Depth and Stanley depth of powers of the path ideal of a path graph

Let $I_{n, m} : = (x_{1} x_{2} \dots x_{m}, x_{2} x_{3} \dots x_{m + 1}, \dots, x_{n - m + 1} \dots x_{n})$ be the $m$ -path ideal of the path graph of length $n$ , in the ring $S = K [x_{1}, \dots, x_{n}]$ . We prove that: $d e p t h (S / I_{n, m}^{t}) = {\begin{cases} n - t + 2 - ⌊ \frac{n - t + 2}{m + 1} ⌋ - ⌈ \frac{n - t + 2}{m + 1} ⌉, & t \leq n + 1 - m \\ m - 1, & t > n + 1 - m \end{cases}, for all t \geq 1.$ Also, we prove that $d e p t h (S / I_{n, m}) \geq s d e p t h (S / I_{n, m}^{t}) \geq d e p t h (S / I_{n, m}^{t})$ and $s d e p t h (I_{n, m}^{t}) \geq d e p t h (I_{n, m}^{t})$ , for all $t \geq 1$ .

Depth and Stanley depth of powers of the path ideal of a path graph

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Depth and Stanley depth of powers of the path ideal of a path graph

文献摘要

讨论

讲解博客

AI智能校验

Depth and Stanley depth of powers of the path ideal of a path graph

文献摘要

讨论

|讲解博客

|AI智能校验

Depth and Stanley depth of powers of the path ideal of a path graph

文献摘要

讨论

|讲解博客

|AI智能校验

讲解博客

AI智能校验

讲解博客

AI智能校验