\ldots, g_t)$ and the given generators are minimal. Let $s = \min\{\deg{g_i} {\; | \;}1 \leq i \leq t\}$. Then $h(R/I)$, the $h$-vector of $R/I$, strictly increases by one from $h_0$ to $h_{s-1}$ and $h_{s-1} \geq h_s$, thus the positive part of the first difference of $h(R/I)$, ${\Delta^{\hspace{-0.333em} +}\hspace{-0.167em}}h(R/I)$, is $s$ ones. Moreover, for a general linear form $\ell \in R$, $R/(I,\ell) \cong K[x]/J$ where $J = (x^s)$ so $h(R/(I, \ell))$ is $s$ ones, that is, ${\Delta^{\hspace{-0.333em} +}\hspace{-0.167em}}h(R/I) = h(R/(I,\ell))$. Hence $R/I$ has the weak Lefschetz property with Lefschetz element $\ell$.

Almost complete intersections {#sec:aci}
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Here we restrict to artinian monomial almost complete intersections in three variables. These are the ideals discussed in [@BK Corollary 7.3] and [@MMN Section 6].

Let $K$ be an infinite field, and consider the ideal $$I_{a,b,c,\alpha,\beta,\gamma} = (x^a, y^b, z^c, x^\alpha y^\beta z^\gamma)$$ in $R = K[x,y,z]$, where $0 \leq \alpha < a, 0 \leq \beta < b,$ and $0 \leq \gamma < c$. If $\alpha = \beta = \gamma = 0$, then we define $I_{a,b,c,0,0,0}$ to be $(x^a, y^b, z^c)$ which is a complete intersection and is studied extensively in [@LZ] and [@CGJL]. Assume at most one of $\alpha, \beta,$ and $\gamma$ is 