Multiplicity of powers of path ideals of a line graph

Let S=K[x₁,…,xₙ] and let I be the t-path ideal of the line graph Lₙ with n-vertices. It is shown that the set of associated prime ideals of Iˢ is equal to the set of minimal prime ideals of I for all s≥ 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/Iˢ for all s≥ 1, revealing that it is a polynomial in s from the beginning.