A non-partitionable Cohen–Macaulay simplicial complex

Art M. Duval; Bennet Goeckner; Caroline J. Klivans; Jeremy Martin

doi:10.46298/dmtcs.6325

Art M. Duval ; Bennet Goeckner ; Caroline J. Klivans ; Jeremy Martin - A non-partitionable Cohen–Macaulay simplicial complex

dmtcs:6325 - Discrete Mathematics & Theoretical Computer Science, April 22, 2020, DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016) - https://doi.org/10.46298/dmtcs.6325

A non-partitionable Cohen–Macaulay simplicial complexArticle

Authors: Art M. Duval ¹; Bennet Goeckner ²; Caroline J. Klivans ³; Jeremy Martin ²

1 University of Texas [El Paso]
2 Department of Mathematics [Kansas]
3 Department of Mathematics

A long-standing conjecture of Stanley states that every Cohen–Macaulay simplicial complex is partition- able. We disprove the conjecture by constructing an explicit counterexample. Due to a result of Herzog, Jahan and Yassemi, our construction also disproves the conjecture that the Stanley depth of a monomial ideal is always at least its depth.

https://doi.org/10.46298/dmtcs.6325

Source: HAL:hal-02173381v1

Volume: DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016)

Published on: April 22, 2020

Imported on: July 4, 2016

Keywords: [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO]

Bibliographic References

30 Documents citing this article

Pavel Paták;Martin Tancer, arXiv (Cornell University), Shellability Is Hard Even for Balls, 2023, Orlando FL USA, 10.1145/3564246.3585152, https://arxiv.org/abs/2211.07978.

Raheleh Jafari;Ali Akbar Yazdan Pour, 2023, SHEDDING VERTICES AND ASS-DECOMPOSABLE MONOMIAL IDEALS, arXiv (Cornell University), 53, 1, 10.1216/rmj.2023.53.89, https://arxiv.org/abs/2111.10851.

Andrés D. Santamaría-Galvis, 2022, Partitioning the projective plane and the dunce hat, arXiv (Cornell University), 106, pp. 103584, 10.1016/j.ejc.2022.103584, https://arxiv.org/abs/2109.01154.

Bakhtawar Shaukat;Ahtsham Ul Haq;Muhammad Ishaq, 2022, Some algebraic Invariants of the residue class rings of the edge ideals of perfect semiregular trees, arXiv (Cornell University), 51, 6, pp. 2364-2383, 10.1080/00927872.2022.2159968, https://arxiv.org/abs/2110.15031.

Joseph Doolittle;Bennet Goeckner;Alexander Lazar, 2022, Partition and Cohen–Macaulay extenders, arXiv (Cornell University), 102, pp. 103488, 10.1016/j.ejc.2021.103488, https://arxiv.org/abs/1911.12791.

Malik Muhammad Suleman Shahid;Muhammad Ishaq;Anuwat Jirawattanapanit;Khanyaluck Subkrajang, 2022, Depth and Stanley depth of the edge ideals of multi triangular snake and multi triangular ouroboros snake graphs, AIMS mathematics, 7, 9, pp. 16449-16463, 10.3934/math.2022900, https://doi.org/10.3934/math.2022900.

Masahiro Hachimori, 2021, Sequential Partitions of Nonpure Simplicial Complexes, Graphs and combinatorics, 37, 5, pp. 1891-1904, 10.1007/s00373-021-02339-0.

Naeem Ud Din;Muhammad Ishaq;Zunaira Sajid, 2021, Values and bounds for depth and Stanley depth of some classes of edge ideals, AIMS mathematics, 6, 8, pp. 8544-8566, 10.3934/math.2021496, https://doi.org/10.3934/math.2021496.

Somayeh Bandari;Raheleh Jafari, 2021, Minimal depth of monomial ideals via associated radical ideals, Archiv der Mathematik, 117, 5, pp. 495-507, 10.1007/s00013-021-01656-3.

Zahid Iqbal;Muhammad Ishaq;Muhammad Ahsan Binyamin, 2021, Depth and Stanley depth of the edge ideals of the strong product of some graphs, Hacettepe journal of mathematics and statistics, 50, 1, pp. 92-109, 10.15672/hujms.638033, https://doi.org/10.15672/hujms.638033.

Hailong Dao;Joseph Doolittle;Justin Lyle, 2020, Minimal Cohen--Macaulay Simplicial Complexes, arXiv (Cornell University), 34, 3, pp. 1602-1608, 10.1137/19m1275164, https://arxiv.org/abs/1905.05043.

S. A. Seyed Fakhari, Springer proceedings in mathematics & statistics, Homological and Combinatorial Properties of Powers of Cover Ideals of Graphs, pp. 143-159, 2020, 10.1007/978-3-030-52111-0_11.

Jia-Bao Liu;Mobeen Munir;Raheel Farooki;Muhammad Imran Qureshi;Quratulien Muneer, 2019, Stanley Depth of Edge Ideals of Some Wheel-Related Graphs, Mathematics, 7, 2, pp. 202, 10.3390/math7020202, https://doi.org/10.3390/math7020202.

