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.
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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.
V. Chvátal, On certain polytopes associated with graphs, J. Combin. Theory Ser B 18 (1975), 138–154.
L. Comtet, Advanced Combinatorics, Reidel, Dordrecht-Boston, 1974.
E. E. Doberkat, Problem 84-20, SIAM Review 26 (1984), 580.
B. Dreesen, W. Poguntke, and P. Winkler, Comparability invariance of the fixed point property, preprint.
L. Geissinger, A polytope associated to a finite ordered set, preprint.
M. C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, 1980.
M. Habib, Comparability invariants, Ann. Discrete Math. 23 (1984), 371–386.
J. Kahn and M. Saks, Balancing poset extensions, Order 1 (1984), 113–126.
I. G. Macdonald and R. B. Nelsen (independently), Solution to E2701, Amer. Math. Monthly 86 (1979), 396.
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.
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.
R. Stanley, Ordered structures and partitions, Mem. Amer. Math. Society, no. 119, 1972.
R. Stanley, Eulerian partitions of a unit hypercube, in Higher Combinatorics (M. Aigner, ed.), Reidel, Dordrecht-Boston, 1977, p. 49.
R. Stanley, Decompositions of rational convex polytopes, Ann. Discrete Math. 6 (1980), 333–342.
R. Stanley, Two combinatorial applications of the Aleksandrov-Fenchel inequalities, J. Combin. Theory Ser A 31 (1981), 56–65.
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Partially supported by NSF Grant No. 8104855-MCS and by a Guggenheim Fellowship.
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Stanley, R.P. Two poset polytopes. Discrete Comput Geom 1, 9–23 (1986). https://doi.org/10.1007/BF02187680
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DOI: https://doi.org/10.1007/BF02187680