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In this paper, we study the powers of the generalized binomial edge ideal Formula: see text of a path graph Formula: see text. We explicitly compute their regularities and determine the limit of their depths. We also show that these ordinary powers coincide with their symbolic powers. Additionally, we study the Rees algebra and the special fiber ring of Formula: see text utilizing Sagbi basis theory. In particular, we obtain precise formulas for the regularity of these blowup algebras.
Shen et al. (Mon,) studied this question.