S. A. Seyed Fakhari, 2019, Stability of depth and Stanley depth of symbolic powers of squarefree monomial ideals, Proceedings of the American Mathematical Society, 148, 5, pp. 1849-1862, 10.1090/proc/14864, https://doi.org/10.1090/proc/14864.

S. A. Seyed Fakhari, 2019, On the depth and Stanley depth of the integral closure of powers of monomial ideals, Collectanea mathematica, 70, 3, pp. 447-459, 10.1007/s13348-019-00240-x.

Seyed Fakhari, 2019, On the Stanley Depth of Powers of Monomial Ideals, Mathematics, 7, 7, pp. 607, 10.3390/math7070607, https://doi.org/10.3390/math7070607.

Zahid Iqbal;Muhammad Ishaq, 2019, Depth and Stanley depth of the edge ideals of the powers of paths and cycles, Analele Universităţii "Ovidius" Constanţa. Seria Matematică, 27, 3, pp. 113-135, 10.2478/auom-2019-0037, https://doi.org/10.2478/auom-2019-0037.

Zahid Iqbal;Muhammad Ishaq, 2019, Depth and Stanley depth of edge ideals associated to some line graphs, AIMS mathematics, 4, 3, pp. 686-698, 10.3934/math.2019.3.686, https://doi.org/10.3934/math.2019.3.686.

James Murdock;Theodore Murdock, 2018, Block Stanley Decompositions II. Greedy Algorithms, Applications, and Open Problems, Būlitan-i anjuman-i riyāz̤ī-i Īrān., 45, 1, pp. 127-172, 10.1007/s41980-018-0123-9.

Masahiro Hachimori, Indian statistical institute series, Optimization Problems on Acyclic Orientations of Graphs, Shellability of Simplicial Complexes, and Acyclic Partitions, pp. 49-66, 2018, 10.1007/978-981-13-3059-9_3.

Mircea Cimpoeas, 2018, On the Stanley Depth of Powers of Some Classes of Monomial Ideals, arXiv (Cornell University), 44, 3, pp. 739-747, 10.1007/s41980-018-0049-2, http://arxiv.org/abs/1512.08195.

S. A. Seyed Fakhari, Springer proceedings in mathematics & statistics, On the Stanley Depth and the Schmitt–Vogel Number of Squarefree Monomial Ideals, pp. 77-82, 2018, 10.1007/978-3-319-90493-1_3.

Bogdan Ichim;Lukas Katthän;Julio José Moyano-Fernández, 2017, Stanley depth and the lcm-lattice, Journal of combinatorial theory. Series A, 150, pp. 295-322, 10.1016/j.jcta.2017.03.005, https://doi.org/10.1016/j.jcta.2017.03.005.

Mina Bigdeli;Ali Akbar Yazdan Pour;Rashid Zaare-Nahandi, 2017, Stability of Betti numbers under reduction processes: Towards chordality of clutters, Journal of combinatorial theory. Series A, 145, pp. 129-149, 10.1016/j.jcta.2016.06.001, https://doi.org/10.1016/j.jcta.2016.06.001.

Mircea Cimpoeas, 2017, A class of square-free monomial ideals associated to two integer sequences, arXiv (Cornell University), 46, 3, pp. 1179-1187, 10.1080/00927872.2017.1339060, https://arxiv.org/abs/1604.02933.

S. A. Seyed Fakhari, 2017, ON THE STANLEY DEPTH AND SIZE OF MONOMIAL IDEALS, Glasgow mathematical journal, 59, 3, pp. 705-715, 10.1017/s0017089516000495.

S.A. Seyed Fakhari, 2017, Depth and Stanley depth of symbolic powers of cover ideals of graphs, Journal of algebra, 492, pp. 402-413, 10.1016/j.jalgebra.2017.08.032, https://doi.org/10.1016/j.jalgebra.2017.08.032.

S.A. Seyed Fakhari, 2017, On the Stanley depth of powers of edge ideals, Journal of algebra, 489, pp. 463-474, 10.1016/j.jalgebra.2017.07.011, https://doi.org/10.1016/j.jalgebra.2017.07.011.

Zahid Iqbal;Muhammad Ishaq;Muhammad Aamir, 2017, Depth and Stanley depth of the edge ideals of square paths and square cycles, arXiv (Cornell University), 46, 3, pp. 1188-1198, 10.1080/00927872.2017.1339068.

Art M. Duval;Caroline J. Klivans;Jeremy L. Martin, KU ScholarWorks (University of Kansas), Simplicial and cellular trees, pp. 713-752, 2016, 10.1007/978-3-319-24298-9_28, http://hdl.handle.net/1808/21425.

Sources : OpenCitations, OpenAlex & Crossref

Share and export

Consultation statistics

This page has been seen 153 times.

This article's PDF has been downloaded 304 times.