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Inventiones mathematicae

, Volume 101, Issue 1, pp 1–17 | Cite as

Cranks andt-cores

  • Frank Garvan
  • Dongsu Kim
  • Dennis Stanton
Article

Summary

New statistics on partitions (calledcranks) are defined which combinatorially prove Ramanujan's congruences for the partition function modulo 5, 7, 11, and 25. Explicit bijections are given for the equinumerous crank classes. The cranks are closely related to thet-core of a partition. Usingq-series, some explicit formulas are given for the number of partitions which aret-cores. Some related questions for self-conjugate and distinct partitions are discussed.

Keywords

Partition Function Explicit Formula Related Question Distinct Partition Function Modulo 
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.

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References

  1. 1.
    Andrews, G.E.: The theory of partitions. Encyclopedia of Mathematics and Its Applications, Vol. 2. Rota, G.-C. (ed.), Reading, MA:, Addison-Wesley 1976 (reissued by Cambridge Univ. Press, London and New York, 1985)Google Scholar
  2. 2.
    Andrews, G.E.: Applications of basic hypergeometric functions SIAM Rev.16, 441–484 (1975)Google Scholar
  3. 3.
    Andrews, G.E., Garvan, F.G.: Dyson's crank of a partition. Bull. Am. Math. Soc.18, 167–171 (1988)Google Scholar
  4. 4.
    Atkin, A.O.L.: Proof of a conjecture of Ramanujan. Glasgow Math. J.8, 14–32 (1967)Google Scholar
  5. 5.
    Atkin, A.O.L., Swinnerton-Dyer, P.: Some properties of partitions. Proc. Lond. Math. Soc., III. Ser.4, 84–106 (1954)Google Scholar
  6. 6.
    Bailey, W.N.: A note on two of Ramanujan's formulae. Q. J. Math. Oxf. II. Ser.,3, 29–31 (1952)Google Scholar
  7. 7.
    Dyson, F.: Some guesses in the theory of partitions. Eureka (Cambridge),8, 10–15 (1944)Google Scholar
  8. 8.
    Fine, N.: On a system of modular functions connected with the Ramanujan identities. Tohoku Math. J.8, 149–164 (1956)Google Scholar
  9. 9.
    Garvan, F.: New combinatorial interpretations of Ramanujan's partition congruences mod 5, 7 and 11. Trans. Am. Math. Soc.305, 47–77 (1988)Google Scholar
  10. 10.
    Garvan, F.: The crank of partitions mod 8, 9 and 10. PreprintGoogle Scholar
  11. 11.
    Garvan, F., Stanton, D.: Sieved partition functions andq-binomial coefficients. Math. Comput. (to appear)Google Scholar
  12. 12.
    Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers. London: Oxford Univ. Press 1979Google Scholar
  13. 13.
    James, G., Kerber, A.: The Representation Theory of the Symmetric Group. Reading, MA: Addison-Wesley 1981Google Scholar
  14. 14.
    Koblitz, N.: Introduction to Elliptic Curves and Modular Forms. New York: Springer, 1984Google Scholar
  15. 15.
    Kolberg, O.: Some identities involving the partition function. Math. Scand5, 77–92, (1957)Google Scholar
  16. 16.
    Kolberg, O.: An elementary discussion of certain modular forms. Univ. Bergen Arb. naturv. r. Nr. 19. (1959)Google Scholar
  17. 17.
    Morris, A., Yaseen, K.: Some combinatorial results for shifted Young diagrams. Math. Proc. Camb. Philos. Soc.,99, 23–31 (1986)Google Scholar
  18. 18.
    Olsson, J.: Frobenius symbols for partitions and degrees of spin characters. Math. Scand.61, 223–247 (1987)Google Scholar
  19. 19.
    Ramanujan, S.: Collected Papers of S. Ramanujan. London, New York: Cambridge Univ. Press 1927 (reprinted by Chelsea, New York, 1962)Google Scholar
  20. 20.
    Watson, G.N.: Ramanujans Vermutung über Zerfällungsanzahlen. J. Reine Angew. Math.179, pp. 97–128 (1938)Google Scholar
  21. 21.
    Watson, G.N.: Proof of certain identities in combinatory analysis. J. Ind. Math. Soc.20, 57–69 (1933)Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Frank Garvan
    • 1
  • Dongsu Kim
    • 2
  • Dennis Stanton
  1. 1.School of Mathematics, Physics, Computing and ElectronicsMacquarie UniversitySydneyAustralia
  2. 2.School of MathematicsUniversity of MinnesotaMinneapolisUSA

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