  For a standard Artinian $k$-algebra $A=R/I$, we give equivalent conditions
for $A$ to have the weak (or strong) Lefschetz property or the strong Stanley
property in terms of the minimal system of generators of the generic initial
ideal $\mathrm{gin}(I)$ of $I$ under the reverse lexicographic order. Using the
equivalent condition for the weak Lefschetz property, we show that some graded
Betti numbers of $\mathrm{gin}(I)$ are determined just by the the Hilbert
function of $I$ if $A$ has the weak Lefschetz property. Furthermore, for the
case that $A$ is a standard Artinian $k$-algebra of codimension 3, we show that
every graded Betti numbers of $\mathrm{gin}(I)$ are determined by the graded
Betti numbers of $I$ if $A$ has the weak Lefschetz property. And if $A$ has the
strong Lefschetz (resp. Stanley) property, then we show that the minimal system
of generators of $\mathrm{gin}(I)$ is determined by the graded Betti numbers
(resp. by the Hilbert function) of $I$.
