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.

- George E. Andrews,
*Applications of basic hypergeometric functions*, SIAM Rev.**16**(1974), 441–484. MR**352557**, DOI https://doi.org/10.1137/1016081 - George E. Andrews,
*The theory of partitions*, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. Encyclopedia of Mathematics and its Applications, Vol. 2. MR**0557013** - George E. Andrews,
*An introduction to Ramanujan’s “lost” notebook*, Amer. Math. Monthly**86**(1979), no. 2, 89–108. MR**520571**, DOI https://doi.org/10.2307/2321943 - George E. Andrews,
*Partitions: yesterday and today*, New Zealand Mathematical Society, Wellington, 1979. With a foreword by J. C. Turner. MR**557539** - George E. Andrews,
*Hecke modular forms and the Kac-Peterson identities*, Trans. Amer. Math. Soc.**283**(1984), no. 2, 451–458. MR**737878**, DOI https://doi.org/10.1090/S0002-9947-1984-0737878-3 - George E. Andrews,
*Generalized Frobenius partitions*, Mem. Amer. Math. Soc.**49**(1984), no. 301, iv+44. MR**743546**, DOI https://doi.org/10.1090/memo/0301 - A. O. L. Atkin and P. Swinnerton-Dyer,
*Some properties of partitions*, Proc. London Math. Soc. (3)**4**(1954), 84–106. MR**60535**, DOI https://doi.org/10.1112/plms/s3-4.1.84 - A. O. L. Atkin and S. M. Hussain,
*Some properties of partitions. II*, Trans. Amer. Math. Soc.**89**(1958), 184–200. MR**103872**, DOI https://doi.org/10.1090/S0002-9947-1958-0103872-3 - A. O. L. Atkin,
*A note on ranks and conjugacy of partitions*, Quart. J. Math. Oxford Ser. (2)**17**(1966), 335–338. MR**202688**, DOI https://doi.org/10.1093/qmath/17.1.335 - A. O. L. Atkin,
*Note on a paper of Cheema and Gordon*, Duke Math. J.**34**(1967), 57–58. MR**207671** - A. O. L. Atkin,
*Proof of a conjecture of Ramanujan*, Glasgow Math. J.**8**(1967), 14–32. MR**205958**, DOI https://doi.org/10.1017/S0017089500000045 - M. S. Cheema and Basil Gordon,
*Some remarks on two- and three-line partitions*, Duke Math. J.**31**(1964), 267–273. MR**160770**
F. J. Dyson, - F. G. Garvan,
*A simple proof of Watson’s partition congruences for powers of $7$*, J. Austral. Math. Soc. Ser. A**36**(1984), no. 3, 316–334. MR**733905**
---, - Michael D. Hirschhorn and David C. Hunt,
*A simple proof of the Ramanujan conjecture for powers of $5$*, J. Reine Angew. Math.**326**(1981), 1–17. MR**622342**, DOI https://doi.org/10.1515/crll.1981.326.1 - M. D. Hirschhorn,
*A simple proof of an identity of Ramanujan*, J. Austral. Math. Soc. Ser. A**34**(1983), no. 1, 31–35. MR**683175**
---, - J. J. Sylvester and F. Franklin,
*A Constructive Theory of Partitions, Arranged in Three Acts, an Interact and an Exodion*, Amer. J. Math.**5**(1882), no. 1-4, 251–330. MR**1505328**, DOI https://doi.org/10.2307/2369545
G. N. Watson, - Lasse Winquist,
*An elementary proof of $p(11m+6)\equiv 0\,({\rm mod}\ 11)$*, J. Combinatorial Theory**6**(1969), 56–59. MR**236136**

*Some guesses in the theory of partitions*, Eureka (Cambridge)

**8**(1944), 10-15.

*Generalizations of Dyson’s rank*, Ph. D. thesis, Pennsylvania State University, 1986, 127 pp.

*A generalization of Winquist’s identity and a conjecture of Ramanujan*, J.I.M.S. Ramanujan Centenary Volume. 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. 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.

*A new proof of the Rogers-Ramanujan identities*, J. London Math. Soc.

**4**(1929), 4-9. ---,

*Ramanujans Vermutung über Zerfällungsanzahlen*, J. Reine Angew. Math.

**179**(1938), 97-128.

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Keywords:
Partition congruences,
Dyson’s rank,
crank,
Ramanujan’s "lost" notebook

Article copyright:
© Copyright 1988
American Mathematical Society