{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T16:48:00Z","timestamp":1775839680879,"version":"3.50.1"},"reference-count":29,"publisher":"Springer Science and Business Media LLC","issue":"1-3","license":[{"start":{"date-parts":[[2007,9,13]],"date-time":"2007-09-13T00:00:00Z","timestamp":1189641600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2008,3]]},"DOI":"10.1007\/s00454-007-9020-3","type":"journal-article","created":{"date-parts":[[2007,9,12]],"date-time":"2007-09-12T10:28:39Z","timestamp":1189592919000},"page":"123-137","source":"Crossref","is-referenced-by-count":4,"title":["Enumeration in Convex Geometries and Associated Polytopal Subdivisions of Spheres"],"prefix":"10.1007","volume":"39","author":[{"given":"Louis J.","family":"Billera","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Samuel K.","family":"Hsiao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J. Scott","family":"Provan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2007,9,13]]},"reference":[{"key":"9020_CR1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1112\/S0010437X0500165X","volume":"142","author":"M. Aguiar","year":"2006","unstructured":"Aguiar, M., Bergeron, N., Sottile, F.: Combinatorial Hopf algebras and generalized Dehn\u2013Sommerville relations. Compos. Math. 142, 1\u201330 (2006)","journal-title":"Compos. Math."},{"key":"9020_CR2","doi-asserted-by":"crossref","first-page":"62","DOI":"10.4153\/CJM-1990-004-4","volume":"42","author":"M. Bayer","year":"1990","unstructured":"Bayer, M., Sturmfels, B.: Lawrence polytopes. Can. J. Math. 42, 62\u201379 (1990)","journal-title":"Can. J. Math."},{"key":"9020_CR3","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1006\/jcta.1997.2797","volume":"80","author":"L.J. Billera","year":"1997","unstructured":"Billera, L.J., Ehrenborg, R., Readdy, M.: The c-2d-index of oriented matroids. J. Comb. Theory Ser. A 80, 79\u2013105 (1997)","journal-title":"J. Comb. Theory Ser. A"},{"key":"9020_CR4","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1016\/0001-8708(90)90077-Z","volume":"83","author":"L.J. Billera","year":"1990","unstructured":"Billera, L.J., Filliman, P., Sturmfels, B.: Constructions and complexity of secondary polytopes. Adv. Math. 83, 155\u2013179 (1990)","journal-title":"Adv. Math."},{"key":"9020_CR5","doi-asserted-by":"crossref","first-page":"248","DOI":"10.1016\/S0001-8708(02)00067-1","volume":"176","author":"L.J. Billera","year":"2003","unstructured":"Billera, L.J., Hsiao, S.K., van Willigenburg, S.: Peak quasisymmetric functions and Eulerian enumeration. Adv. Math. 176, 248\u2013276 (2003)","journal-title":"Adv. Math."},{"key":"9020_CR6","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1016\/S0195-6698(81)80037-6","volume":"2","author":"L.J. Billera","year":"1981","unstructured":"Billera, L.J., Lee, C.W.: The numbers of faces of polytope pairs and unbounded polyhedra. Eur. J. Comb. 2, 307\u2013322 (1981)","journal-title":"Eur. J. Comb."},{"key":"9020_CR7","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1016\/0196-8858(85)90021-1","volume":"6","author":"A. Bj\u00f6rner","year":"1985","unstructured":"Bj\u00f6rner, A., Korte, B., Lov\u00e1sz, L.L.: Homotopy properties of greedoids. Adv. Appl. Math. 6, 447\u2013494 (1985)","journal-title":"Adv. Appl. Math."},{"key":"9020_CR8","volume-title":"Oriented Matroids","author":"A. Bj\u00f6rner","year":"1993","unstructured":"Bj\u00f6rner, A., Las Vergnas, M., Sturmfels, B., White, N., Ziegler, G.M.: Oriented Matroids. Cambridge University Press, New York (1993)"},{"key":"9020_CR9","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1007\/s00454-004-1144-0","volume":"34","author":"A. Bj\u00f6rner","year":"2005","unstructured":"Bj\u00f6rner, A., Paffenholz, A., Sj\u00f6strand, J., Ziegler, G.: Bier spheres and posets. Discret. Comput. Geom. 34, 71\u201386 (2005)","journal-title":"Discret. Comput. Geom."