Advertisement

Discrete & Computational Geometry

, Volume 1, Issue 1, pp 9–23 | Cite as

Two poset polytopes

  • Richard P. Stanley
Article

Abstract

Two convex polytopes, called theorder polytope ϑ(P) andchain polytope ℒ(P), are associated with a finite posetP. There is a close interplay between the combinatorial structure ofP and the geometric structure of ϑ(P). For instance, the order polynomial Ω(P, m) ofP and Ehrhart polynomiali(ϑ(P),m) of ϑ(P) are related by Ω(P, m+1)=i(ϑ(P),m). A “transfer map” then allows us to transfer properties of ϑ(P) to ℒ(P). In particular, we transfer known inequalities involving linear extensions ofP to some new inequalities.

Keywords

Discrete Comput Geom Linear Extension Convex Polytopes Mixed Volume Abstract Simplicial Complex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    A. Björner, A. Garsia, and R. Stanley, An introduction to the theory of Cohen-Maculay partially ordered sets, in Ordered Sets (I. Rival, ed.), Ridel, Dordrecht-Boston-London, 1982, 583–616.CrossRefGoogle Scholar
  2. 2.
    V. Chvátal, On certain polytopes associated with graphs, J. Combin. Theory Ser B 18 (1975), 138–154.CrossRefzbMATHGoogle Scholar
  3. 3.
    L. Comtet, Advanced Combinatorics, Reidel, Dordrecht-Boston, 1974.CrossRefzbMATHGoogle Scholar
  4. 4.
    E. E. Doberkat, Problem 84-20, SIAM Review 26 (1984), 580.CrossRefGoogle Scholar
  5. 5.
    B. Dreesen, W. Poguntke, and P. Winkler, Comparability invariance of the fixed point property, preprint.Google Scholar
  6. 6.
    L. Geissinger, A polytope associated to a finite ordered set, preprint.Google Scholar
  7. 7.
    M. C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, 1980.Google Scholar
  8. 8.
    M. Habib, Comparability invariants, Ann. Discrete Math. 23 (1984), 371–386.MathSciNetGoogle Scholar
  9. 9.
    J. Kahn and M. Saks, Balancing poset extensions, Order 1 (1984), 113–126.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    I. G. Macdonald and R. B. Nelsen (independently), Solution to E2701, Amer. Math. Monthly 86 (1979), 396.MathSciNetGoogle Scholar
  11. 11.
    J. S. Provan and L. J. Billera, Simplicial Complexes Associated with Convex Polyhedra, I: Constructions and Combinatorial Examples, Technical Rept. no. 402, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York, January 1979.Google Scholar
  12. 12.
    R. Stanley, A chromatic-like polynomial for ordered sets., in Proc. Second Chapel Hill conference on Combinatorial Mathematics and Its Applications (May, 1970), Univ. of North Carolina, Chapel Hill, 421–427.Google Scholar
  13. 13.
    R. Stanley, Ordered structures and partitions, Mem. Amer. Math. Society, no. 119, 1972.Google Scholar
  14. 14.
    R. Stanley, Eulerian partitions of a unit hypercube, in Higher Combinatorics (M. Aigner, ed.), Reidel, Dordrecht-Boston, 1977, p. 49.Google Scholar
  15. 15.
    R. Stanley, Decompositions of rational convex polytopes, Ann. Discrete Math. 6 (1980), 333–342.MathSciNetCrossRefGoogle Scholar
  16. 16.
    R. Stanley, Two combinatorial applications of the Aleksandrov-Fenchel inequalities, J. Combin. Theory Ser A 31 (1981), 56–65.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1986

Authors and Affiliations

  • Richard P. Stanley
    • 1
  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridge

Personalised recommendations