Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The crank of partitions mod $ 8,\;9$ and $ 10$


Author: Frank G. Garvan
Journal: Trans. Amer. Math. Soc. 322 (1990), 79-94
MSC: Primary 11P83; Secondary 05A17, 11P81
DOI: https://doi.org/10.1090/S0002-9947-1990-1012520-8
MathSciNet review: 1012520
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Recently new combinatorial interpretations of Ramanujan's partition congruences modulo $ 5$, $ 7$ and $ 11$ were found. These were in terms of the crank. A refinement of the congruence modulo $ 5$ is proved. The number of partitions of $ 5n + 4$ with even crank is congruent to 0 modulo $ 5$. The residue of the even crank modulo $ 10$ divides these partitions into five equal classes. Other relations for the crank modulo $ 8$, $ 9$ and $ 10$ are also proved. The dissections of certain generating functions associated with these results are calculated. All of the results are proved by elementary methods.


References [Enhancements On Off] (What's this?)

  • [A] 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.) MR 0557013 (58:27738)
  • [A-G1] G. E. Andrews and F. G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Adv. in Math. 73 (1989), 242-255. MR 987276 (90d:11115)
  • [A-G2] -, Dyson's crank of a partition, Bull. Amer. Math. Soc. 18 (1988), 167-171. MR 929094 (89b:11079)
  • [A-S] A. O. L. Atkin and H. P. F. Swinnerton-Dyer, Some properties of partitions, Proc. London Math. Soc. (3) 4 (1954), 84-106. MR 0060535 (15:685d)
  • [Bia] A. J. F. Biagioli, A proof of some identities of Ramanujan using modular forms, Glasgow Math. J. 31 (1989), 271-295. MR 1021804 (90m:11060)
  • [Bir] B. J. Birch, A look back at Ramanujan's manuscripts, Math. Proc. Cambridge Philos. Soc. 78 (1975), 73-79. MR 0379372 (52:277)
  • [D1] F. J. Dyson, Some guesses in the theory of partitions, Eureka (Cambridge) 8 (1944), 10-15.
  • [D2] -, Missed opportunities, Bull. Amer. Math. Soc. 78 (1972), 635-652. MR 0522147 (58:25442)
  • [D3] -, 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] -, Mappings and symmetries of partitions, J. Combin. Theory Ser. A 51 (1989), 169-180. MR 1001259 (90f:05009)
  • [G1] F. G. Garvan, Some congruence properties of the partition function, M. Sc. thesis, University of New South Wales, 1982.
  • [G2] -, New combinatorial interpretations of Ramanujan's partition congruences $ \operatorname{mod} 5, 7$   and$ 11$, Trans. Amer. Math. Soc. 305 (1988), 47-77. MR 920146 (89b:11081)
  • [G3] -, 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.
  • [G-K-S] F. G. Garvan, D. Kim and D. Stanton, Cranks and $ t$-cores, Invent. Math. (to appear). MR 1055707 (91h:11106)
  • [G-S] F. G. Garvan, and D. Stanton, Sieved partition functions and $ q$-binomial coefficients, Math. Comp. (to appear). MR 1023761 (90j:11102)
  • [H-W] G. H. Hardy and E. M. Wright, An Introduction to the theory of numbers, Oxford Univ. Press, London, 1968.
  • [Hic1] D. Hickerson, A proof of the mock theta conjectures, Invent. Math. 94 (1988), 639-660. MR 969247 (90f:11028a)
  • [Hic2] -, On the seventh order mock theta functions, Invent. Math. 94 (1988), 661-677. MR 969248 (90f:11028b)
  • [Hir1] M. D. Hirschhorn, A simple proof of an identity of Ramanujan, J. Austral. Math. Soc. (Ser. A) 34 (1983), 31-35. MR 683175 (84h:10067)
  • [Hir2] -, A generalisation of Winquist's identity and a conjecture of Ramanujan, J. Indian Math. Soc. 51 (1987), 49-55. MR 988308 (90e:11152)
  • [Hir3] -, A birthday present to Ramanujan, Amer. Math. Monthly (to appear). MR 1048912 (91e:11116)
  • [Hir4] -, private communication.
  • [K-F] F. Klein and R. Fricke, Vorlesungen über die Theorie der elliptischen Modulfunktionen, Vol. 2, Teubner, Leipzig, 1892.
  • [M] I. G. Macdonald, Affine root systems and Dedekind's $ \eta $-function, Invent. Math. 15 (1972), 91-143. MR 0357528 (50:9996)
  • [Ra1] S. Ramanujan, Some properties of $ p(n)$, the number of partitions of $ n$, Paper $ 25$ of Collected Papers of S. Ramanujan, Cambridge Univ. Press, London and New York, 1927; reprinted: Chelsea, New York, 1962.
  • [Ra2] -, The lost notebook and other unpublished papers, with an introduction by G. E. Andrews, Narosa Publishing House, New Delhi, 1988, (North American and European distribution: Springer-Verlag). MR 947735 (89j:01078)
  • [Rø] Ø. Rødseth, Dissections of the generating functions of $ q(n)$ and $ {q_0}(n)$, Univ. Bergen Årb. Naturv. Serie 1969, No 13.
  • [St1] D. Stanton, Sign variations of the Macdonald identities, SIAM J. Math. Anal. 17 (1986), 1454-1460. MR 860926 (88c:17020)
  • [St2] -, An elementary approach to the Macdonald identities, $ q$-Series and Partitions, IMA Volumes in Math. and its Applications, Vol. 18, Springer-Verlag, New York, 1989. MR 1019844 (90j:05026c)
  • [Wa] G. N. Watson, Proof of certain identities in combinatory analysis, J. Indian Math. Soc. 20 (1933), 57-69.
  • [Wi] L. Winquist, An elementary proof of $ p(11m + 6) \equiv 0({\operatorname{mod}}11)$, J. Combin. Theory 6 (1969), 56-59. MR 0236136 (38:4434)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 11P83, 05A17, 11P81

Retrieve articles in all journals with MSC: 11P83, 05A17, 11P81


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1990-1012520-8
Keywords: Congruences, crank, dissections, generating functions, Macdonald identities, modular functions, partitions, quadratic forms, symmetry group, theta functions, Ramanujan
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society