References
[A1] G. E. Andrews, Problems and prospects for basic hypergeometric series, “The Theory and Application of Special Functions”, Academic, New York, 1975.
[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).
[A3] G. E. Andrews, “Partitions: Yesterday and Today”, New Zealand Math. Soc., Wellington, 1979.
[A4] G. E. Andrews, Partitions and Durfee dissection,Amer. J. Math. 101 (1979), 735–742.
[A5] G. E. Andrews, Multiple series Rogers-Ramanujan type identities,Pacific J. Math. 114 (1984), 267–283.
[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.
[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.
[A-G1] G. E. Andrews and F. G. Garvan, Dyson's crank of a partition,Bull. Amer. Math. Soc. 18 (1988), 167–171.
[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.
[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.
[D1] F. J. Dyson, Some guesses in the theory of partitions,Eureka (Cambridge) 8 (1944), 10–15.
[D2] F. J. Dyson, A new symmetry for partitions,J. Combin. Theory 7 (1969), 56–61.
[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.
[D4] F. J. Dyson, Mappings and symmetries of partitions,J. Combin. Theory Ser. A 51 (1989), 169–180.
[Ga1] F. G. Garvan, “Generalizations of Dyson's Rank”, Ph. D. thesis, Pennsylvania State University, 1986.
[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.
[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.
[Ga4] F. G. Garvan, The crank of partitions mod 8, 9 and 10,Trans. Amer. Math. Soc. 322 (1990), 79–94.
[G-K-S] F. G. Garvan, D. Kim and D. Stanton, Cranks andt-cores,Inventiones math. 101 (1990), 1–17.
[Go] B. Gordon, A combinatorial generalization of the Rogers-Ramanujan identities,Amer. J. Math. 83 (1961), 393–399.
[H1] D. Hickerson, A proof of the mock theta conjectures,Invent. Math. 94 (1988), 639–660.
[H1] D. Hickerson, On the seventh order mock theta functions,Invent. Math. 94 (1988), 661–677.
[L1] R. P. Lewis, On some relations between the rank and the crank,J. Combin. Theory Ser. A 59 (1992), 104–110.
[L2] R. P. Lewis, On the ranks of partitions modulo 9,Bull. London Math. Soc. 23 (1991), 417–421.
[L3] R. P. Lewis, Relations between the rank and the crank moduli 9,?London Math. Soc. 45 (1992), 222–231.
[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.
[SG1] N. Santa-Gadea, “On the Rank and Crank Moduli 8, 9 and 12”, Ph. D. thesis, Pennsylvania State University, 1990.
[SG2] N. Santa-Gadea, On some relations for the rank moduli 9 and 12,J. Number Theory 40 (1992), 130–145.
[S1] L. J. Slater, A new proof of Rogers's transformations of infinite series,Proc. London Math. Soc. (2) (1951), 460–475.
[S2] L. J. Slater, Further identities of the Rogers-Ramanujan type,Proc. London Math. Soc. (2)54 (1952), 147–167.
[S-W] D. Stanton and D. White, “Constructive Combinatorics”, Springer-Verlag, New York, 1986.
[W1] G. N. Watson, A new proof of the Rogers-Ramanujan identities,J. London Math. Soc. 4 (1929), 4–9.
[W2] G. N. Watson, The final problem: An account of the mock theta functions, J. London Math. Soc.11 (1936), 55–80.
Author information
Authors and Affiliations
Additional information
The author was supported in part by NSF Grant DMS-9208813
This article was processed by the author using the Springer-Verlag TEX QPMZGHB macro package 1991.
Rights and permissions
About this article
Cite this article
Garvan, F.G. Generalizations of Dyson's rank and non-Rogers-Ramanujan partitions. Manuscripta Math 84, 343–359 (1994). https://doi.org/10.1007/BF02567461
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02567461