Depth and Stanley depth of the path ideal associated to an nn-cyclic graph
Depth and Stanley depth of the path ideal associated to an nn-cyclic graph
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.
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- [1] Bir´o C, Howard DM, Keller MT, Trotter WT, Young SJ. Interval partitions and Stanley depth. J Comb Theory A 2010; 117: 475-482.
- [2] Cimpoea¸s M. Several inequalities regarding Stanley depth. Romanian Journal of Math and Computer Science 2012; 2: 28-40.
- [3] Cimpoea¸s M. On the Stanley depth of edge ideals of line and cyclic graphs. Romanian Journal of Math and Computer Science 2015; 5: 70-75.
- [4] Cimpoea¸s M. Stanley depth of the path ideal associated to a line graph. Preprint. ArXiv:1508.07540V1 [math. AC].
- [5] CoCoATeam. CoCoA: A System for Doing Computations in Commutative Algebra. Available at http://cocoa.dima.unige.it.
- [6] Duval AM, Goeckneker B, Klivans CJ, Martine JL. A non-partitionable Cohen-Macaulay simplicial complex. Invent Math 2015; 299: 381-395.
- [7] Herzog J. A survey on Stanley depth. Lect Notes Math 2013; 2083: 3-45.
- [8] Herzog J, Vl˘adoiu M, Zheng X. How to compute the Stanley depth of a monomial ideal. J Algebra 2009; 322: 3151-3169.
- [9] Morey S. Depths of powers of the edge ideal of a tree. Comm Algebra 2010; 38: 4042-4055.
- [10] Rauf A. Stanley decompositions, pretty clean filtrations and reductions modulo regular elements. Bull Math Soc Sci Math Roumanie 2007; 50: 347-354.
- [11] Rauf A. Depth and sdepth of multigraded module. Comm Algebra 2010; 38: 773-784.
- [12] Rinaldo G. An algorithm to compute the Stanley depth of monomials ideals. Le Mathematiche 2008; 63: 243-256.
- [13] Stanley RP. Linear Diophantine equations and local cohomology. Invent Math 1982; 68: 175-193.
- [14] S¸tefan A. Stanley depth of powers of the path ideal. Preprint. ArXiv:1409.6072V1 [math. AC].
- [15] Vasconcelos WV. Arithmetic of Blowup Algebras. London Mathematical Society Lecture Note Series, Vol. 195. Cambridge, UK: Cambridge University Press, 1994.
- [16] Villarreal RH. Monomial Algebras. New York, NY, USA: Marcel Dekker, 2001.
- [17] Zhu GJ. Depth and Stanley depth of the edge ideals of some m-line graphs and m-cyclic graphs with a common vertex. Romanian Journal of Math and Computer Science 2015; 5: 118-129.