Skip to main content
SpringerLink
Log in

On the Foundations of Combinatorial Theory

I. Theory of Möbius Functions

  • Chapter
Classic Papers in Combinatorics

Part of the book series: Modern Birkhäuser Classics ((MBC))

Abstract

One of the most useful principles of enumeration in discrete probability and combinatorial theory is the celebrated principle of·inclusion-exclusion (ef. Feller*, FrÉchet, Riordan, Ryser). When skillfully applied, this principle has yielded the solution to many a combinatorial problem. Its mathematical foundations were thoroughly investigated not long ago in a monograph by FrÉchet, and it might at first appear that, after such exhaustive work, little else could be said on the subject.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Auslander. L., and H. M. Trent: Incidence matrices and linear graphs. J. Math. Mech. 8. 827–835 (1959).

    MATH  MathSciNet  Google Scholar 

  • Bell, E. T.: Algebraic Arithmetic. New York: Amer. Math. Soc. (1927),

    MATH  Google Scholar 

  • Bell, E. T.: Exponential polynomials. Ann. of Math., II. Ser. 35, 258–277 (1934).

    Article  Google Scholar 

  • Berge, C.: Théorie des graphes et ses applications. Paris: Dounod 1958.

    Google Scholar 

  • Birkhoff, Garrett: Lattice Theory, third preliminary edition. Harvard University. 1963.

    Google Scholar 

  • Birkhoff, Garrett: Lattice Theory, revised edition. American Mathematical Society, 1948.

    Google Scholar 

  • Birkhoff, G. D.: A determinant formula for the number of ways of coloring a map. Ann. of Math., II. Ser. 14. 42–46 (1913).

    Article  Google Scholar 

  • Birkhoff, G. D., and D. C. Lewis: Chromatic polynomials. Trans. Amer. math. Soc. 60, 355–451 (1946).

    MATH  MathSciNet  Google Scholar 

  • Bletcher, M. N., and G. B. Preston: Abstract linear dependence relations. Publ. Math., Debrecen 8, 55–63 (1961).

    MathSciNet  Google Scholar 

  • Bougayev, N. V.: Theory of numerical derivatives. Moscow, 1870–1873, pp. 1–222.

    Google Scholar 

  • Bruijn, N. G. DE: Generalization of Polya’s fundamental theorem in enumerative combinatorial analysis. Indagationes math. 21, 59–69 (1959).

    Google Scholar 

  • 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).

    Article  MATH  MathSciNet  Google Scholar 

  • Dedekind, R.: Gesammelte Mathematische Werke, volls. I–II–111. Hamburg: Deutsche Math. Verein. (1930).

    Google Scholar 

  • Delsarte, S.: Fonctions de Möbius sur les groupes abéliens finis. Ann. of Math., II. Ser. 49, 600–609 (1948).

    Article  MathSciNet  Google Scholar 

  • Dilworth, R. P.: Proof of a conjecture on finite modular lattices. Ann. of Math.. II. Ser. 60, 359–364 (1954).

    Article  MathSciNet  Google Scholar 

  • Dirac, G. A.: On the four-color conjecture. Proc. London math. Society, III. Ser. 13, 193 to 218 (1963).

    Article  MATH  MathSciNet  Google Scholar 

  • Dowker, C. H.: Homology groups of relations. Ann. of Math., II. Ser. 56, 84–95 (1952).

    Article  MathSciNet  Google Scholar 

  • 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.

    Google Scholar 

  • Eilenberg. S., and N. Steenrod: Foundations of algebraic topology. Princeton: University Press 1952.

    MATH  Google Scholar 

  • Fary, I.: On straight-line representation of planar graphs. Acta Sei. math. Szeged 11. 229–233 (1948).

    MathSciNet  Google Scholar 

  • Feller, W.: An introduction to probability theory and its applications, second edition. New York: Wiley 1960.

    Google Scholar 

  • Franklin, P.: The four-color problem. Amer. J. Math. 44, 225–236 (1922).

    Article  MATH  MathSciNet  Google Scholar 

  • Fréchet, M.: Les probabilités associées à un système d’évenements compatibles et dépendants. Actualitées scientifiques et industrielles, nos. 859 et 942. Paris: Hermann 1940 et 1943.

    Google Scholar 

  • Frontera Marqués, B.: Una funcio´n numériea en los reticulos finitos que se anula para Ion retículos reducibles. Actas de la 2a, Reunión de matemáticos españoles. Zaragoza 103- III 1962.

    Google Scholar 

  • Frucht, R., and G.-C. Rota: La funeión de Möbius para el retículo di particiones de un conjunto finito. To appear in Scientia (Chile).

