Keller, M.T., Shen, Y., Streib, N. et al. J Algebr Comb (2011) 33: 313. doi:10.1007/s10801-010-0249-1
Let K be a field and S=K[x1,…,xn]. 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 In,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 (In,d)=⌊(n−d)/(d+1)⌋+d, and if d≥1 and n≥5d+4, then d+3≤sdepth (In,d)≤⌊(n−d)/(d+1)⌋+d.
Stanley depthSquarefree monomial idealInterval partitionSquarefree Veronese ideal