2011/11/11 by Alex Zhai, Zhai, Alex · 2 citations
Decision Sciences · Mathematics · #Commutative Algebra and Its Applications #Graph theory and applications #Scheduling and Timetabling Solutions #math.CO
paper · pdf · doi:10.48550/arxiv.1111.2779
9 pages, submitted to Semigroup Forum
arxiv created 2011/11/11 · arxiv updated 2011/11/14
Let Λ be a numerical semigroup with embedding dimension e(Λ). Define c(Λ) to be one plus the largest integer not in Λ, and define c'(Λ) to be the number of elements in Λ less than c(Λ). It was asked by Wilf whether (c'(Λ))/(c(Λ)) ≥ (1)/(e(Λ)) always holds. We prove an asymptotic version of this conjecture: we show that for a fixed positive integer k and any ε> 0, the inequality (c'(Λ))/(c(Λ)) ≥ (1)/(k) - ε holds for all but finitely many numerical semigroups Λ satisfying e(Λ) = k.