Euler's relation, möbius functions and matroid identities
Article
Received:
- 45 Downloads
- 10 Citations
Abstract
We give a short combinatorial proof of the Euler relation for convex polytopes in the context of oriented matroids. Using counting arguments we derive from the Euler relation several identities holding in the lattice of flats of an oriented matroid. These identities are proven for any matroid by Möbius inversion.
Keywords
Convex Polytopes Counting Argument Combinatorial Proof Oriented Matroid Euler Relation
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.
Preview
Unable to display preview. Download preview PDF.
Bibliography
- 1.Bland, R., Las Vergnas, M.: ‘Orientability of Matroids’, J. Combinatorial Theory (B) 24 (1978), 94–123.Google Scholar
- 2.Brugesser, H., and Mani, P.: ‘Shellable Decompositions of Cells and Spheres’, Math. Scand. 29 (1971), 197–205.Google Scholar
- 3.Crapo, H. H.: ‘Möbius Inversion in Lattices’, Archiv der Math. 19 (1968), 595–607.Google Scholar
- 4.Edmonds, J., and Mandel, A.: (abstract), Notices Amer. Math. Soc. 25 (1978), A-510.Google Scholar
- 5.Edmonds, J., Fukuda, K., and Mandel, A.: ‘Topology of Oriented Matroids’, in preparation.Google Scholar
- 6.Folkman, J., and Lawrence, J.: ‘Oriented Matroids’, J. Combinatorial Theory (B) 25 (1978), 199–236.Google Scholar
- 7.Grünbaum, B.: Convex polytopes, Interscience Publishers, London, New York, Sidney, 1967.Google Scholar
- 8.Hadwiger, H.: ‘Eulers Charakteristik und kombinatorische Geometrie’, J. reine angew. Math. 194 (1955), 87–94.Google Scholar
- 9.Klee, V.: ‘The Euler Characteristic in Combinatorial Geometry’, Amer. Math. Monthly 70 (1963), 119–127.Google Scholar
- 10.Las Vergnas, M.: ‘Convexity in Oriented Matroids’, J. Combinatorial Theory (B) 29 (1980), 231–243, announced in: Matroides orientables. C. R. Acad. Sci. Paris (A) 280 (1975), 61–64.Google Scholar
- 11.Las Vergnas, M.: ‘Acyclic and Totally Cyclic Orientations of Combinatorial Geometries’, Discrete Math. 20 (1977), 51–61.Google Scholar
- 12.Las Vergnas, M.: ‘Sur les activités des orientations d'une géométrie combinatoire’, Proc. Colloq. Mathématiques Discrètes: Codes et Hypergraphes (Bruxelles 1978), Cahiers C. E. R. O. (Bruxelles) 20 (1978), 293–300.Google Scholar
- 13.Las Vergnas, M.: ‘On the Tutte Polynomial of a Morphism of Matroids’, Proc. Joint Canada-France Combinatorial Colloquium, Montréal 1979, Annals Discrete Math. 8 (1980), 7–20.Google Scholar
- 14.Lindström, B.: ‘On the Realization of Convex Polytopes, Euler's Formula and Möbius Functions’, Aeq. Math. 6 (1971), 235–240.Google Scholar
- 15.Rota, G.-C.: ‘On the Foundations of Combinatorial Theory. I: Theory of Möbius Functions’, Z. Für Wahrscheinlichkeitstheorie und verw. Gebiete 2 (1964), 340–368.Google Scholar
- 16.Rota, G-C.: ‘On the Combinatorics of Euler Characteristic’, Studies in Pure Mathematics (presented to Richard Rado), Academic Press, London, 1971, pp. 221–233.Google Scholar
- 17.Stanley, R. P.: ‘Modular Elements of Geometric Lattices’, Algebra Universalis 1 (1971), 214–217.Google Scholar
- 18.Stanley, R. P.: ‘Combinatorial Reciprocity Theorems’, Advances in Math. 14 (1974), 194–253.Google Scholar
- 19.Tverberg, H.: ‘How to Cut a Convex Polytope into Simplices’, Geom. Dedicata 3 (1974), 239–240.Google Scholar
- 20.Welsh, D. J. A.: Matroid Theory, Academic Press, London, 1976.Google Scholar
- 21.Zaslavsky, T.: ‘Facing up to Arrangements: Face-Count Formulas for Partitions of Spaces by Hyperplanes’, Memoirs Amer. Math. Soc. 154 (1975).Google Scholar
Copyright information
© D. Reidel Publishing Co 1982