Matching powers of monomial ideals and edge ideals of weighted oriented graphs

We introduce the concept of matching powers of monomial ideals. Let $I$ be a monomial ideal of $S=K[x_1,\dots ,x_n]$, where $K$ is a field. The $k$th matching power of $I$ is the monomial ideal $I^{[k]}$ generated by the products $u_1\cdots u_k$ where $u_1,\dots ,u_k$ is a sequence of support-disjoint monomials contained in $I$. This concept naturally generalizes the notion of squarefree powers of squarefree monomial ideals. We study the normalized depth function of matching powers of monomial ideals and provide bounds for the regularity and projective dimension of edge ideals of weighted oriented graphs. When $I$ is a non-quadratic edge ideal of a weighted oriented graph that contains no even cycles, we characterize when $I^{[k]}$ has a linear resolution.