On the Weak Lefschetz Property for Hilbert functions of almost complete intersections

It is known that all complete intersection Artinian standard graded algebras of codimension 3 have the Weak Lefschetz Property. Unfortunately, this property does not continue to be true when you increase the number of minimal generators for the ideal defining the algebra. For instance, it is not more valid for almost complete intersection Artinian standard graded algebras of codimension 3. On the other hand, the Hilbert functions of all Weak Lefschetz Artinian graded algebras are unimodal and with the positive part of their first difference forming an $O$-sequence (i.e. are Weak Lefschetz sequences). In this paper we show that all the Hilbert functions of the almost complete intersection Artinian standard graded algebras of codimension 3 are Weak Lefschetz sequences.