Stanley depth of weakly polymatroidal ideals

AbstractLet S = K[x1 , . . . , xn] be the polynomial ring in n-variables over a ûeld K and I a monomial ideal of S. According to one standard primary decomposition of I, we get a Stanley decomposition of the monomial factor algebra S/I. Using this Stanley decomposition, one can estimate the Stanley depth of S/I. It is proved that sdepthS(S/I) ≤ sizeS(I). When I is squarefree and bigsizeS(I) ≤ 2, the Stanley conjecture holds for S/I, i.e., sdepthS(S/I) ≥ depthS(S/I).

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Stanley depth of edge ideals

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.49.2012.4.1223 ◽

2012 ◽

Vol 49 (4) ◽

pp. 501-508 ◽

Cited By ~ 1

Author(s):

Muhammad Ishaq ◽

Muhammad Qureshi

Keyword(s):

Complete Graph ◽

Upper Bound ◽

Edge Ideal ◽

Stanley Depth ◽

Edge Ideals

We give an upper bound for the Stanley depth of the edge ideal I of a k-partite complete graph and show that Stanley’s conjecture holds for I. Also we give an upper bound for the Stanley depth of the edge ideal of a s-uniform complete bipartite hypergraph.

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Stanley depth and Stanley support-regularity of monomial ideals

Collectanea mathematica ◽

10.1007/s13348-015-0140-4 ◽

2015 ◽

Vol 67 (2) ◽

pp. 227-246

Author(s):

Yi-Huang Shen

Keyword(s):

Monomial Ideals ◽

Stanley Depth

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Stanley’s conjecture for critical ideals

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.2010.1160 ◽

2011 ◽

Vol 48 (2) ◽

pp. 220-226

Author(s):

Azeem Haider ◽

Sardar Khan

Keyword(s):

Polynomial Ring ◽

Hilbert Function ◽

Monomial Ideal ◽

Monomial Ideals ◽

Stanley Depth

Let S = K[x1,…,xn] be a polynomial ring in n variables over a field K. Stanley’s conjecture holds for the modules I and S/I, when I ⊂ S is a critical monomial ideal. We calculate the Stanley depth of S/I when I is a canonical critical monomial ideal. For non-critical monomial ideals we show the existence of a Stanley ideal with the same depth and Hilbert function.

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Weakly Polymatroidal Ideals

Algebra Colloquium ◽

10.1142/s1005386706000666 ◽

2006 ◽

Vol 13 (04) ◽

pp. 711-720 ◽

Cited By ~ 9

Author(s):

Masako Kokubo ◽

Takayuki Hibi

Keyword(s):

Betti Numbers ◽

Partially Ordered Sets ◽

Finite Sequence ◽

Ordered Sets ◽

Partially Ordered ◽

Graded Betti Numbers ◽

Linear Resolution ◽

Fundamental Fact ◽

Polymatroidal Ideals ◽

Polymatroidal Ideal

The concept of the weakly polymatroidal ideal, which generalizes both the polymatroidal ideal and the prestable ideal, is introduced. A fundamental fact is that every weakly polymatroidal ideal has a linear resolution. One of the typical examples of weakly polymatroidal ideals arises from finite partially ordered sets. We associate each weakly polymatroidal ideal with a finite sequence, alled the polymatroidal sequence, which will be useful for the computation of graded Betti numbers of weakly polymatroidal ideals as well as for the construction of weakly polymatroidal ideals.

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On stability properties of powers of polymatroidal ideals

Collectanea mathematica ◽

10.1007/s13348-018-0234-x ◽

2018 ◽

Vol 70 (3) ◽

pp. 357-365

Author(s):

Shokoufe Karimi ◽

Amir Mafi

Keyword(s):

Stability Properties ◽

Polymatroidal Ideals

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Stanley depth of squarefree Veronese ideals

Analele Universitatii Ovidius Constanta - Seria Matematica ◽

10.2478/auom-2013-0043 ◽

2013 ◽

Vol 21 (3) ◽

pp. 67-72 ◽

Cited By ~ 1

Author(s):

Mircea Cimpoeas

Keyword(s):

Quotient Ring ◽

Stanley Depth

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.

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