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Combinatorial reciprocity theorems. (English) Zbl 0294.05006


MSC:

05A15 Exact enumeration problems, generating functions
52Bxx Polytopes and polyhedra
05A19 Combinatorial identities, bijective combinatorics
11A15 Power residues, reciprocity
06A06 Partial orders, general
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[1] Ahlfors, L. V., Complex Analysis (1966), McGraw-Hill: McGraw-Hill New York · Zbl 0154.31904
[2] Birkhoff, G., Lattice Theory (1967), American Mathematical Society: American Mathematical Society Providence, R.I · Zbl 0126.03801
[3] Brylawski, T. H., A decomposition for combinatorial geometries, Trans. Amer. Math. Soc., 171, 235-282 (1972) · Zbl 0224.05007
[4] Brylawski, T. H.; Lucas, D., Uniquely representable combinatorial geometries, (Proc. International Colloquium on Combinatorial Theory. Proc. International Colloquium on Combinatorial Theory, Accad. Naz. dei Lincei (September 3-15, 1973)), (to appear) · Zbl 0392.51007
[5] Crapo, H. H., The Tutte polynomial, Aequationes Math., 3, 211-229 (1969) · Zbl 0197.50202
[6] Dean, R. A.; Keller, G., Natural partial orders, Canad. J. Math., 20, 535-554 (1968) · Zbl 0174.29701
[7] Dines, L. L., Systems of linear inequalities, Ann. of Math., 20, 2, 191-199 (1918-1919) · JFM 47.0091.01
[8] Ehrhart, E., Sur un problème de géométrie diophantienne linéaire I, II, J. Reine Angew. Math., 227, 25-49 (1967)
[9] Ehrhart, E., Démonstration de la loi de réciprocité du polyèdre rationnel, C. R. Acad. Sci. Paris, 265A, 91-94 (1967) · Zbl 0148.27503
[10] Ehrhart, E., Sur la loi de réciprocité des polyèdres rationnels, C. R. Acad. Sci. Paris, 266A, 696-697 (1968) · Zbl 0155.37502
[11] Ehrhart, E., Une extension de la loi de réciprocité des polyèdres rationnels, C. R. Acad. Sci. Paris, 277A, 575-577 (1973) · Zbl 0267.10042
[12] Elliott, E. R., On linear homogeneous diophantine equations, Quart. J. Pure Appl. Math., 34, 348-377 (1903) · JFM 34.0219.01
[13] Grünbaum, B., Convex Polytopes (1967), Wiley (Interscience): Wiley (Interscience) London · Zbl 0163.16603
[14] Harary, F., Graph Theory (1969), Addison-Wesley: Addison-Wesley Reading, Mass · Zbl 0797.05064
[15] Kreweras, G., Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers, Cahiers du BURO, 6 (1965)
[16] Kreweras, G., Sur les partitions noncroissées d’un cycle, Discrete Math., 1, 333-350 (1972) · Zbl 0231.05014
[17] Lindström, B., On the realization of convex polytopes, Euler’s formula and Möbius functions, Aequationes Math., 6, 235-240 (1971) · Zbl 0294.52006
[18] Macdonald, I. G., The volume of a lattice polyhedron, (Proc. Cambridge Philos. Soc., 59 (1963)), 719-726 · Zbl 0126.18103
[19] Macdonald, I. G., Polynomials associated with finite cell-complexes, J. London Math. Soc., 4, 2, 181-192 (1971) · Zbl 0216.45205
[20] MacMahon, P. A., (Combinatory Analysis, Vols. I and II (1916), Cambridge University Press), reprinted in one volume by Chelsea, New York, 1960 · JFM 23.0186.03
[21] Minty, G., On the axiomatic foundations of directed linear graphs, electrical networks, and network-programming, J. Math. Mech., 15, 485-520 (1966) · Zbl 0141.21601
[22] Muir, T., A Treatise on the Theory of Determinants (1933), Longmans, Greens, and Co: Longmans, Greens, and Co New York, (revised and enlarged by W. H. Metzler)
[23] Popoviciu, T., Studie şi cercetari ştiintifice, Acad. R.P.R., Filiala Cluj, 4, 8 (1953)
[24] Reeve, J. E., On the volume of lattice polyhedra, Proc. London Math. Soc., 7, 3, 378-395 (1957) · Zbl 0080.26701
[25] Reeve, J. E., A further note on the volume of lattice polyhedra, J. London Math. Soc., 34, 57-62 (1959) · Zbl 0111.34904
[26] Riordan, J., An Introduction to Combinatorial Analysis (1958), Wiley: Wiley New York · Zbl 0078.00805
[27] Rota, G.-C, On the foundations of combinatorial theory, I. Theory of Möbius functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 2, 340-368 (1964) · Zbl 0121.02406
[28] Rota, G.-C, On the combinatorics of the Euler characteristic, (Studies in Pure Mathematics (presented to Richard Rado) (1971), Academic Press: Academic Press London), 221-233
[29] Smith, D., Incidence functions as generalized arithmetic functions I, Duke Math. J., 34, 617-634 (1967) · Zbl 0168.26302
[30] Stanley, R., A chromatic-like polynomial for ordered sets, (Proc. Second Chapel Hill Conference on Combinatorial Mathematics and its Applications (May, 1970)), 421-427
[31] Stanley, R., Ordered structures and partitions, (Ph.D. dissertation (January, 1971), Harvard University) · Zbl 0246.05007
[32] Stanley, R., Ordered structures and partitions, Mem. Amer. Math. Soc., 119 (1972) · Zbl 0246.05007
[33] Stanley, R., A Brylawski decomposition for finite ordered sets, Discrete Math., 4, 77-82 (1973) · Zbl 0256.06001
[34] Stanley, R., Acyclic orientations of graphs, Discrete Math., 5, 171-178 (1973) · Zbl 0258.05113
[35] Stanley, R., Linear homogeneous diophantine equations and magic labelings of graphs, Duke Math. J., 40, 607-632 (1973) · Zbl 0269.05109
[36] Stiemke, E., Über positive Lösungen homogener linearer Gleichungen, Math. Ann., 76, 340-342 (1915) · JFM 45.0168.04
[37] Ehrhart, E., Sur les polyèdres entiers à \(n\) dimensions, C. R. Acad. Sci. Paris, 248, 1281-1284 (1959) · Zbl 0083.26204
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