{"pk":64809,"title":"Resolving Stanley's conjecture on $k$-fold acyclic complexes","subtitle":null,"abstract":"In 1993 Stanley showed that if a simplicial complex is acyclic over some field, then its face poset can be decomposed into disjoint rank $1$ boolean intervals whose minimal faces together form a subcomplex. Stanley further conjectured that complexes with a higher notion of acyclicity could be decomposed in a similar way using boolean intervals of higher rank. We provide an explicit counterexample to this conjecture. We also prove a version of the conjecture for boolean trees and show that the original conjecture holds when this notion of acyclicity is as high as possible.\nMathematics Subject Classifications: 05E45, 55U10","language":"en","license":{"name":"Creative Commons Attribution 4.0","short_name":"CC BY 4.0","text":"Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.","url":"https://creativecommons.org/licenses/by/4.0"},"keywords":[],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/0d89b1cc","frozenauthors":[{"first_name":"Joseph","middle_name":"","last_name":"Doolittle","name_suffix":"","institution":"Institut für Geometrie, Technische Universität Graz, Austria","department":""},{"first_name":"Bennet","middle_name":"","last_name":"Goeckner","name_suffix":"","institution":"Department of Mathematics, University of Washington, U.S.A.","department":""}],"date_submitted":"2021-11-12T21:36:21Z","date_accepted":"2021-11-12T21:36:21Z","date_published":"2021-12-15T08:00:00Z","render_galley":null,"galleys":[{"label":"","type":"pdf","path":"https://journalpub.escholarship.org/combinatorial_theory/article/64809/galley/49619/download/"}]}