},{"key":"9020_CR10","doi-asserted-by":"crossref","first-page":"284","DOI":"10.1017\/CBO9780511662041.009","volume-title":"Matroid Applications","author":"A. Bj\u00f6rner","year":"1992","unstructured":"Bj\u00f6rner, A., Ziegler, G.: Introduction to greedoids. In: White, N. (ed.) Matroid Applications, pp. 284\u2013357. Cambridge University Press, Cambridge (1992)"},{"issue":"2","key":"9020_CR11","doi-asserted-by":"crossref","first-page":"771","DOI":"10.2307\/1968857","volume":"41","author":"R.P. Dilworth","year":"1940","unstructured":"Dilworth, R.P.: Lattices with unique irreducible decompositions. Ann. Math. 41(2), 771\u2013777 (1940)","journal-title":"Ann. Math."},{"key":"9020_CR12","doi-asserted-by":"crossref","first-page":"290","DOI":"10.1007\/BF02482912","volume":"10","author":"P.H. Edelman","year":"1980","unstructured":"Edelman, P.H.: Meet-distributive lattices and the anti-exchange closure. Algebra Universalis 10, 290\u2013299 (1980)","journal-title":"Algebra Universalis"},{"key":"9020_CR13","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1007\/BF00149365","volume":"19","author":"P.H. Edelman","year":"1985","unstructured":"Edelman, P.H., Jamison, R.E.: The theory of convex geometries. Geometriae Dedicata 19, 247\u2013270 (1985)","journal-title":"Geometriae Dedicata"},{"key":"9020_CR14","series-title":"Contemporary Mathematics","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1090\/conm\/057\/856235","volume-title":"Combinatorics and Ordered Sets (Arcata, 1985)","author":"P.H. Edelman","year":"1986","unstructured":"Edelman, P.H.: Abstract convexity and meet-distributive lattices. In: Rival, I. (ed.) Combinatorics and Ordered Sets (Arcata, 1985). Contemporary Mathematics, vol.\u00a057, pp. 127\u2013150. American Mathematical Society, Providence (1986)"},{"key":"9020_CR15","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1007\/BF00143895","volume":"5","author":"P.H. Edelman","year":"1988","unstructured":"Edelman, P.H., Saks, M.E.: Combinatorial Representation and convex dimension of convex geometries. Order 5, 23\u201332 (1988)","journal-title":"Order"},{"issue":"1","key":"9020_CR16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1006\/aima.1996.0026","volume":"119","author":"R. Ehrenborg","year":"1996","unstructured":"Ehrenborg, R.: On posets and Hopf algebras. Adv. Math. 119(1), 1\u201325 (1996)","journal-title":"Adv. Math."},{"key":"9020_CR17","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1007\/s00454-005-1171-5","volume":"34","author":"S. Gal","year":"2005","unstructured":"Gal, S.: Real root conjecture fails for five- and higher-dimensional spheres. Discret. Comput. Geom. 34, 269\u2013284 (2005)","journal-title":"Discret. Comput. Geom."},{"key":"9020_CR18","doi-asserted-by":"crossref","first-page":"493","DOI":"10.1007\/s00454-004-1115-5","volume":"32","author":"G. Hetyei","year":"2004","unstructured":"Hetyei, G.: Tchebyshev posets. Discret. Comput. Geom. 32, 493\u2013520 (2004)","journal-title":"Discret. Comput. Geom."