The toric $h$-vectors of partially ordered sets
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- by Margaret M. Bayer and Richard Ehrenborg PDF
- Trans. Amer. Math. Soc. 352 (2000), 4515-4531 Request permission
Abstract:
An explicit formula for the toric $h$-vector of an Eulerian poset in terms of the $\mathbf {cd}$-index is developed using coalgebra techniques. The same techniques produce a formula in terms of the flag $h$-vector. For this, another proof based on Fine’s algorithm and lattice-path counts is given. As a consequence, it is shown that the Kalai relation on dual posets, $g_{n/2}(P)=g_{n/2}(P^*)$, is the only equation relating the $h$-vectors of posets and their duals. A result on the $h$-vectors of oriented matroids is given. A simple formula for the $\mathbf {cd}$-index in terms of the flag $h$-vector is derived.References
- Margaret M. Bayer, Face numbers and subdivisions of convex polytopes, Polytopes: abstract, convex and computational (Scarborough, ON, 1993) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 440, Kluwer Acad. Publ., Dordrecht, 1994, pp. 155–171. MR 1322061
- Margaret M. Bayer and Louis J. Billera, Counting faces and chains in polytopes and posets, Combinatorics and algebra (Boulder, Colo., 1983) Contemp. Math., vol. 34, Amer. Math. Soc., Providence, RI, 1984, pp. 207–252. MR 777703, DOI 10.1090/conm/034/777703
- Margaret M. Bayer and Andrew Klapper, A new index for polytopes, Discrete Comput. Geom. 6 (1991), no. 1, 33–47. MR 1073071, DOI 10.1007/BF02574672
- Louis J. Billera, Richard Ehrenborg, and Margaret Readdy, The $c$-$2d$-index of oriented matroids, J. Combin. Theory Ser. A 80 (1997), no. 1, 79–105. MR 1472106, DOI 10.1006/jcta.1997.2797
- Louis J. Billera, Richard Ehrenborg, and Margaret Readdy, The $cd$-index of zonotopes and arrangements, Mathematical essays in honor of Gian-Carlo Rota (Cambridge, MA, 1996) Progr. Math., vol. 161, Birkhäuser Boston, Boston, MA, 1998, pp. 23–40. MR 1627367
- Louis J. Billera and Carl W. Lee, A proof of the sufficiency of McMullen’s conditions for $f$-vectors of simplicial convex polytopes, J. Combin. Theory Ser. A 31 (1981), no. 3, 237–255. MR 635368, DOI 10.1016/0097-3165(81)90058-3
- Francisco Brenti, Twisted incidence algebras and Kazhdan-Lusztig-Stanley functions, Adv. Math. 148 (1999), 44–74.
- Richard Ehrenborg and Margaret Readdy, Coproducts and the $cd$-index, J. Algebraic Combin. 8 (1998), no. 3, 273–299. MR 1651249, DOI 10.1023/A:1008614816374
- S. A. Joni and G.-C. Rota, Coalgebras and bialgebras in combinatorics, Stud. Appl. Math. 61 (1979), no. 2, 93–139. MR 544721, DOI 10.1002/sapm197961293
- Joseph P. S. Kung (ed.), Gian-Carlo Rota on combinatorics, Contemporary Mathematicians, Birkhäuser Boston, Inc., Boston, MA, 1995. Introductory papers and commentaries. MR 1392961
- P. McMullen, The maximum numbers of faces of a convex polytope, Mathematika 17 (1970), 179–184. MR 283691, DOI 10.1112/S0025579300002850
- D. M. Y. Sommerville, The relations connecting the angle-sums and volume of a polytope in space of $n$ dimensions, Proc. Roy. Soc. London Ser. A 115 (1927), 103–119.
- Richard P. Stanley, The number of faces of a simplicial convex polytope, Adv. in Math. 35 (1980), no. 3, 236–238. MR 563925, DOI 10.1016/0001-8708(80)90050-X
- Radu Bǎdescu, On a problem of Goursat, Gaz. Mat. 44 (1939), 571–577. MR 0000087
- Richard P. Stanley, Flag $f$-vectors and the $cd$-index, Math. Z. 216 (1994), no. 3, 483–499. MR 1283084, DOI 10.1007/BF02572336
- Dennis Stanton and Dennis White, Constructive combinatorics, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1986. MR 843332, DOI 10.1007/978-1-4612-4968-9
Additional Information
- Margaret M. Bayer
- Affiliation: Department of Mathematics, University of Kansas, Lawrence, Kansas 66045
- MR Author ID: 32915
- ORCID: 0000-0002-8519-5438
- Email: bayer@math.ukans.edu
- Richard Ehrenborg
- Affiliation: School of Mathematics, Institute for Advanced Study, Olden Lane, Princeton, New Jersey 08540
- Address at time of publication: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
- Email: jrge@ms.uky.edu
- Received by editor(s): February 15, 1998
- Published electronically: June 13, 2000
- Additional Notes: The first author was supported in part at MSRI by NSF grant #DMS 9022140
This work was begun when the second author was an H. C. Wang Assistant Professor at Cornell University and was completed at IAS under the partial support of NSF grant #DMS 97-29992 and NEC Research Institute, Inc. - © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 352 (2000), 4515-4531
- MSC (2000): Primary 06A07; Secondary 52B05
- DOI: https://doi.org/10.1090/S0002-9947-00-02657-X
- MathSciNet review: 1779486