Stanley depth of factors of polymatroidal ideals and the edge ideal of forests

2015 ◽  
Vol 105 (4) ◽  
pp. 323-332 ◽  
Author(s):  
A. Alipour ◽  
S. A. Seyed Fakhari ◽  
S. Yassemi
Keyword(s):  
Edge Ideal ◽  
Stanley Depth ◽  
Polymatroidal Ideals

scholarly journals Stanley depth of edge ideals

2012 ◽  
Vol 49 (4) ◽  
pp. 501-508 ◽  
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.

scholarly journals Stanley depth of powers of the edge ideal of a forest

2013 ◽  
Vol 141 (10) ◽  
pp. 3327-3336 ◽  
Author(s):  
M. R. Pournaki ◽  
S. A. Seyed Fakhari ◽  
S. Yassemi
Keyword(s):  
Edge Ideal ◽  
Stanley Depth

scholarly journals On the Stanley depth of weakly polymatroidal ideals

2013 ◽  
Vol 100 (2) ◽  
pp. 115-121 ◽  
Author(s):  
M. R. Pournaki ◽  
S. A. Seyed Fakhari ◽  
S. Yassemi
Keyword(s):  
Stanley Depth ◽  
Polymatroidal Ideals

scholarly journals Stanley depth of weakly polymatroidal ideals

2014 ◽  
Vol 103 (3) ◽  
pp. 229-233 ◽  
Author(s):  
S. A. Seyed Fakhari
Keyword(s):  
Stanley Depth ◽  
Polymatroidal Ideals

scholarly journals Depth and Stanley depth of the edge ideals of the powers of paths and cycles

2019 ◽  
Vol 27 (3) ◽  
pp. 113-135
Author(s):  
Zahid Iqbal ◽  
Muhammad Ishaq
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Quotient Ring ◽  
Edge Ideal ◽  
Stanley Depth ◽  
Edge Ideals

AbstractLet k be a positive integer. We compute depth and Stanley depth of the quotient ring of the edge ideal associated to the kth power of a path on n vertices. We show that both depth and Stanley depth have the same values and can be given in terms of k and n. If n≣0, k + 1, k + 2, . . . , 2k(mod(2k + 1)), then we give values of depth and Stanley depth of the quotient ring of the edge ideal associated to the kth power of a cycle on n vertices and tight bounds otherwise, in terms of n and k. We also compute lower bounds for the Stanley depth of the edge ideals associated to the kth power of a path and a cycle and prove a conjecture of Herzog for these ideals.

On Stanley Depths of Certain Monomial Factor Algebras

2015 ◽  
Vol 58 (2) ◽  
pp. 393-401
Author(s):  
Zhongming Tang
Keyword(s):  
Polynomial Ring ◽  
Monomial Ideal ◽  
Primary Decomposition ◽  
Factor Algebra ◽  
Stanley Depth ◽  
Stanley Conjecture ◽  
Stanley Decomposition

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).

scholarly journals Comparing powers of edge ideals

2019 ◽  
Vol 18 (10) ◽  
pp. 1950184 ◽  
Author(s):  
Mike Janssen ◽  
Thomas Kamp ◽  
Jason Vander Woude
Keyword(s):  
Symbolic Power ◽  
Edge Ideal ◽  
Homogeneous Ideal ◽  
Edge Ideals ◽  
Ideal Formula ◽  
Number Of Generators ◽  
Recent Interest ◽  
Odd Cycle ◽  
Ordinary Power

Given a nontrivial homogeneous ideal [Formula: see text], a problem of great recent interest has been the comparison of the [Formula: see text]th ordinary power of [Formula: see text] and the [Formula: see text]th symbolic power [Formula: see text]. This comparison has been undertaken directly via an exploration of which exponents [Formula: see text] and [Formula: see text] guarantee the subset containment [Formula: see text] and asymptotically via a computation of the resurgence [Formula: see text], a number for which any [Formula: see text] guarantees [Formula: see text]. Recently, a third quantity, the symbolic defect, was introduced; as [Formula: see text], the symbolic defect is the minimal number of generators required to add to [Formula: see text] in order to get [Formula: see text]. We consider these various means of comparison when [Formula: see text] is the edge ideal of certain graphs by describing an ideal [Formula: see text] for which [Formula: see text]. When [Formula: see text] is the edge ideal of an odd cycle, our description of the structure of [Formula: see text] yields solutions to both the direct and asymptotic containment questions, as well as a partial computation of the sequence of symbolic defects.

scholarly journals Stanley’s conjecture for critical ideals

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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