On the Stanley depth of squarefree Veronese ideals

摘要

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 J?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≤ d≤ n<5d+4, then sdepth∈(I n,d)=?(n-d)/(d+1)?+d, and if d≤ 1 and n≤ 5d+4, then d+3≤ sdepth∈(I_(n,d))?(n-d)/(d+1)?+d.