, Volume 33, Issue 2, pp 313-324,
Open Access This content is freely available online to anyone, anywhere at any time.
Date: 05 Aug 2010

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)≤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 JI 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≤dn<5d+4, then sdepth (I n,d )=⌊(nd)/(d+1)⌋+d, and if d≥1 and n≥5d+4, then d+3≤sdepth (I n,d )≤⌊(nd)/(d+1)⌋+d.