Graded Betti numbers of powers of path ideals of paths

TL;DR

This paper computes the graded Betti numbers for all powers of path ideals of paths, providing detailed algebraic invariants that deepen understanding of their algebraic and combinatorial properties.

Contribution

It explicitly determines the graded Betti numbers of all powers of path ideals of paths, a new comprehensive calculation in combinatorial commutative algebra.

Findings

All graded Betti numbers of the powers are explicitly computed.

The results reveal patterns in the algebraic structure of path ideals.

Provides formulas that can be applied to similar combinatorial ideals.

Abstract

Let $I_{n,m} = (x_1\cdots x_{m},x_2 \cdots x_{m+1},\ldots,x_{n+1}x_{n+2}\cdots x_{n+m})$ be the $m$-path ideal of a path of length $n + m-1$ over a polynomial ring $S = \mathrm{k}[x_1,\ldots,x_{n+m}]$. We compute all the graded Betti numbers of all powers of $I_{n,m}$.