frobenius number and maximal embedding dimension.
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