On the Stanley Depth of Squarefree Veronese Ideals

Abstract

Let $K$ be a field and $S=K[x_1,...,x_n]$. In 1982, Stanley defined what is now called the Stanley depth of an $S$-module $M$, denoted $\sdepth(M)$, and conjectured that $\depth(M) \le \sdepth(M)$ for all finitely generated $S$-modules $M$. This conjecture remains open for most cases. However, Herzog, Vladoiu and Zheng recently proposed a method of attack in the case when $M = I / J$ with $J \subset I$ being monomial $S$-ideals. Specifically, their method associates $M$ with a partially ordered set. In this paper we take advantage of this association by using combinatorial tools to analyze squarefree Veronese ideals in $S$. In particular, if $I_{n,d}$ is the squarefree Veronese ideal generated by all squarefree monomials of degree $d$, we show that if $1\le d\le n < 5d+4$, then $\sdepth(I_{n,d})= \floor{\binom{n}{d+1}\Big/\binom{n}{d}}+d$, and if $d\geq 1$ and $n\ge 5d+4$, then $d+3\le \sdepth(I_{n,d}) \le \floor{\binom{n}{d+1}\Big/\binom{n}{d}}+d$.