The Ramanujan Journal

, Volume 7, Issue 1–3, pp 343–366 | Cite as

Relations Between the Ranks and Cranks of Partitions

  • A.O.L. Atkin
  • F.G. Garvan


New identities and congruences involving the ranks and cranks of partitions are proved. The proof depends on a new partial differential equation connecting their generating functions.

partitions rank crank Ramanujan congruences Eisenstein series modular forms 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    G.E. Andrews, The Theory of Partitions, Encyclopedia of Mathematics and its Applications, Vol. 2 (G.-C. Rota, ed.), Addison-Wesley, Reading, MA, 1976. (Reissued: Cambridge Univ. Press, London and New York, 1985).Google Scholar
  2. 2.
    G.E. Andrews and F.G. Garvan, “Dyson's crank of a partition,” Bull. Amer. Math. Soc. (N.S.) 18 (1988), 167-171.Google Scholar
  3. 3.
    T.M. Apostol, Modular Functions and Dirichlet Series in Number Theory, 2nd. edn., Springer, New York, 1990.Google Scholar
  4. 4.
    A.O.L. Atkin and S.M. Hussain, “Some properties of partitions II,” Trans. Amer. Math. Soc. 89 (1958), 184-200.Google Scholar
  5. 5.
    A.O.L. Atkin and P. Swinnerton-Dyer, “Some properties of partitions,” Proc. London Math. Soc. 4 (1954), 84-106.Google Scholar
  6. 6.
    B.C. Berndt, Ramanujan's Notebooks, Part II, Springer, New York, 1989.Google Scholar
  7. 7.
    A. Berkovich and F.G. Garvan, “Some observations on Dyson's new symmetries of partitions,” J. Combin. Theory Ser. A 100 (2002), 61-93.Google Scholar
  8. 8.
    F.J. Dyson “Some guesses in the theory of partitions,” Eureka (Cambridge) 8 (1944), 10-15.Google Scholar
  9. 9.
    F.J. Dyson, “Mappings and symmetries of partitions,” J. Combin. Theory Ser. A 51 (1989), 169-180.Google Scholar
  10. 10.
    F.J. Dyson, “Selected papers of Freeman Dyson with commentary,” Amer. Math. Soc., Providence, RI, 1996.Google Scholar
  11. 11.
    F.G. Garvan, “New combinatorial interpretations of Ramanujan's partition congruences mod 5, 7 and 11,” Trans. Amer. Math. Soc. 305 (1988), 47-77.Google Scholar
  12. 12.
    F.G. Garvan, “Combinatorial interpretations of Ramanujan's partition congruences,” in “Ramanujan Revisited: Proc. of the Centenary Conference,” Univ. of Illinois at Urbana-Champaign, June 1-5, 1987, Academic Press, San Diego, 1988.Google Scholar
  13. 13.
    F.G. Garvan, “The crank of partitions mod 8, 9 and 10,” Trans. Amer. Math. Soc. 322 (1990), 79-94.Google Scholar
  14. 14.
    M. Kaneko and D.B. Zagier, “A generalized Jacobi theta function and quasimodular forms,” in “The moduli space of curves,” Progr. Math., 129, Birkhauser, Boston, MA, 1995, 165-172.Google Scholar
  15. 15.
    N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer, New York, 1993.Google Scholar
  16. 16.
    R.P. Lewis, “On some relations between the rank and the crank,” J. Combin. Theory Ser. A 59 (1992), 104-110.Google Scholar
  17. 17.
    R.P. Lewis, “On the ranks of partitions modulo 9,” Bull. London Math. Soc. 23 (1991), 417-421.Google Scholar
  18. 18.
    R.P. Lewis, “Relations between the rank and the crank modulo 9,” J. London Math. Soc. 45 (1992), 222-231.Google Scholar
  19. 19.
    R.P. Lewis and N. Santa-Gadea, “On the rank and the crank moduli 4 and 8,” Trans. Amer. Math. Soc. 341 (1994), 449-465.Google Scholar
  20. 20.
    J.N. O'Brien, “Some properties of partitions with special reference to primes other than 5, 7 and 11,” Ph.D. thesis, Univ. of Durham, England, 1966.Google Scholar
  21. 21.
    S. Ramanujan, “On certain arithmetic functions,” Trans Cambridge Philos. Soc. XXII (1916), 159-184.Google Scholar
  22. 22.
    R.A. Rankin, Modular Forms and Functions, Cambridge Univ. Press, Cambridge, 1977.Google Scholar
  23. 23.
    N. Santa-Gadea, “On the rank and crank moduli 8, 9 and 12,” Ph.D. thesis, Pennsylvania State University, 1990.Google Scholar
  24. 24.
    N. Santa-Gadea, “On some relations for the rank moduli 9 and 12,” J. Number Theory 40 (1992), 130-145.Google Scholar
  25. 25.
    J.-P. Serre, A Course in Arithmetic, Springer, New York, 1973.Google Scholar
  26. 26.
    H.P.F. Swinnerton-Dyer, “On l-adic representations and congruences for coefficients of modular forms,” Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture Notes in Math., vol. 476, Springer, Berlin, 1975, pp. 1-55.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • A.O.L. Atkin
    • 1
  • F.G. Garvan
    • 2
  1. 1.Department of MathematicsUniversity of Illinois at ChicagoChicago
  2. 2.Department of MathematicsUniversity of FloridaGainesville

Personalised recommendations