Multiplicity of Powers of Path Ideals of A Line Graph

Let \(R=\mathbb {K}[x_1,\ldots ,x_n]\) $R=\mathbb{K}[x_1,\dots ,x_n]$ be the polynomial ring over a field \(\mathbb {K}\) $\mathbb{K}$ , and let \(I\subseteq R\) $I\subseteq R$ be the t-path ideal of the line graph \(L_n\) $L_n$ with n-vertices. It is shown that the set of associated prime ideals of \(I^s\) $I^s$ is equal to the set of minimal prime ideals of I for all \(s\ge 1\) $s\ge 1$ , and we provide an explicit description of these prime ideals. Additionally, as the main contribution of this paper, we derive an explicit formula for the multiplicity of \(R/I^s\) $R/I^s$ for all \(s\ge 1\) $s\ge 1$ , revealing that it is a polynomial in s from the beginning.