$R/I$ has the weak Lefschetz property if and only if $\det{N} \not\equiv 0 \pmod{\operatorname{char}{K}}$, and

3.  the entries of $N$ are given by $$(N)_{i,j} = \left\{
                        \begin{array}{ll}
                            \displaystyle \binom{c}{A + j - i} & \mbox{if } 1 \leq i \leq C, \\[0.8em]
                            \displaystyle \binom{\gamma}{A+C-\beta + j - i} & \mbox{if } C + 1 \leq i \leq C+M.
                         \end{array}
                    \right.$$

Notice that $R/(I,x+y+z) \cong S/J$, where $S = K[x,y]$ and $$J = (x^a, y^b, (x+y)^c, x^\alpha y^\beta (x+y)^\gamma).$$ Thus the sequence $$[R/I]_d \xrightarrow{\times(x+y+z)} [R/I]_{d+1} \rightarrow [R/(I, x+y+z)]_{d+1} \rightarrow 0$$ implies that $\times(x+y+z): [R/I]_s \rightarrow [R/I]_{s+1}$ is injective exactly when $[S/J]_{s+1} = 0$. Hence it suffices to show that all $s+2$ monomials of the form $x^iy^j$ where $i+j=s+1$ are in $J$.

Clearly if $i \geq a$ or $j \geq b$, then $x^iy^j$ is in $J$. This leaves $s+2 - (s+2-a) - (s+2-b) = s+2 - A - B = C+M$ monomials that are not trivially in $J$. Thus there are $C+M$ equations and unknowns, all of which only involve the non-monomial terms (after reduction by the monomial terms). Associated to this system of equations is a square integer matrix of size $C+M$, call it $N$. Then $N$ is invertible if and 