New combinatorial interpretations of Ramanujan's partition congruences mod and

Author:
F. G. Garvan

Journal:
Trans. Amer. Math. Soc. **305** (1988), 47-77

MSC:
Primary 11P76; Secondary 05A17, 05A19

MathSciNet review:
920146

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Abstract: Let denote the number of unrestricted partitions of . The congruences referred to in the title are , and (, and , respectively). Dyson conjectured and Atkin and Swinnerton-Dyer proved combinatorial results which imply the congruences and . These are in terms of the rank of partitions. Dyson also conjectured the existence of a "crank" which would likewise imply the congruence . In this paper we give a crank which not only gives a combinatorial interpretation of the congruence but also gives new combinatorial interpretations of the congruences and . 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 are encapsulated in this second identity. We give a number of relations for the crank of vector partitions and , as well as some new inequalities for the rank of ordinary partitions and . Our methods are elementary relying for the most part on classical identities of Euler and Jacobi.

**[1]**George E. Andrews,*Applications of basic hypergeometric functions*, SIAM Rev.**16**(1974), 441–484. MR**0352557****[2]**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****[3]**George E. Andrews,*An introduction to Ramanujan’s “lost” notebook*, Amer. Math. Monthly**86**(1979), no. 2, 89–108. MR**520571**, 10.2307/2321943**[4]**George E. Andrews,*Partitions: yesterday and today*, New Zealand Mathematical Society, Wellington, 1979. With a foreword by J. C. Turner. MR**557539****[5]**George E. Andrews,*Hecke modular forms and the Kac-Peterson identities*, Trans. Amer. Math. Soc.**283**(1984), no. 2, 451–458. MR**737878**, 10.1090/S0002-9947-1984-0737878-3**[6]**George E. Andrews,*Generalized Frobenius partitions*, Mem. Amer. Math. Soc.**49**(1984), no. 301, iv+44. MR**743546**, 10.1090/memo/0301**[7]**A. O. L. Atkin and P. Swinnerton-Dyer,*Some properties of partitions*, Proc. London Math. Soc. (3)**4**(1954), 84–106. MR**0060535****[8]**A. O. L. Atkin and S. M. Hussain,*Some properties of partitions. II*, Trans. Amer. Math. Soc.**89**(1958), 184–200. MR**0103872**, 10.1090/S0002-9947-1958-0103872-3**[9]**A. O. L. Atkin,*A note on ranks and conjugacy of partitions*, Quart. J. Math. Oxford Ser. (2)**17**(1966), 335–338. MR**0202688****[10]**A. O. L. Atkin,*Note on a paper of Cheema and Gordon*, Duke Math. J.**34**(1967), 57–58. MR**0207671****[11]**A. O. L. Atkin,*Proof of a conjecture of Ramanujan*, Glasgow Math. J.**8**(1967), 14–32. MR**0205958****[12]**M. S. Cheema and Basil Gordon,*Some remarks on two- and three-line partitions*, Duke Math. J.**31**(1964), 267–273. MR**0160770****[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), no. 3, 316–334. MR**733905****[15]**-,*Generalizations of Dyson's rank*, Ph. D. thesis, Pennsylvania State University, 1986, 127 pp.**[16]**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****[17]**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****[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*,*and*, Ph. D. thesis, Univ. of Durham, England, 1966, 95 pp.**[20]**S. Ramanujan,*Some properties of*,*the number of partitions of*, 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 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**, 10.2307/2369545**[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]**Lasse Winquist,*An elementary proof of 𝑝(11𝑚+6)≡0(𝑚𝑜𝑑11)*, J. Combinatorial Theory**6**(1969), 56–59. MR**0236136**

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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