A Survey on Stanley Depth

Herzog, Jürgen

doi:10.1007/978-3-642-38742-5_1

Cited by 55 publications

(76 citation statements)

References 47 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [19], Rauf showed that if I is a complete intersection monomial ideal, then sdepth(I) > sdepth(S/I). Subsequently, Herzog's survey article [5] stated the weaker inequality sdepth(I) ≥ sdepth(S/I) (for arbitrary monomial ideals) as Conjecture 64, noting that the inequality is strict in all known cases. The strict version of the inequality was conjectured by Popescu and Qureshi in [17].…”

Section: Introductionmentioning

confidence: 99%

“…Following Herzog's survey article [5], we give the following restricted definition that suffices for our purposes. In this work we focus on the relationship between sdepth I and sdepth S/I for an arbitrary monomial ideal I.…”

Section: Introductionmentioning

confidence: 99%

“…(A more general version of Conjecture 64 in [5] is presented there as Question 63, which involves the relationship between a module and its syzygy module. That question has been answered in the negative by Ichim et al in [8].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Depth and Stanley depth of the edge ideals of square paths and square cycles

Iqbal

Ishaq

Aamir

2017

Communications in Algebra

View full text Add to dashboard Cite

ABSTRACT. We develop combinatorial tools to study the relationship between the Stanley depth of a monomial ideal I and the Stanley depth of its compliment, S/I. Using these results we are able to prove that if S is a polynomial ring with at most 5 indeterminates and I is a square-free monomial ideal, then the Stanley depth of S/I is strictly larger than the Stanley depth of I. Using a computer search, we are able to extend this strict inequality up to polynomial rings with at most 7 indeterminates. This partially answers questions asked by Propescu and Qureshi as well as Herzog.

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Depth and Stanley depth of the edge ideals of square paths and square cycles

Iqbal

Ishaq

Aamir

2017

Communications in Algebra

View full text Add to dashboard Cite

show abstract

“…However, it [7]. Let I n,m and J n,m be the path ideals of length m associated to the n -line, respectively n -cyclic, graph.…”

Section: Introductionmentioning

confidence: 99%

Depth and Stanley depth of the path ideal associated to an $n$-cyclic graph

Zhu¹

2017

Turk J Math

View full text Add to dashboard Cite

We compute the depth and Stanley depth for the quotient ring of the path ideal of length 3 associated to a n -cyclic graph, given some precise formulas for the depth when n ̸ ≡ 1 (mod 4) , tight bounds when n ≡ 1 (mod 4) , and for Stanley depth when n ≡ 0, 3 (mod 4) , tight bounds when n ≡ 1, 2 (mod 4) . We also give some formulas for the depth and Stanley depth of a quotient of the path ideals of length n − 1 and n .

show abstract

“…In this note we consider the Stanley depth of S/I and of I , which is a combinatorial invariant. We refer the reader to Pournaki et al (2009) for a short introduction to the subject and to Herzog (2013) for a comprehensive survey.…”

Section: Introductionmentioning

confidence: 99%