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, Volume 84, Issue 1, pp 343–359 | Cite as

Generalizations of Dyson's rank and non-Rogers-Ramanujan partitions

  • Frank G. Garvan
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References

  1. [A1] G. E. Andrews, Problems and prospects for basic hypergeometric series, “The Theory and Application of Special Functions”, Academic, New York, 1975.Google Scholar
  2. [A2] G. E. Andrews, “The Theory of Partitions”, Encyclopedia of Mathematics and Its Applications, Vol. 2(G.-C. Rota, ed.), Addison-Wesley, Reading, Mass., 1976. (Reissued: Cambridge Univ. Press, London and New York, 1985).Google Scholar
  3. [A3] G. E. Andrews, “Partitions: Yesterday and Today”, New Zealand Math. Soc., Wellington, 1979.MATHGoogle Scholar
  4. [A4] G. E. Andrews, Partitions and Durfee dissection,Amer. J. Math. 101 (1979), 735–742.MATHCrossRefMathSciNetGoogle Scholar
  5. [A5] G. E. Andrews, Multiple series Rogers-Ramanujan type identities,Pacific J. Math. 114 (1984), 267–283.MATHMathSciNetGoogle Scholar
  6. [A6] G. E. Andrews, “q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics and Computer Algebra”, CBMS Regional Conf. Series in Math., No. 66, Amer. Math. Soc., Providence, 1986.Google Scholar
  7. [A7] G. E. Andrews, J. J. Sylvester, Johns Hopkins and partitions, “A Century of Mathematics in America, Part I”,Hist. Math., 1, Amer. Math. Soc., Providence, 1988.Google Scholar
  8. [A-G1] G. E. Andrews and F. G. Garvan, Dyson's crank of a partition,Bull. Amer. Math. Soc. 18 (1988), 167–171.MATHMathSciNetCrossRefGoogle Scholar
  9. [A-G2] G. E. Andrews and F. G. Garvan, Ramanujan's “lost” notebook VI: The mock theta conjectures,Advances in Math. 73 (1989), 242–255.MATHCrossRefMathSciNetGoogle Scholar
  10. [A-SD] A.O.L. Atkin and H.P.F. Swinnerton-Dyer, Some properties of partitions,Proc. London Math. Soc. (3)4 (1954), 84–106.MATHMathSciNetGoogle Scholar
  11. [D1] F. J. Dyson, Some guesses in the theory of partitions,Eureka (Cambridge) 8 (1944), 10–15.Google Scholar
  12. [D2] F. J. Dyson, A new symmetry for partitions,J. Combin. Theory 7 (1969), 56–61.MATHMathSciNetGoogle Scholar
  13. [D3] F. J. Dyson, A walk through Ramanujan's garden, “Ramanujan Revisted: Proc. of the Centenary Conference”, Univ. of Illinois at Urbana-Champaign, June 1–5, 1987, Academic Press, San Diego, 1988.Google Scholar
  14. [D4] F. J. Dyson, Mappings and symmetries of partitions,J. Combin. Theory Ser. A 51 (1989), 169–180.MATHCrossRefMathSciNetGoogle Scholar
  15. [Ga1] F. G. Garvan, “Generalizations of Dyson's Rank”, Ph. D. thesis, Pennsylvania State University, 1986.Google Scholar
  16. [Ga2] 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
  17. [Ga3] F. G. Garvan, Combinatorial interpretations of Ramanujan's partition congruences, “Ramanujan Revisted: Proc. of the Centenary Conference”, Univ. of Illinois at Urbana-Champaign, June 1–5, 1987, Academic Press, San Diego, 1988.Google Scholar
  18. [Ga4] F. G. Garvan, The crank of partitions mod 8, 9 and 10,Trans. Amer. Math. Soc. 322 (1990), 79–94.MATHCrossRefMathSciNetGoogle Scholar
  19. [G-K-S] F. G. Garvan, D. Kim and D. Stanton, Cranks andt-cores,Inventiones math. 101 (1990), 1–17.MATHCrossRefMathSciNetGoogle Scholar
  20. [Go] B. Gordon, A combinatorial generalization of the Rogers-Ramanujan identities,Amer. J. Math. 83 (1961), 393–399.MATHCrossRefMathSciNetGoogle Scholar
  21. [H1] D. Hickerson, A proof of the mock theta conjectures,Invent. Math. 94 (1988), 639–660.MATHCrossRefMathSciNetGoogle Scholar
  22. [H1] D. Hickerson, On the seventh order mock theta functions,Invent. Math. 94 (1988), 661–677.MATHCrossRefMathSciNetGoogle Scholar
  23. [L1] R. P. Lewis, On some relations between the rank and the crank,J. Combin. Theory Ser. A 59 (1992), 104–110.MATHCrossRefMathSciNetGoogle Scholar
  24. [L2] R. P. Lewis, On the ranks of partitions modulo 9,Bull. London Math. Soc. 23 (1991), 417–421.MATHMathSciNetGoogle Scholar
  25. [L3] R. P. Lewis, Relations between the rank and the crank moduli 9,?London Math. Soc. 45 (1992), 222–231.MATHMathSciNetGoogle Scholar
  26. [L-SG1] R. P. Lewis and N. Santa-Gadea, On the rank and the crank moduli 4 and 8,Trans. Amer. Math. Soc., to appear.Google Scholar
  27. [SG1] N. Santa-Gadea, “On the Rank and Crank Moduli 8, 9 and 12”, Ph. D. thesis, Pennsylvania State University, 1990.Google Scholar
  28. [SG2] N. Santa-Gadea, On some relations for the rank moduli 9 and 12,J. Number Theory 40 (1992), 130–145.CrossRefMathSciNetGoogle Scholar
  29. [S1] L. J. Slater, A new proof of Rogers's transformations of infinite series,Proc. London Math. Soc. (2) (1951), 460–475.MathSciNetGoogle Scholar
  30. [S2] L. J. Slater, Further identities of the Rogers-Ramanujan type,Proc. London Math. Soc. (2)54 (1952), 147–167.MATHMathSciNetGoogle Scholar
  31. [S-W] D. Stanton and D. White, “Constructive Combinatorics”, Springer-Verlag, New York, 1986.MATHGoogle Scholar
  32. [W1] G. N. Watson, A new proof of the Rogers-Ramanujan identities,J. London Math. Soc. 4 (1929), 4–9.CrossRefGoogle Scholar
  33. [W2] G. N. Watson, The final problem: An account of the mock theta functions, J. London Math. Soc.11 (1936), 55–80.MATHGoogle Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Frank G. Garvan
    • 1
  1. 1.Departmet of MathematicsUniversity of FloridaGainesvilleUSA

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