Variants on a question of Wilf

TL;DR

This paper investigates Wilf's conjecture on the density of omitted numbers in numerical semigroups, relating it to weight distributions and providing affirmative results in specific cases.

Contribution

It connects Wilf's problem to weight distributions on certain sets and offers affirmative answers in special cases, extending previous related results.

Findings

Affirmative answers in specific cases of Wilf's conjecture.

Relation of Wilf's problem to weight distribution on sets.

Extension of previous results by Fr"oberg, Gottlieb, and H"aggkvist.

Abstract

Let $S\neq\mathbb N$ be a numerical semigroup generated by $e$ elements. In his paper (A Circle-Of-Lights Algorithm for the "Money-Changing Problem", Amer. Math. Monthly 85 (1978), 562--565), H.~S.~Wilf raised the following question: Let $\Omega$ be the number of positive integers not contained in $S$ and $c-1$ the largest such element. Is it true that the fraction $\frac\Omega c$ of omitted numbers is at most $1-\frac1e$? Let $B\subseteq\mathbb N^{e-1}$ be the complement of an artinian $\mathbb N^{e-1}$-ideal. Following a concept of A.~Zhai (An asymptotic result concerning a question of Wilf, arXiv:1111.2779v1 [math.CO]) we relate Wilf's problem to a more general question about the weight distribution on $B$ with respect to a positive weight vector. An affirmative answer is given in special cases, similar to those considered by R.~Fr\"oberg, C.~Gottlieb, R.~H\"aggkvist (On numerical semigroups, Semigroup Forum, Vol.~35, Issue 1, 1986/1987, 63--83) for Wilf's question.