Stanley depth of square free Veronese ideals

Abstract

We compute the Stanley depth for the quotient ring of a square free Veronese ideal and we give some bounds for the Stanley depth of a square free Veronese ideal. In particular, it follows that both satisfy the Stanley's conjecture.

On the Stanley depth of squarefree Veronese ideals

“…This formula was later confirmed by Biró, Howard, Keller, Trotter and Young in [4] using combinatorial techniques upon which this paper builds. The Stanley depth of I n,d in the other situation remains unexplored in general, although Cimpoeas ¸considered a similar problem in [7]. In this paper, we prove an exact formula for the Stanley depth of I n,d for certain values of n and d and a bound for others.…”

Gröbner bases of syzygies and Stanley depth

“…But then sdepth I j ≥ p + 1. In fact, Cimpoeaş [4,Corollary 1.5] showed that the Stanley depth of any Z n -graded torsionfree S-module is at least 1. Hence the asserted inequality for the Stanley depth of I j follows from [8,Lemma 3.6].…”

“…Remark 3.6. In [4] it is shown that the Stanley depth of the squarefree Veronese ideal generated by squarefree monomials of degree d has Stanley depth less or equal to…”

On a Conjecture of Stanley Depth of Squarefree Veronese Ideals

“…where each u i ∈ M is homogeneous, and Z i is a subset of {x In [20], Stanley conjectures that depth M ≤ sdepth M for all finitely generated Z ngraded S-module M. Although this conjecture remains open in general, it has been confirmed in several special cases, see, for example, [1][2][3], [5][6][7][8][9][10][11][12], [17] and [18]. In [11], Herzog, Vladoiu and Zheng proved that the Stanley depth of M = I/J can be computed in finite number of steps, where J ⊂ I are monomial ideals of S. They associated I/J with a poset P I/J and showed that the Stanley depth of I/J is determined by partitioning P I/J into suitable intervals.…”