},{"key":"9020_CR19","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1016\/j.jcta.2005.03.003","volume":"113","author":"S.K. Hsiao","year":"2006","unstructured":"Hsiao, S.K.: A signed analog of the Birkhoff transform. J. Comb. Theory Ser. A 113, 251\u2013272 (2006)","journal-title":"J. Comb. Theory Ser. A"},{"key":"9020_CR20","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-58191-5","volume-title":"Greedoids","author":"B. Korte","year":"1991","unstructured":"Korte, B., Lov\u00e1sz, L.L., Schrader, R.: Greedoids. Springer, Berlin (1991)"},{"key":"9020_CR21","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1007\/BF00582748","volume":"1","author":"B. Monjardet","year":"1985","unstructured":"Monjardet, B.: A use for frequently rediscovering a concept. Order 1, 415\u2013417 (1985)","journal-title":"Order"},{"key":"9020_CR22","unstructured":"Provan, J.S.: Decompositions, shellings, and diameters of simplicial complexes and convex polyhedra. Ph.D. Dissertation, Cornell University, Ithaca (1977)"},{"key":"9020_CR23","doi-asserted-by":"crossref","first-page":"576","DOI":"10.1287\/moor.5.4.576","volume":"5","author":"J.S. Provan","year":"1980","unstructured":"Provan, J.S., Billera, L.J.: Decompositions of simplicial complexes related to diameters of convex polyhedra. Math. Oper. Res. 5, 576\u2013594 (1980)","journal-title":"Math. Oper. Res."},{"key":"9020_CR24","series-title":"Cambridge Studies in Advanced Mathematics","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511805967","volume-title":"Enumerative Combinatorics, vol.\u00a01","author":"R. Stanley","year":"1997","unstructured":"Stanley, R.: Enumerative Combinatorics, vol.\u00a01. Cambridge Studies in Advanced Mathematics, vol.\u00a049. Cambridge University Press, Cambridge (1997)"},{"key":"9020_CR25","series-title":"Cambridge Studies in Advanced Mathematics","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511609589","volume-title":"Enumerative Combinatorics, vol. 2","author":"R. Stanley","year":"1999","unstructured":"Stanley, R.: Enumerative Combinatorics, vol. 2. Cambridge Studies in Advanced Mathematics, vol.\u00a062. Cambridge University Press, Cambridge (1999)"},{"issue":"2","key":"9020_CR26","doi-asserted-by":"crossref","first-page":"763","DOI":"10.1090\/S0002-9947-97-01804-7","volume":"349","author":"J.R. Stembridge","year":"1997","unstructured":"Stembridge, J.R.: Enriched P-partitions. Trans. Am. Math. Soc. 349(2), 763\u2013788 (1997)","journal-title":"Trans. Am. Math. Soc."},{"key":"9020_CR27","doi-asserted-by":"crossref","first-page":"1115","DOI":"10.1090\/S0002-9947-06-04271-1","volume":"359","author":"J.R. Stembridge","year":"2007","unstructured":"Stembridge, J.R.: Counterexamples to the poset conjectures of Neggers, Stanley, and Stembridge. Trans. Am. Math. Soc. 359, 1115\u20131128 (2007)","journal-title":"Trans. Am. Math. Soc."},{"key":"9020_CR28","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1090\/S0894-0347-02-00413-7","volume":"16","author":"E. Swartz","year":"2003","unstructured":"Swartz, E.: Topological representations of matroids. J. Am. Math. Soc. 16, 427\u2013442 (2003)","journal-title":"J. Am. Math. Soc."},{"key":"9020_CR29","doi-asserted-by":"crossref","unstructured":"Zaslavsky, T.: Facing up to arrangements: face count formulas for partitions of space by hyperplanes. Mem. Am. Math. Soc. 1(154) (1975)","DOI":"10.1090\/memo\/0154"}],"container-title":["Discrete &amp; Computational Geometry"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-007-9020-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00454-007-9020-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-007-9020-3","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,28]],"date-time":"2019-05-28T19:47:34Z","timestamp":1559072854000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00454-007-9020-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,9,13]]},"references-count":29,"journal-issue":{"issue":"1-3","published-print":{"date-parts":[[2008,3]]}},"alternative-id":["9020"],"URL":"https:\/\/doi.org\/10.1007\/s00454-007-9020-3","relation":{},"ISSN":["0179-5376","1432-0444"],"issn-type":[{"value":"0179-5376","type":"print"},{"value":"1432-0444","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,9,13]]}}}