Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The toric $h$-vectors of partially ordered sets

Authors: Margaret M. Bayer and Richard Ehrenborg
Journal: Trans. Amer. Math. Soc. 352 (2000), 4515-4531
MSC (2000): Primary 06A07; Secondary 52B05
Published electronically: June 13, 2000
MathSciNet review: 1779486
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information


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 [Enhancements On Off] (What's this?)

  • 1. Margaret M. Bayer, Face numbers and subdivisions of convex polytopes, Polytopes: Abstract, Convex, and Computational (Dordrecht-Boston) (T. Bisztriczky, P. McMullen, R. Schneider, and A. Ivic Weiss, eds.), NATO Advanced Science Institute Series, vol. C 440, Kluwer Academic Publishers, 1994, pp. 155-171. MR 96a:52012
  • 2. Margaret M. Bayer and Louis J. Billera, Generalized Dehn-Sommerville relations for polytopes, spheres and Eulerian partially ordered sets, Invent. Math. 79 (1985), 143-157. MR 86f:52010b
  • 3. Margaret M. Bayer and Andrew Klapper, A new index for polytopes, Discrete Comput. Geom. 6 (1991), 33-47. MR 91k:52024
  • 4. Louis J. Billera, Richard Ehrenborg, and Margaret Readdy, The ${\bf c}\mbox{-}2{\bf d}$-index of oriented matroids, J. Combin. Theory Ser. A 80 (1997), 79-105. MR 98h:05051
  • 5. -, The cd-index of zonotopes and arrangements, Mathematical essays in honor of Gian-Carlo Rota (Bruce E. Sagan and Richard P. Stanley, eds.), Birkhäuser Boston, Boston, 1998, pp. 23-40. MR 99h:52011
  • 6. Louis J. Billera and Carl W. Lee, A proof of the sufficiency of McMullen's conditions for $f$-vectors of simplicial polytopes, J. Combin. Theory Ser. A 31 (1981), 237-255. MR 82m:52006
  • 7. Francisco Brenti, Twisted incidence algebras and Kazhdan-Lusztig-Stanley functions, Adv. Math. 148 (1999), 44-74.
  • 8. Richard Ehrenborg and Margaret Readdy, Coproducts and the ${\bf cd}$-index, J. Algebraic Combin. 8 (1998), 273-299. MR 2000b:52009
  • 9. S. A. Joni and G.-C. Rota, Coalgebras and bialgebras in combinatorics, Stud. Appl. Math. 61 (1979), 93-139. MR 81c:05002
  • 10. Joseph P. S. Kung (ed.), Gian-Carlo Rota on combinatorics. Introductory papers and commentaries, Birkhäuser Boston, Boston, 1995. MR 99b:01027
  • 11. Peter McMullen, The maximum number of faces of a convex polytope, Mathematika 17 (1970), 179-184. MR 44:921
  • 12. 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.
  • 13. Richard P. Stanley, The number of faces of simplicial convex polytopes, Adv. Math. 35 (1980), 236-238. MR 81f:52014
  • 14. -, Enumerative combinatorics, Vol. I, Wadsworth and Brooks/Cole, Pacific Grove, 1986. MR 87j:05003; Corrected reprint of 1986 original, Cambridge Univ. Press, 1997. MR 98a:05001
  • 15. -, Flag $f$-vectors and the $cd$-index, Math. Z. 216 (1994), 483-499. MR 96b:06006
  • 16. Dennis Stanton and Dennis White, Constructive combinatorics, Springer-Verlag, New York, 1986. MR 88a:05001

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 06A07, 52B05

Retrieve articles in all journals with MSC (2000): 06A07, 52B05

Additional Information

Margaret M. Bayer
Affiliation: Department of Mathematics, University of Kansas, Lawrence, Kansas 66045

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

Keywords: Partially ordered set, $h$-vector, $\mathbf{c}\mathbf{d}$-index, coalgebra
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.
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society