Distributions of weights and a question of Wilf

Abstract

Let $S$ be a numerical semigroup of embedding dimension $e$ and conductor $c$. The question of Wilf is, if $\#(\mathbb{N}\setminus S)/c\le e-1/e$. \noindent In (An asymptotic result concerning a question of Wilf, arXiv:1111.2779v1 [math.CO], 2011, Lemma 3), Zhai has shown an analogous inequality for the distribution of weights $x\cdot \gamma$, $x\in {\mathbb{N}}^d$, w.\,r. to a positive weight vector $\gamma$: \noindent Let $B\subseteq {\mathbb{N}}^d$ be finite and the complement of an ${\mathbb{N}}^d$-ideal. Denote by $\mathrm{mean}(B\cdot \gamma )$ the average weight of $B$. Then \[\operatorname{mean}(B\cdotγ)/\max(B\cdotγ)\leq d/d+1.\] $\bullet$ For the family ${\Delta }_n:=\{x\in {\mathbb{N}}^d|x\cdot \gamma <n+1\}$ of such sets we are able to show, that $\mathrm{mean}({\Delta }_n\cdot \gamma )/\max ({\Delta }_n\cdot \gamma )$ converges to $d/d+1$, as $n$ goes to infinity. $\bullet$ Applying Zhai's Lemma 3 to the Hilbert function of a positively graded Artinian algebra yields a new class of numerical semigroups satisfying Wilf's inequality.

7 pages

References in corpus (2)

• An asymptotic result concerning a question of Wilf
• Variants on a question of Wilf