On the Stanley Depth of Squarefree Veronese Ideals

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 $1le dle n < 5d+4$, then $sdepth(I_{n,d})= floor{binom{n}{d+1}Big/binom{n}{d}}+d$, and if $dgeq 1$ and $nge 5d+4$, then $d+3le sdepth(I_{n,d}) le floor{binom{n}{d+1}Big/binom{n}{d}}+d$.