On the foundations of combinatorial theory I. Theory of Möbius Functions

Springer Science and Business Media LLC - Tập 2 - Trang 340-368 - 1964
Gian -Carlo Rota1
1Department of Mathematics, Massachusetts Institute of Technology, Cambridge 39

Tài liệu tham khảo

Auslander, L., and H. M. Trent: Incidence matrices and linear graphs. J. Math. Mech. 8, 827–835 (1959). Bell, E. T.: Algebraic Arithmetic. New York: Amer. Math. Soc. (1927), —: Exponential polynomials. Ann. of Math., II. Ser. 35, 258–277 (1934). Berge, C.: Théorie des graphes et ses applications. Paris: Dounod 1958. Birkhoff, Garrett: Lattice Theory, third preliminary edition. Harvard University, 1963. - Lattice Theory, revised edition. American Mathematical Society, 1948. Birkhoff, G. D.: A determinant formula for the number of ways of coloring a map. Ann. of Math., II. Ser. 14, 42–46 (1913). —, and D. C. Lewis: Chromatic polynomials. Trans. Amer. math. Soc. 60, 355–451 (1946). Bleicher, M. N., and G. B. Preston: Abstract linear dependence relations. Publ. Math., Debrecen 8, 55–63 (1961). Bougayev, N. V.: Theory of numerical derivatives. Moscow, 1870–1873, pp. 1–222. Bruijn, N. G. De: Generalization of Polya's fundamental theorem in enumerative combinatorial analysis. Indagationes math. 21, 59–69 (1959). Chung, K.-L., and L. T. C. Hsu: A combinatorial formula with its application to the theory of probability of arbitrary events. Ann. math. Statistics 16, 91–95 (1945). Dedekind, R.: Gesammelte Mathematische Werke, volls. I–II–III. Hamburg: Deutsche Math. Verein. (1930). Delsarte, S.: Fonctions de Möbius sur les groupes abéliens finis. Ann. of Math., II. Ser. 49, 600–609 (1948). Dilworth, R. P.: Proof of a conjecture on finite modular lattices. Ann. of Math., II. Ser. 60, 359–364 (1954). Dirac, G. A.: On the four-color conjecture. Proc. London math. Society, III. Ser. 13, 193 to 218 (1963). Dowker, C. H.: Homology groups of relations. Ann. of Math., II. Ser. 56, 84–95 (1952). Dubreil-Jacotin, M.-L., L. Lesieur et R. Croisot: LeÇons sur la théorie des treilles des structures algebriques ordonnées et des treilles géometriques. Paris: Gauthier-Villars 1953. Eilenberg, S., and N. Steenrod: Foundations of algebraic topology. Princeton: University Press 1952. Fary, I.: On straight-line representation of planar graphs. Acta Sci. math. Szeged 11, 229–233 (1948). Feller, W.: An introduction to probability theory and its applications, second edition. New York: Wiley 1960. Franklin, P.: The four-color problem. Amer. J. Math. 44, 225–236 (1922). Fréchet, M.: Les probabilités associées à un système d'événements compatibles et dépendants. Actualitées scientifiques et industrielles, nos. 859 et 942. Paris: Hermann 1940 et 1943. Frontera Marqués, B.: Una función numérica en los retículos finitos que se anula para los retícules reducibles. Actas de la 2a, Reunión de matemáticos españoles. Zaragoza 103–111 1962. Frucht, R., and G.-C. Rota: La función de Möbius para el retículo di particiones de un conjunto finito. To appear in Scientia (Chile). Goldberg, K., M. S. Green and R. E. Nettleton: Dense subgraphs and connectivity. Canadian J. Math. 11 (1959). Golomb, S. W.: A mathematical theory of discrete classification. Fourth Symposium in Information Theory, London, 1961. Green, M. S., and R. E. Nettleton: Möbius function on the lattice of dense subgraphs. J. Res. nat. Bur. Standards 64B, 41–47 (1962). — —: Expression in terms of modular distribution functions for the entropy density in an infinite system. J. Chemical Physisc 29, 1365–1370 (1958). Hadwiger, H.: Eulers Charakteristik und kombinatorische Geometrie. J. reine angew. Math. 194, 101–110 (1955). Hall, Philip: A contribution to the theory of groups of prime power order. Proc. London math. Soc., II. Ser. 36, 39–95 (1932). - The Eulerian functions of a group. Quart. J. Math. Oxford Ser. 134–151, 1936. Harary, F.: Unsolved problems in the enumeration of graphs. Publ. math. Inst. Hungar. Acad. Sci. 5, 63–95 (1960). Hardy, G. H.: Ramanujan. Cambridge: University Press 1940. —, and E. M. Wright: An introduction to the theory of numbers. Oxford: University Press 