Counting numerical sets with no small atoms

2008/05/22 by Marzuola, Jeremy, Miller, Andy · 1 citation
#05A15 #05A16 #05A17 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.0805.3493

Abstract

A numerical set S with Frobenius number g is a set of integers with min(S) = 0 and max(\Zbb - S)=g, and its atom monoid is A(S) = \setpresn ∈ \Zbbn+s ∈ S for all s ∈ S. Let γ_g be the number of numerical sets S having A(S) = \set0 ∪ (g,∞) divided by the total number of numerical sets with Frobenius number g. We show that the sequence \setγ_g is decreasing and converges to a number γ_∞ ≈ .4844 (with accuracy to within .0050). We also examine the singularities of the generating function for \setγ_g. Parallel results are obtained for the ratio \gsymmg of the number of symmetric numerical sets S with A(S) = \set0 ∪ (g,∞) by the number of symmetric numerical sets with Frobenius number g. These results yield information regarding the asymptotic behavior of the number of finite additive 2-bases.

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