  We introduce the $k$-strong Lefschetz property ($k$-SLP) and the $k$-weak
Lefschetz property ($k$-WLP) for graded Artinian $K$-algebras, which are
generalizations of the Lefschetz properties. The main results obtained in this
paper are as follows:
  1. Let $I$ be a graded ideal of $R=K[x_1, x_2, x_3]$ whose quotient ring
$R/I$ has the SLP. Then the generic initial ideal of $I$ is the unique almost
revlex ideal with the same Hilbert function as $R/I$.
  2. Let $I$ be a graded ideal of $R=K[x_1, x_2, ..., x_n]$ whose quotient ring
$R/I$ has the $n$-SLP. Suppose that all $k$-th differences of the Hilbert
function of $R/I$ are quasi-symmetric. Then the generic initial ideal of $I$ is
the unique almost revlex ideal with the same Hilbert function as $R/I$.
  3. We give a sharp upper bound on the graded Betti numbers of Artinian
$K$-algebras with the $k$-WLP and a fixed Hilbert function.