1954. Hartmanis, J.: Lattice theory of generalized partitions. Canadian J. Math. 11, 97–106 (1959). Hille, E.: The inversion problems of Möbius. Duke math. J. 3, 549–568 (1937). Hsu, L. T. C.: Abstract theory of inversion of iterated summation. Duke math. J. 14, 465 to 473 (1947). —: On Romanov's device of orthogonalization. Sci. Rep. Nat. Tsing Hua Univ. 5, 1–12 (1948). —: Note on an abstract inversion principle. Proc. Edinburgh math. Soc. (2) 9, 71–73 (1954). Jackson, F. H.: Series connected with the enumeration of partitions. Proc. London math. Soc., II. Ser. 1, 63–88 (1904). —: The q-form of Taylor's theorem. Messenger of Mathematics 38, 57–61 (1909). Jónsson, B.: Lattice-theoretic approach to projective and affine geometry. Symposium on the Axiomatic Method. Amsterdam, North-Holland Publishing Company, 1959, 188–205. —, and A. Tarski: Direct decomposition of finite algebraic systems. Notre Dame Mathematical lectures, no. 5. Indiana: Notre Dame 1947. Kac, M., and J. C. Ward: A combinatorial solution of the two-dimensional Ising model. Phys. Review 88, 1332–1337 (1952). Kaplanski, I., and J. Riordan: The problème des ménages. Scripta math. 12, 113–124 (1946). Klee, V.: The Euler characteristic in combinatorial geometry. Amer. math. Monthly 70, 119–127 (1963). Lazarson, T.: The representation problem for independence functions. J. London math. Soc. 33, 21–25 (1958). MacLane, S.: A lattice formulation of transcendence degrees and p-bases. Duke math. J. 4, 455–468 (1938). MacMillan, B.: Absolutely monotone functions. Ann. of Math., II. Ser. 60, 467–501 (1954). Möbius, A. F.: über eine besondere Art von Umkehrung der Reihen. J. reine angew. Math. 9, 105–123 (1832). Ore, O.: Theory of graphs. Providence: American Mathematical Society 1962. Polya, G.: Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen. Acta math. 68, 145–253 (1937). Rado, R.: Note on independence functions. Proc. London math. Soc., III. Ser. 7, 300–320 (1957). Read, R. C.: The enumeration of locally restricted graphs, I. J. London math. Soc. 34, 417 to 436 (1959). Redfield, J. H.: The theory of group-reduced distributions. Amer. J. Math. 49, 433–455 (1927). Revuz, André: Fonctions croissantes et mesures sur les espaces topologiques ordonnés. Ann. Inst. Fourier 6 187–268 (1955). Riordan, J.: An introduction to combinatorial analysis. New York: Wiley 1958. Romanov, N. P.: On a special orthonormal system and its connection with the theory of primes. Math. Sbornik, N. S. 16, 353–364 (1945). Rota, G.-C.: Combinatorial theory and Möbius functions. To appear in Amer. math. Monthly. - The number of partitions of a set. To appear in Amer. math. Monthly. Ryser, H. J.: Combinatorial Mathematics. Buffalo: Mathematical Association of America 1963. Schützenberger, M. P.: Contribution aux applications statistiques de la théorie de l'information. Publ. Inst. Stat. Univ. Paris, 3, 5–117 (1954). Tarski, A.: Ordinal algebras. Amsterdam: North-Holland Publishing Company 1956. Touchard, J.: Sur un problème de permutations. C. r. Acad. Sci., Paris, 198, 631–633 (1934). Tutte, W. T.: A contribution to the theory of chromatic polynomials. Canadian J. Math. 6, 80–91 (1953). —: A class of Abelian group. Canadian J. Math. 8, 13–28 (1956). —: A homotopy theorem for matroids, I. and II. Trans. Amer. math. Soc. 88, 144–140 (1958). —: Matroids and graphs. Trans. Amer. math. Soc. 90, 527–552 (1959). Ward, M.: The algebra of lattice functions. Duke math. J. 5, 357–371 (1939). Weisner, L.: Abstract theory of inversion of finite series. Trans. Amer. math. Soc. 38, 474–484 (1935). —: Some properties of prime-power groups. Trans. Amer. math. Soc. 38, 485–492 (1935). Whitney, H.: A logical expansion in mathematics. Bull. Amer. math. Soc. 38, 572–579 (1932). —: Characteristic functions and the algebra of logic. Ann. of Math., II. Ser. 34, 405–414 (1933). —: The abstract properties of linear dependence. Amer. J. Math. 57, 507–533 (1935). Wielandt, H.: Beziehungen zwischen den Fixpunktzahlen von Automorphismengruppen einer endliehen Gruppe. Math. Z. 73, 146–158 (1960). Wintner, A.: Eratosthenian Averages. Baltimore (privately printed) 1943.