the weak Lefschetz property in characteristic zero. Moreover, the formulae are used to generate families of algebras which never, or always, have the weak Lefschetz property, regardless of field characteristic. Finally, we determine (in one case, depending on the presence of the weak Lefschetz property) the splitting type of the syzygy bundle of an artinian monomial almost complete intersection in three variables, when the characteristic of the base field is zero.

    Our results convey an intriguing interplay between problems in algebra, combinatorics, and algebraic geometry, which raises new questions and deserves further investigation.
address: 'Department of Mathematics, University of Kentucky, 715 Patterson Office Tower, Lexington, KY 40506-0027, USA'
author:
- 'David Cook II${}^{\star}$, Uwe Nagel'
title: Enumerations deciding the weak Lefschetz property
---

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Introduction {#sec:intro}
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The starting point of this paper has been an intriguing conjecture in [@MMN] on the weak Lefschetz property of certain algebras. Though the presence of this property implies considerable restrictions on invariants of the algebra, many algebras are expected to have the weak Lefschetz property. However, establishing this property is often rather difficult. In this paper we make progress on the above conjecture and illustrate 