The Multiplicity of Powers of a Class of Non-Square-free Monomial Ideals

Let $R=\mathbb{K}[x_1,\dots ,x_n]$ be a polynomial ring over a field $\mathbb{K}$, and let $I\subseteq R$ be a monomial ideal of height $h$. We provide a formula for the multiplicity of the powers of $I$ when all the primary ideals of height $h$ in the irredundant reduced primary decomposition of $I$ are irreducible. This is a generalization of \cite[Theorem 1.1]{TV}. Furthermore, we present a formula for the multiplicity of powers of special powers of monomial ideals that satisfy the aforementioned conditions. Here, for an integer $m>0$, the $m$-th special power of a monomial ideal refers to the ideal generated by the $m$-th powers of all its minimal generators. Finally, we explicitly provide a formula for the multiplicity of powers of special powers of edge ideals of weighted oriented graphs.