    Google Scholar 

  • Goldberg, K., M.S. Green and R. E. Nettleton: Dense subgraphs and connectivity. Canadian J. Math. 11 (1959).

    Google Scholar 

  • Golomb. S. W.: A mathematical theory of discrete classification. Fourth Symposium in Information Theory, London, 1961.

    Google Scholar 

  • Green, M. S., and R. E. Nettleton: Möbius function on the lattice of dense subgraphs. J. Res. nat. Bur. Standards 64B, 41–47 (1962).

    MathSciNet  Google Scholar 

  • Green, M. S., and R. E. Nettleton: Expression in terms of modular distribution functions for the entropy density in an infinite system. J. Chemical Physisc 29, 1365-1370 (1958).

    Article  Google Scholar 

  • Hadwiger, H.: Eulers Charakteristik und kombinatorische Geometrie. J. reine angew. Math. 194, 101-110 (1955).

    MATH  MathSciNet  Google Scholar 

  • Hall, Philip: A contribution to the theory of groups of prime power order. Proc. London math. Soc., II. Ser. 36, 39–95 (1932).

    Google Scholar 

  • Hall, Philip: The Eulerian functions of a group. Quart. J. Math. Oxford Ser. 134–151, 1936.

    Google Scholar 

  • Harary. F.: Unsolved problems in the enumeration of graphs. Publ. math. Inst. Hungar Acad. Sci. 5, 63–95 (1960).

    MATH  Google Scholar 

  • Hardy, G. H.: Ramanujan. Cambridge: University Press 1940.

    Google Scholar 

  • Hardy, G. H., and E. M. Wright: An introduction to the theory of numbers, Oxford: University Press 1954.

    MATH  Google Scholar 

  • Hartmants, J.: Lattice theory of generalized partitions. Canadian J. Math. II. 97–106 (1959).

    Google Scholar 

  • Hille. E.: The inversion problems of Möbius. Duke math. J. 3, 549–568 (1937).

    Article  MathSciNet  Google Scholar 

  • Hsu, L. T. C.: Abstract theory of inversion of iterated summation. Duke math. J. 14. 465 to 473 (1947).

    Article  MATH  MathSciNet  Google Scholar 

  • Hsu, L. T. C.: On Romanov’s device of orthogonalization. Sci. Rep. Nat. Tsing Hua Univ. 5, 1–12 (1948).

    MathSciNet  Google Scholar 

  • Hsu, L. T. C.: Note on an abstract inversion principle. Proe. Edinburgh math. Soc. (2) 9, 71–73 (1954).

    Google Scholar 

  • Jackson, F. H.: Series connected with the enumeration of partitions. Proc. London math. Soc., II. Ser. 1, 63—88 (1904).

    Google Scholar 

  • Jackson, F. H.: The q-form of Taylor’s theorem. Messenger of Mathematics 38. 57—61 (1909).

    Google Scholar 

  • Jónsson, B.: Lattice-theoretic approach to protective and affine geometry. Symposium on the Axiomatic Method. Amsterdam, North-Holland Publishing Company, 1959, 188 — 205.

    Google Scholar 

  • Jónsson, B. and A. Tarski: Direct decomposition of finite algebraic systems. Notre Dame Mathematical lectures, no. 5. Indiana: Notre Dame 1947.

    Google Scholar 

  • Kac. M., and J. C. Ward: A combinatorial solution of the two-dimensional Ising model. Phvs. Review 88, 1332—1337 (1952).

    Article  MATH  Google Scholar 

  • Kaplanski, I., and J. Riordan: The probleme des menages. Scripta math. 12, 113—124 (1946).

    MathSciNet  Google Scholar 

  • Klee, V.: The Kuler characteristic in combinatorial geometry. Amer. math. Monthly 70, 119—127 (1963).

    Article  MATH  MathSciNet  Google Scholar 

  • Lazarson, T.: The representation problem for independence functions. J. London math. Soc. 88, 21-25 (1958).

    Article  MathSciNet  Google Scholar 

  • MacLane, S.: A lattice formulation of transcendence degrees and p-bases. Duke math. J. 4, 455-468 (1938).

    Article  MATH  MathSciNet  Google Scholar 

  • MacMillan, B.: Absolutely monotone functions. Ann. of Math., II. Ser. 60, 467-501 (1954).

    Article  Google Scholar 

  • Möbius, A. F.: Über eine besondere Art von Umkehrung der Reihen. J. reine angew. Math. 9, 105-123 (1832).

    Google Scholar 

  • Ore, O.: Theory of graphs. Providence: American Mathematical Society 1962.

    MATH  Google Scholar 

  • Polya, G.: Kombinatorische Anzahlbestimmungen für Gruppen, Graphen lind chemische Verbindungen. Acta math. 68, 145-253 (1937).

