Stanley depth and simplicial spanning trees

TL;DR

This paper links the Stanley conjecture to simplicial spanning trees, demonstrating that verifying the conjecture can be reduced to a special class of monomial ideals associated with these trees, and confirms the conjecture for certain cases.

Contribution

It establishes a bijection between monomial ideals relevant to the Stanley conjecture and simplicial spanning trees, simplifying the verification process.

Findings

Stanley conjecture verified for monomial ideals with up to six generators

Partial verification for ideals with seven generators

Reduction of the conjecture to a class of ideals linked to simplicial spanning trees

Abstract

We show that for proving the Stanley conjecture, it is sufficient to consider a very special class of monomial ideals. These ideals (or rather their lcm lattices) are in bijection with the simplicial spanning trees of skeletons of a simplex. We apply this result to verify the Stanley conjecture for quotients of monomial ideals with up to six generators. For seven generators we obtain a partial result.