When will the Stanley depth increase

TL;DR

This paper establishes a new lower bound for the Stanley depth of certain squarefree monomial ideals based on their minimal generators, advancing understanding of their combinatorial properties.

Contribution

It provides a novel criterion linking the number of minimal generators to the Stanley depth, offering a new lower bound for specific classes of ideals.

Findings

If the number of degree d minimal generators is below a certain threshold, the Stanley depth is at least d+1.

The bound involves combinatorial expressions related to binomial coefficients.

This result improves the understanding of the relationship between generators and Stanley depth in monomial ideals.

Abstract

Let $I\subset S=\KK[x_1,...,x_n]$ be an ideal generated by squarefree monomials of degree $\ge d$. If the number of degree $d$ minimal generating monomials $\mu_d(I)\le \min(\binom{n}{d+1},\sum_{j=1}^{n-d}\binom{2j-1}{j})$, then the Stanley depth $\sdepth_S(I)\ge d+1$.