{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T15:48:47Z","timestamp":1758815327812,"version":"3.37.3"},"reference-count":24,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2021,7,26]],"date-time":"2021-07-26T00:00:00Z","timestamp":1627257600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,7,26]],"date-time":"2021-07-26T00:00:00Z","timestamp":1627257600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100003151","name":"Fonds de Recherche du Qu\u00e9bec - Nature et Technologies","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100003151","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Agence Natinoale de la Recherche","award":["15 CE40 0004 01"],"award-info":[{"award-number":["15 CE40 0004 01"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2022,1]]},"DOI":"10.1007\/s00454-021-00312-5","type":"journal-article","created":{"date-parts":[[2021,7,26]],"date-time":"2021-07-26T14:02:46Z","timestamp":1627308166000},"page":"166-202","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["The Facial Weak Order on Hyperplane Arrangements"],"prefix":"10.1007","volume":"67","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3193-3431","authenticated-orcid":false,"given":"Aram","family":"Dermenjian","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christophe","family":"Hohlweg","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Thomas","family":"McConville","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vincent","family":"Pilaud","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,7,26]]},"reference":[{"issue":"4","key":"312_CR1","doi-asserted-by":"publisher","first-page":"1179","DOI":"10.1090\/S0002-9939-1992-1140664-6","volume":"116","author":"A Bj\u00f6rner","year":"1992","unstructured":"Bj\u00f6rner, A.: Essential chains and homotopy type of posets. Proc. Am. Math. Soc. 116(4), 1179\u20131181 (1992)","journal-title":"Proc. Am. Math. Soc."},{"key":"312_CR2","doi-asserted-by":"crossref","unstructured":"Bj\u00f6rner, A.: The homology and shellability of matroids and geometric lattices. In: Matroid Applications. Encyclopedia of Mathematics and Its Applications, vol. 40, pp. 226\u2013283. Cambridge University Press, Cambridge (1992)","DOI":"10.1017\/CBO9780511662041.008"},{"key":"312_CR3","unstructured":"Bj\u00f6rner, A.: Topological methods. In: Handbook of Combinatorics, vol.\u00a02, pp. 1819\u20131872. Elsevier, Amsterdam (1995)"},{"key":"312_CR4","unstructured":"Bj\u00f6rner, A., Brenti, F.: Combinatorics of Coxeter Groups. Graduate Texts in Mathematics, vol. 231. Springer, New York (2005)"},{"issue":"3","key":"312_CR5","doi-asserted-by":"publisher","first-page":"263","DOI":"10.1007\/BF02187790","volume":"5","author":"A Bj\u00f6rner","year":"1990","unstructured":"Bj\u00f6rner, A., Edelman, P.H., Ziegler, G.M.: Hyperplane arrangements with a lattice of regions. Discrete Comput. Geom. 5(3), 263\u2013288 (1990)","journal-title":"Discrete Comput. Geom."},{"key":"312_CR6","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511586507","volume-title":"Oriented Matroids. Encyclopedia of Mathematics and Its Applications","author":"A Bj\u00f6rner","year":"1999","unstructured":"Bj\u00f6rner, A., Las Vergnas, M., Sturmfels, B., White, N., Ziegler, G.M.: Oriented Matroids. Encyclopedia of Mathematics and Its Applications, vol. 46. Cambridge University Press, Cambridge (1999)"},{"key":"312_CR7","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511809088","volume-title":"Introduction to Lattices and Order","author":"BA Davey","year":"2002","unstructured":"Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order. Cambridge University Press, New York (2002)"},{"issue":"2","key":"312_CR8","doi-asserted-by":"publisher","first-page":"1469","DOI":"10.1090\/tran\/7307","volume":"370","author":"A Dermenjian","year":"2018","unstructured":"Dermenjian, A., Hohlweg, Ch., Pilaud, V.: The facial weak order and its lattice quotients. Trans. Am. Math. Soc. 370(2), 1469\u20131507 (2018)","journal-title":"Trans. Am. Math. Soc."},{"key":"312_CR9","doi-asserted-by":"crossref","unstructured":"Edelman, P.H.: A partial order on the regions of $${\\bf R}^n$$ dissected by hyperplanes. Trans. Am. Math. Soc. 283(2), 617\u2013631 (1984)","DOI":"10.1090\/S0002-9947-1984-0737888-6"},{"issue":"1","key":"312_CR10","doi-asserted-by":"publisher","first-page":"99","DOI":"10.1007\/s00454-001-0055-6","volume":"27","author":"PH Edelman","year":"2002","unstructured":"Edelman, P.H., Reiner, V., Welker, V.: Convex, acyclic, and free sets of an oriented matroid. Discrete Comput. Geom. 27(1), 99\u2013116 (2002)","journal-title":"Discrete Comput. Geom."},{"issue":"2","key":"312_CR11","doi-asserted-by":"publisher","first-page":"221","DOI":"10.1090\/S0002-9939-1985-0784167-3","volume":"94","author":"PH Edelman","year":"1985","unstructured":"Edelman, P.H., Walker, J.W.: The homotopy type of hyperplane posets. Proc. Am. Math. Soc. 94(2), 221\u2013225 (1985)","journal-title":"Proc. Am. Math. Soc."