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)

 
 

 

New combinatorial interpretations of Ramanujan's partition congruences mod $ 5,7$ and $ 11$


Author: F. G. Garvan
Journal: Trans. Amer. Math. Soc. 305 (1988), 47-77
MSC: Primary 11P76; Secondary 05A17, 05A19
DOI: https://doi.org/10.1090/S0002-9947-1988-0920146-8
MathSciNet review: 920146
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ p(n)$ denote the number of unrestricted partitions of $ n$. The congruences referred to in the title are $ p(5n + 4)$, $ p(7n + 5)$ and $ p(11n + 6) \equiv 0$ ($ \bmod 5$, $ 7$ and $ 11$, respectively). Dyson conjectured and Atkin and Swinnerton-Dyer proved combinatorial results which imply the congruences $ \bmod 5$ and $ 7$. These are in terms of the rank of partitions. Dyson also conjectured the existence of a "crank" which would likewise imply the congruence $ \bmod 11$. In this paper we give a crank which not only gives a combinatorial interpretation of the congruence $ \bmod 11$ but also gives new combinatorial interpretations of the congruences $ \bmod 5$ and $ 7$. However, our crank is not quite what Dyson asked for; it is in terms of certain restricted triples of partitions, rather than in terms of ordinary partitions alone.

Our results and those of Dyson, Atkin and Swinnerton-Dyer are closely related to two unproved identities that appear in Ramanujan's "lost" notebook. We prove the first identity and show how the second is equivalent to the main theorem in Atkin and Swinnerton-Dyer's paper. We note that all of Dyson's conjectures $ \bmod 5$ are encapsulated in this second identity. We give a number of relations for the crank of vector partitions $ \bmod 5$ and $ 7$, as well as some new inequalities for the rank of ordinary partitions $ \bmod 5$ and $ 7$. Our methods are elementary relying for the most part on classical identities of Euler and Jacobi.


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

  • [1] G. E. Andrews, Applications of basic hypergeometric functions, SIAM Rev. 16 (1974), 441-484. MR 0352557 (50:5044)
  • [2] -, 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)
  • [3] -, An introduction to Ramanujan's "lost" notebook, Amer. Math. Monthly 86 (1979), 89-108. MR 520571 (80e:01018)
  • [4] -, Partitions: Yesterday and today, New Zealand Math. Soc., Wellington, 1979, 55 pp. MR 557539 (81b:01015)
  • [5] -, Hecke modular forms and the Kac-Petersen identities, Trans. Amer. Math. Soc. 283 (1984), 451-458. MR 737878 (85e:11031)
  • [6] -, Generalized Frobenius Partitions, Mem. Amer. Math. Soc. No. 301, 1984. MR 743546 (85m:11063)
  • [7] 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)
  • [8] A. O. L. Atkin and S. M. Hussain, Some properties of partitions $ (2)$, Trans. Amer. Math. Soc. 89 (1958), 184-200. MR 0103872 (21:2635)
  • [9] A. O. L. Atkin, A note on ranks and conjugacy of partitions, Quart. J. Math. Oxford (2) 17 (1966), 335-8. MR 0202688 (34:2548)
  • [10] -, Note on a paper of Chemma and Gordon, Duke Math. J. 34 (1967), 57-58. MR 0207671 (34:7486)
  • [11] -, Proof of a conjecture of Ramanujan, Glasgow Math. J. 8 (1967), 14-32. MR 0205958 (34:5783)
  • [12] M. S. Cheema and Basil Gordon, Some remarks on two- and three-line partitions, Duke Math. J. 31 (1964), 267-273. MR 0160770 (28:3981)
  • [13] F. J. Dyson, Some guesses in the theory of partitions, Eureka (Cambridge) 8 (1944), 10-15.
  • [14] F. G. Garvan, A simple proof of Watson's partition congruences for powers of $ 7$, J. Austral. Math. Soc. Ser. A 36 (1984), 316-334. MR 733905 (85f:11072)
  • [15] -, Generalizations of Dyson's rank, Ph. D. thesis, Pennsylvania State University, 1986, 127 pp.
  • [16] M. D. Hirschhorn and D. C. Hunt, A simple proof of the Ramanujan conjecture for powers of $ 5$, J. Reine Angew. Math. 336 (1981), 1-17. MR 622342 (82m:10023)
  • [17] 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)
  • [18] -, A generalization of Winquist's identity and a conjecture of Ramanujan, J.I.M.S. Ramanujan Centenary Volume.
  • [19] 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, 95 pp.
  • [20] 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.
  • [21] J. J. Sylvester, A constructive theory of partitions in three acts, an interact and an exodian, Amer. J. Math. 5 (1882), 251-330. MR 1505328
  • [22] G. N. Watson, A new proof of the Rogers-Ramanujan identities, J. London Math. Soc. 4 (1929), 4-9.
  • [23] -, Ramanujans Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math. 179 (1938), 97-128.
  • [24] L. Winquist, An elementary proof of $ p(11m + 6) \equiv 0(\bmod 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: 11P76, 05A17, 05A19

Retrieve articles in all journals with MSC: 11P76, 05A17, 05A19


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1988-0920146-8
Keywords: Partition congruences, Dyson's rank, crank, Ramanujan's "lost" notebook
Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society