Multiplicity of powers of path ideals of a line graph (Q6661080)

Given a finite simple graph \(G\) with \(V(G)=\{x_1,\dots,x_n\}\), a \textit{\(t\)-path} in \(G\) is a path of length \(t-1\). The \textit{\(t\)-path ideal} \(I_t(G)\), introduced by \N\textit{A. Conca} and \textit{E. De Negri} [J. Algebra 211, No. 2, 599--624 (1999; Zbl 0924.13012)], is the monomial ideal\N\[\NI_t(G):=(x_{i_1}x_{i_2}\cdots x_{i_t}\mid \{x_{i_1},x_{i_2},\dots,x_{i_t}\}\text{ is a }t\text{-path of }G)\N\]\Nin the polynomial ring \(S=\mathbb{K}[x_1,\dots,x_n]\) over some field \(\mathbb{K}\). This notion can be considered as a generalization of the well-known \textit{edge ideal} of the graph \(G\).\N\NLet \(G\) be a line graph. For the powers of \(I_t(G)\), an explicit formula for the depth is known through the efforts of \N\textit{S. Bălănescu} and \textit{M. Cimpoeaş} [``Depth and Stanley depth of powers of the path ideal of a path graph'', Preprint, \url{arXiv:2303.01132}]. In turn, the regularity of such power was presented by \Nthe first and the third author of the paper under review [Commun. Algebra 52, No. 12, 5215--5223 (2024; Zbl 1548.13037)]. As the title suggests, the main task of this paper is to derive an explicit formula for the multiplicity. This is done by starting with the study of the minimal prime ideals of \(I_t(G)\). Somewhat related, this paper also shows that \(I_t(G)\) is normally torsion-free, i.e., \(I_t(G)\) and all its powers have the same set of associated prime ideals.