A Survey of Multipartitions Congruences and Identities

Part of the Developments in Mathematics book series (DEVM, volume 17)


The concept of a multipartition of a number, which has proved so useful in the study of Lie algebras, is studied for its own intrinsic interest. Following up on the work of Atkin, we shall present an infinite family of congruences for P_k (n), the number of k-component


Mock Theta Function Umbral Calculus Ramanujan Identity Complete Residue System Pentagonal Number Theorem 
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  1. 1.
    G. E. Andrews, An analytic generalization of the Rogers-Ramanujan identities for odd moduli, Proc. Nat. Acad. Sci. USA, 71, (1974), 4082–4085.MATHCrossRefGoogle Scholar
  2. 2.
    G. E. Andrews, Problems and prospects for basic hypergeometric functions, from The Theory and Applications of Special Functions (R. Askey, ed.), pp. 191–224, Academic Press, New York, 1975.Google Scholar
  3. 3.
    G. E. Andrews, The Theory of Partitions, Encycl. of Math. and Its Appl. (G.-C. Rota, ed.), Vol. 2, Addison-Wesley, Reading, 1975 (Reprinted: Cambridge University Press, Cambridge, 1998).Google Scholar
  4. 4.
    G. E. Andrews, Multiple q-series identities, Houston Math. J., 7 (1981), 11–22.MATHGoogle Scholar
  5. 5.
    G. E. Andrews, Multiple series Rogers-Ramanujan identities, Pac. J. Math., 114 (1984), 267–283.MATHGoogle Scholar
  6. 6.
    G. E. Andrews, Umbral calculus, Bailey chains and pentagonal number theorems, J. Comb. Th., Ser. A, 91 (2000), 464–475.MATHCrossRefGoogle Scholar
  7. 7.
    G. E. Andrews and F. G. Garvan, Dyson’s crank of a partition, Bull. Amer. Math. Soc., 18 (1988), 167–171.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    G. E. Andrews and R. Roy, Ramanujan’s method in q-series congruences, Elec. J. Comb., 4(2): R2, 7pp., 1997.Google Scholar
  9. 9.
    A. O. L. Atkin, Proof of a conjecture of Ramanujan, Glasgow Math. J., 8 (1967), 14–32.MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    A. O. L. Atkin, Ramanujan congruences for p-k (n), Canad. J. Math., 20 (1968), 67–78.MATHGoogle Scholar
  11. 11.
    A. Berkovich, The tripentagonal number theorem and related identities, Int. J. Number Theory, (to appear).Google Scholar
  12. 12.
    P. Bouwknegt, Multipartitions, generalized Durfee squares and affine Lie algebra characters, J. Austral. Math. Soc., 72 (2002), 395–408.MATHMathSciNetGoogle Scholar
  13. 13.
    M. Broueacute; and G. Malle, Zyklotomische Heckealgebren, Asteacute;risque, 212 (1993), 119–189.Google Scholar
  14. 14.
    M. S. Cheema and C. T. Haskell, Multirestricted and rowed partitions, Duke Math. J., 34 (1967), 443–451.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    F. J. Dyson, Some guesses in the theory of partitions, Eureka (Cambridge), 8 (1944), 10–15.Google Scholar
  16. 16.
    M. Fayers, Weights of multipartitions and representations of Ariki-Koike algebras, Adv. in Math., 206 (2006), 112–144.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    F. G. Garvan, New combinatorial interpretations of Ramanujan’s partition congruences mod 5, 7 and 11, Trans. Amer. Math. Soc., 305 (1988), 47–77.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    F. G. Garvan, Ranks and cranks for bipartitions \mod 5, (in preparation).Google Scholar
  19. 19.
    G. Gasper and N. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge, 1990.MATHGoogle Scholar
  20. 20.
    H. Gupta, Selected Topics in Number Theory, Abacus Press, Turnbridge Wells, 1980.MATHGoogle Scholar
  21. 21.
    H. Gupta, C. E. Gwyther, and J. C. P. Miller, Tables of Partitions, Royal Society Math. Tables, Vol. 4, Cambridge University Press, Cambridge, 1958.Google Scholar
  22. 22.
    G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 4th ed., Oxford University Press, London, 1960.MATHGoogle Scholar
  23. 23.
    K. Mahlburg, Partition congruences and the Andrews-Garvan-Dyson crank, Proc. Nat. Acad. Sci., U.S.A, 102 (2005), 15373–15376.MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    P. Paule, On identities of the Rogers–Ramanujan type, J. Math. Anal. and Appl., 107 (1985), 255–284.MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    S. Ramanujan, The Lost Notebook and Other Unpublished Papers, intro. by G. E. Andrews}, Narosa, New Delhi, 1987.Google Scholar
  26. 26.
    L. J. Rogers, On two theorems of combinatory analysis and allied identities, Proc. London Math. Soc. (2), 16 (1917), 315–336.Google Scholar
  27. 27.
    L. J. Slater, Further identities of the Rogers-Ramanujan type, Proc. London Math. Soc. (2), 54 (1952), 147–167.MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    J. J. Sylvester, A constructive theory of partitions arranged in three acts, an interact and an exodion, Amer. J. Math., 5 (1882), 251–330.CrossRefMathSciNetGoogle Scholar
  29. 29.
    G. N. Watson, The mock theta functions (2), Proc. London Math. Soc., Ser. 2, 42 (1937), 274–304.CrossRefGoogle Scholar
  30. 30.
    G. N. Watson, A note on Lerch’s functions, Quart. J. Math., Oxford Series, 8 (1937), 44–47.Google Scholar

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© Springer-Verlag New York 2008

Authors and Affiliations

  1. 1.The Pennsylvania State University, University ParkUSA

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