},{"key":"312_CR12","volume-title":"Free Lattices. Mathematical Surveys and Monographs","author":"R Freese","year":"1995","unstructured":"Freese, R., Je\u017eek, J., Nation, J.B.: Free Lattices. Mathematical Surveys and Monographs, vol. 42. American Mathematical Society, Providence (1995)"},{"key":"312_CR13","doi-asserted-by":"crossref","unstructured":"Hohlweg, Ch.: Permutahedra and associahedra: generalized associahedra from the geometry of finite reflection groups. Associahedra, Tamari Lattices and Related Structures. Progress in Mathematical Physics, vol. 299, pp. 129\u2013159. Birkh\u00e4user, Basel (2012)","DOI":"10.1007\/978-3-0348-0405-9_8"},{"key":"312_CR14","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511623646","volume-title":"Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics","author":"JE Humphreys","year":"1990","unstructured":"Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics, vol. 29. Cambridge University Press, Cambridge (1990)"},{"key":"312_CR15","doi-asserted-by":"crossref","unstructured":"Krob, D., Latapy, M., Novelli, J.-Ch., Phan, H.-D., Schwer,\u00a0S.: Pseudo-permutations\u00a0I: first combinatorial and lattice properties. In: 13th International Conference on Formal Power Series and Algebraic Combinatorics (Tempe 2001)","DOI":"10.1007\/978-3-662-04166-6"},{"key":"312_CR16","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1090\/S0002-9947-1971-0279689-2","volume":"159","author":"P McMullen","year":"1971","unstructured":"McMullen, P.: On zonotopes. Trans. Am. Math. Soc. 159, 91\u2013109 (1971)","journal-title":"Trans. Am. Math. Soc."},{"key":"312_CR17","volume-title":"Elements of Algebraic Topology","author":"JR Munkres","year":"1984","unstructured":"Munkres, J.R.: Elements of Algebraic Topology. Addison-Wesley, Menlo Park (1984)"},{"key":"312_CR18","volume-title":"Arrangements of Hyperplanes. Grundlehren der Mathematischen Wissenschaften","author":"P Orlik","year":"1992","unstructured":"Orlik, P., Terao, H.: Arrangements of Hyperplanes. Grundlehren der Mathematischen Wissenschaften, vol. 300. Springer, Berlin (1992)"},{"issue":"2","key":"312_CR19","doi-asserted-by":"publisher","first-page":"648","DOI":"10.1016\/j.jalgebra.2005.09.042","volume":"299","author":"P Palacios","year":"2006","unstructured":"Palacios, P., Ronco, M.O.: Weak Bruhat order on the set of faces of the permutohedron and the associahedron. J. Algebra 299(2), 648\u2013678 (2006)","journal-title":"J. Algebra"},{"issue":"2","key":"312_CR20","doi-asserted-by":"publisher","first-page":"179","DOI":"10.1007\/s00012-003-1834-0","volume":"50","author":"N Reading","year":"2003","unstructured":"Reading, N.: Lattice and order properties of the poset of regions in a hyperplane arrangement. Algebra Universalis 50(2), 179\u2013205 (2003)","journal-title":"Algebra Universalis"},{"key":"312_CR21","doi-asserted-by":"crossref","unstructured":"Reading, N.: Lattice theory of the poset of regions. In: Lattice Theory: Special Topics and Applications, vol.\u00a02, pp. 399\u2013487. Birkh\u00e4user, Cham (2016)","DOI":"10.1007\/978-3-319-44236-5_9"},{"key":"312_CR22","unstructured":"Stanley, R.P.: Enumerative Combinatorics, vol.\u00a01. Cambridge Studies in Advanced Mathematics, vol. 49. Cambridge University Press, Cambridge (2011)"},{"issue":"2","key":"312_CR23","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1016\/S0195-6698(88)80033-7","volume":"9","author":"JW Walker","year":"1988","unstructured":"Walker, J.W.: Canonical homeomorphisms of posets. Eur. J. Combin. 9(2), 97\u2013107 (1988)","journal-title":"Eur. J. Combin."},{"key":"312_CR24","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-8431-1","volume-title":"Lectures on Polytopes. Graduate Texts in Mathematics","author":"GM Ziegler","year":"1995","unstructured":"Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, New York (1995)"}],"container-title":["Discrete &amp; Computational Geometry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-021-00312-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00454-021-00312-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-021-00312-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,6]],"date-time":"2023-11-06T13:27:29Z","timestamp":1699277249000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00454-021-00312-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,26]]},"references-count":24,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2022,1]]}},"alternative-id":["312"],"URL":"https:\/\/doi.org\/10.1007\/s00454-021-00312-5","relation":{},"ISSN":["0179-5376","1432-0444"],"issn-type":[{"type":"print","value":"0179-5376"},{"type":"electronic","value":"1432-0444"}],"subject":[],"published":{"date-parts":[[2021,7,26]]},"assertion":[{"value":"22 November 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 November 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 May 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 July 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}