Journal of Algebraic Combinatorics

, Volume 33, Issue 2, pp 313–324

On the Stanley depth of squarefree Veronese ideals

  • Mitchel T. Keller
  • Yi-Huang Shen
  • Noah Streib
  • Stephen J. Young
Open AccessArticle

DOI: 10.1007/s10801-010-0249-1

Cite this article as:
Keller, M.T., Shen, Y., Streib, N. et al. J Algebr Comb (2011) 33: 313. doi:10.1007/s10801-010-0249-1

Abstract

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 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 In,d is the squarefree Veronese ideal generated by all squarefree monomials of degree d, we show that if 1≤dn<5d+4, then sdepth (In,d)=⌊(nd)/(d+1)⌋+d, and if d≥1 and n≥5d+4, then d+3≤sdepth (In,d)≤⌊(nd)/(d+1)⌋+d.

Keywords

Stanley depthSquarefree monomial idealInterval partitionSquarefree Veronese ideal
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© The Author(s) 2010

Authors and Affiliations

  • Mitchel T. Keller
    • 1
  • Yi-Huang Shen
    • 2
  • Noah Streib
    • 1
  • Stephen J. Young
    • 1
    • 3
  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA
  2. 2.Department of MathematicsUniversity of Science and Technology of ChinaHefeiChina
  3. 3.Department of MathematicsUniversity of California, San DiegoLa JollaUSA