Upper Bounds for the Stanley Depth

Ishaq, Muhammad

doi:10.1080/00927872.2010.523642

Cited by 15 publications

(19 citation statements)

References 20 publications

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“…Finally, let us point out several results that also follow from our main result. The following result was originally proven in Ishaq [Ish12]. See also Seyed Fakhari [SF17] for a different proof.…”

Section: Colon Ideals and Associated Primesmentioning

confidence: 86%

Stanley depth and the lcm-lattice

Ichim

Katthän

Moyano‐Fernández

2017

Journal of Combinatorial Theory, Series A

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ABSTRACT. In this paper we show that the Stanley depth, as well as the usual depth, are essentially determined by the lcm-lattice. More precisely, we show that for quotients I/J of monomial ideals J ⊂ I, both invariants behave monotonic with respect to certain maps defined on their lcm-lattice. This allows simple and uniform proofs of many new and known results on the Stanley depth. In particular, we obtain a generalization of our result on polarization presented in [IKMF15]. We also obtain a useful description of the class of all monomial ideals with a given lcm-lattice, which is independent from our applications to the Stanley depth.

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Section: Colon Ideals and Associated Primesmentioning

confidence: 86%

Stanley depth and the lcm-lattice

Ichim

Katthän

Moyano‐Fernández

2017

Journal of Combinatorial Theory, Series A

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“…Let I ⊂ J ⊂ S be monomial ideals, Herzog et al [10] showed that the invariant Stanley depth of J/I is combinatorial in nature. The strange thing about Stanley depth is that it shares some properties and bounds with homological invariant depth see ( [10,11,19,18]). Until now mathematicians are not too much familiar with Stanley depth as it is hard to compute, for computation and some known results we refer the readers to ( [1,12,13,14,18]).…”

Section: Introductionmentioning

confidence: 99%

Depth and Stanley depth of edge ideals associated to some line graphs

Iqbal¹,

Ishaq

2019

AIMS Mathematics

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In this paper we study depth and Stanley depth of the edge ideals and quotient rings of the edge ideals, associated to classes of graphs obtained by taking the strong product of two graphs. We consider the cases when either both graphs are arbitrary paths or one is an arbitrary path and the other is an arbitrary cycle. We give exact formulae for values of depth and Stanley depth for some subclasses. We also give some sharp upper bounds for depth and Stanley depth in the general cases.

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“…Define sdepth P = min i deg v i and the so called Stanley depth of I/J given by sdepth S I/J = max P sdepth P, where P runs in the set of all partitions of P I\J (see [4], [16]). The Stanley depth is not easy to handle, see [4], [14], [7], [5] for some of its properties.…”

Section: Introductionmentioning

confidence: 99%