Generic initial ideals, graded Betti numbers and $k$-Lefschetz properties

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 $R\mathrm{/}I$ has the SLP. Then the generic initial ideal of $I$ is the unique almost revlex ideal with the same Hilbert function as $R\mathrm{/}I$.
2. Let $I$ be a graded ideal of $R=K[x_1,x_2,\mathrm{.}\mathrm{.}\mathrm{.},x_n]$ whose quotient ring $R\mathrm{/}I$ has the $n$-SLP. Suppose that all $k$-th differences of the Hilbert function of $R\mathrm{/}I$ are quasi-symmetric. Then the generic initial ideal of $I$ is the unique almost revlex ideal with the same Hilbert function as $R\mathrm{/}I$.
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.