$n$-absorbing monomial ideals in polynomial rings

In a commutative ring $R$ with unity, given an ideal $I$ of $R$, Anderson and Badawi in 2011 introduced the invariant $\omega (I)$, which is the minimal integer $n$ for which $I$ is an $n$-absorbing ideal of $R$. In the specific case that $R=k[x_1,\dots ,x_n]$ is a polynomial ring over a field $k$ in $n$ variables $x_1,\dots ,x_n$, we calculate $\omega (I)$ for certain monomial ideals $I$ of $R$.