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 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.
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Alcántar, A.: Rees algebras of square-free Veronese ideals and their a-invariants. Discrete Math. 302, 7–21 (2005)
Apel, J.: On a conjecture of R.P. Stanley. I. Monomial ideals. J. Algebr. Comb. 17(1), 39–56 (2003)
Apel, J.: On a conjecture of R.P. Stanley. II. Quotients modulo monomial ideals. J. Algebr. Comb. 17(1), 57–74 (2003)
Biró, Cs., Howard, D.M., Keller, M.T., Trotter, W.T., Young, S.J.: Interval partitions and Stanley depth. J. Comb. Theory Ser. A. 117(4), 475–482 (2010)
Cimpoeaş, M.: Stanley depth of complete intersection monomial ideals. Bull. Math. Soc. Sci. Math. Roum. (N.S.) 51(99)(3), 205–211 (2008)
Cimpoeaş, M.: A note on Stanley’s conjecture for monomial ideals. arXiv:0906.1303 [math.AC] (2009)
Cimpoeaş, M.: Stanley depth of square free Veronese ideals. arXiv:0907.1232 [math.AC] (2009)
Dress, A.: A new algebraic criterion for shellability. Beitr. Algebra Geom. 34(1), 45–55 (1993)
Herzog, J., Hibi, T.: Cohen-Macaulay polymatroidal ideals. Eur. J. Comb. 27, 513–517 (2006)
Herzog, J., Jahan, A.S., Yassemi, S.: Stanley decompositions and partitionable simplicial complexes. J. Algebr. Comb. 27, 113–125 (2008)
Herzog, J., Vladoiu, M., Zheng, X.: How to compute the Stanley depth of a monomial ideal. J. Algebra 322(9), 3151–3169 (2009)
Keller, M.T., Young, S.J.: Stanley depth of squarefree monomial ideals. J. Algebra 322(10), 3789–3792 (2009)
Nasir, S.: Stanley decompositions and localization. Bull. Math. Soc. Sci. Math. Roum. (N.S.) 51(99)(2), 151–158 (2008)
Okazaki, R.: A lower bound of Stanley depth of monomial ideals. J. Commut. Algebra (2009, to appear)
Popescu, D.: Stanley depth of multigraded modules. J. Algebra 321(10), 2782–2797 (2009)
Shen, Y.H.: Stanley depth of complete intersection monomial ideals and upper-discrete partitions. J. Algebra 321(4), 1285–1292 (2009)
Stanley, R.P.: Linear Diophantine equations and local cohomology. Invent. Math. 68, 175–193 (1982)
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Keller, M.T., Shen, YH., Streib, N. et al. On the Stanley depth of squarefree Veronese ideals. J Algebr Comb 33, 313–324 (2011). https://doi.org/10.1007/s10801-010-0249-1
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DOI: https://doi.org/10.1007/s10801-010-0249-1