    Article  MATH  Google Scholar 

  • Rado, R.: Note on independence functions. Proc. London math. Soc, III. Ser. 7, 300–320 (1957).

    Google Scholar 

  • Read, R. C.: The enumeration of locally restricted graphs, I. J. London math. Soc. 34, 417 to 436 (1959).

    Article  MATH  MathSciNet  Google Scholar 

  • Redfield, J. H.: The theory of group-reduced distributions. Amer. J. Math. 49, 433-455 (1927).

    Article  MATH  MathSciNet  Google Scholar 

  • Revuz. André: Fonctions croissantes et mesures sur les espaces topologiques ordonnés. Ann. Inst. Fourier 6 187-268 (1955).

    Google Scholar 

  • Riordan, J.: An introduction to combinatorial analysis. New York: Wiley 1958.

    MATH  Google Scholar 

  • Romanov, N. P.: On a special orthonormal system and its connection with the theory of primes. Math. Sbornik, N. S. 16, 353–364 (1945).

    Google Scholar 

  • Rota, G.-C,: Combinatorial theory and Möbius functions. To appear in Amer. math. Monthly.

    Google Scholar 

  • Rota, G.-C,: The number of partitions of a set. To appear in Amer. math. Monthly.

    Google Scholar 

  • Ryser, H. J.: Combinatorial Mathematics. Buffalo: Mathematical Association of America 1963.

    MATH  Google Scholar 

  • Schützenberoer. M. P.: Contribution aux applications statistiques de la théorie de l’infor-mation. Publ. Inst. Stat.. Univ. Paris, 3, 5–117 (1954).

    Google Scholar 

  • Tarski, A.: Ordinal algebras. Amsterdam: North-Holland Publishing Company 1956.

    MATH  Google Scholar 

  • Touchard. J.: Sur un probléme de permutations. C. r. Acad. Sci., Paris, 198, 631—633 (1934).

    Google Scholar 

  • Tutte, W. T.: A contribution to the theory of chromatic polynomials. Canadian J. Math. 6, 80–91 (1953).

    MathSciNet  Google Scholar 

  • A class of Ahcliiin group. Canadian. J. Math. 8. 13–28 (1956).

    Google Scholar 

  • A hoinotopy theorem for matroids. I. and II. Trans. Amer. math. Soc. 88, 144–144 (1958).

    Google Scholar 

  • Matroids and graphs. Trans. Amer. math. Soc. 90. 527–552 (1959).

    Google Scholar 

  • Ward. M.: The algebra of lattice functions. Duke math. J. 5, 357–371 (1939).

    Article  MathSciNet  Google Scholar 

  • Weisner, L.: Abstract theory of inversion of finite series. Trans. Amer. math. Soc. 38, 474–484 (1935).

    MATH  MathSciNet  Google Scholar 

  • Weisner, L.: Some properties of prime-power groups. Trans. Amer. math. Soc. 38, 485—492 (1935).

    MATH  MathSciNet  Google Scholar 

  • Whitney, H.: A logical expansion in mathematics. Bull. Amer. math. Soc. 38, 572–579 (1932).

    Article  MathSciNet  Google Scholar 

  • Whitney, H.: Characteristic functions and the algebra of logic. Ann. of Math., II. Ser. 34, 405–414 (1933).

    Article  MathSciNet  Google Scholar 

  • Whitney, H.: The abstract properties of linear dependence. Amer. J. Math. 57, 507–533 (1935).

    Article  MathSciNet  Google Scholar 

  • Wielandt, H.: Beziehungen zwischen den Fixpunktzahlen von Automorphismengruppen einer endlichen Gruppe. Math. Z. 73. 146–158 (1960).

    Article  MATH  MathSciNet  Google Scholar 

  • Wintner, A.: Eratosthenian Averages. Baltimore (privately printed) 1943.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, Cambridge 39, Massachusetts, USA

    Gian-Carlo Rota

Authors
  1. Gian-Carlo Rota
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, Brandeis University, Waltham, MA, 02454, USA

    Ira Gessel

  2. Department of Mathematics, Massachusetts Institute of Technology (MIT), Cambridge, MA, 02139, USA

    Gian-Carlo Rota

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Birkhäuser Boston, a part of Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Rota, GC. (2009). On the Foundations of Combinatorial Theory. In: Gessel, I., Rota, GC. (eds) Classic Papers in Combinatorics. Modern Birkhäuser Classics. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4842-8_25

Download citation

  • DOI: https://doi.org/10.1007/978-0-8176-4842-8_25

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-0-8176-4841-1

  • Online ISBN: 978-0-8176-4842-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation