Multiplicity of powers of path ideals of a line graph

Let $S=K[x_1,\dots ,x_n]$ and let $I$ be the $t$-path ideal of the line graph $L_n$ with $n$-vertices. It is shown that the set of associated prime ideals of $I^s$ is equal to the set of minimal prime ideals of $I$ for all $s\ge 1$, 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 $S\mathrm{/}{I}^s$ for all $s\ge 1$, revealing that it is a polynomial in $s$ from the beginning.