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Journal of Algebraic Combinatorics

, Volume 33, Issue 2, pp 313–324 | Cite as

On the Stanley depth of squarefree Veronese ideals

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

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.

Keywords

Stanley depth Squarefree monomial ideal Interval partition Squarefree Veronese ideal 

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Copyright information

© 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

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