  We give a sharp lower bound for the Hilbert function in degree $d$ of
artinian quotients $\Bbbk[x_1,\ldots,x_n]/I$ failing the Strong Lefschetz
property, where $I$ is a monomial ideal generated in degree $d \geq 2$. We also
provide sharp lower bounds for other classes of ideals, and connect our result
to the classification of the Hilbert functions forcing the Strong Lefschetz
property by Zanello and Zylinski.
