The crank of partitions mod 8,9 and 10

Garvan, Frank G.

doi:10.1090/S0002-9947-1990-1012520-8

The crank of partitions mod $8,\;9$ and $10$
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Trans. Amer. Math. Soc. 322 (1990), 79-94 Request permission

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

George E. Andrews, The theory of partitions, Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. MR 0557013
George E. Andrews and F. G. Garvan, Ramanujan’s “lost” notebook. VI. The mock theta conjectures, Adv. in Math. 73 (1989), no. 2, 242–255. MR 987276, DOI 10.1016/0001-8708(89)90070-4
George E. Andrews and F. G. Garvan, Dyson’s crank of a partition, Bull. Amer. Math. Soc. (N.S.) 18 (1988), no. 2, 167–171. MR 929094, DOI 10.1090/S0273-0979-1988-15637-6
A. O. L. Atkin and P. Swinnerton-Dyer, Some properties of partitions, Proc. London Math. Soc. (3) 4 (1954), 84–106. MR 60535, DOI 10.1112/plms/s3-4.1.84
Anthony J. F. Biagioli, A proof of some identities of Ramanujan using modular forms, Glasgow Math. J. 31 (1989), no. 3, 271–295. MR 1021804, DOI 10.1017/S0017089500007850
B. J. Birch, A look back at Ramanujan’s notebooks, Math. Proc. Cambridge Philos. Soc. 78 (1975), 73–79. MR 379372, DOI 10.1017/S0305004100051501

Some guesses in the theory of partitions

8

Freeman J. Dyson, Missed opportunities, Bull. Amer. Math. Soc. 78 (1972), 635–652. MR 522147, DOI 10.1090/S0002-9904-1972-12971-9

A walk through Ramanujan’s garden

Freeman J. Dyson, Mappings and symmetries of partitions, J. Combin. Theory Ser. A 51 (1989), no. 2, 169–180. MR 1001259, DOI 10.1016/0097-3165(89)90043-5

Some congruence properties of the partition function

F. G. Garvan, New combinatorial interpretations of Ramanujan’s partition congruences mod $5,7$ and $11$, Trans. Amer. Math. Soc. 305 (1988), no. 1, 47–77. MR 920146, DOI 10.1090/S0002-9947-1988-0920146-8

Combinatorial interpretations of Ramanujan’s partition congruences

Frank Garvan, Dongsu Kim, and Dennis Stanton, Cranks and $t$-cores, Invent. Math. 101 (1990), no. 1, 1–17. MR 1055707, DOI 10.1007/BF01231493
Frank Garvan and Dennis Stanton, Sieved partition functions and $q$-binomial coefficients, Math. Comp. 55 (1990), no. 191, 299–311. MR 1023761, DOI 10.1090/S0025-5718-1990-1023761-1

An Introduction to the theory of numbers

Dean Hickerson, A proof of the mock theta conjectures, Invent. Math. 94 (1988), no. 3, 639–660. MR 969247, DOI 10.1007/BF01394279
Dean Hickerson, A proof of the mock theta conjectures, Invent. Math. 94 (1988), no. 3, 639–660. MR 969247, DOI 10.1007/BF01394279
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, DOI 10.1017/S1446788700019728
Michael D. Hirschhorn, A generalisation of Winquist’s identity and a conjecture of Ramanujan, J. Indian Math. Soc. (N.S.) 51 (1987), 49–55 (1988). MR 988308
M. D. Hirschhorn, A birthday present for Ramanujan, Amer. Math. Monthly 97 (1990), no. 5, 398–400. MR 1048912, DOI 10.2307/2324390

Vorlesungen über die Theorie der elliptischen Modulfunktionen

I. G. Macdonald, Affine root systems and Dedekind’s $\eta$-function, Invent. Math. 15 (1972), 91–143. MR 357528, DOI 10.1007/BF01418931

Some properties of

the number of partitions of

Srinivasa Ramanujan, The lost notebook and other unpublished papers, Springer-Verlag, Berlin; Narosa Publishing House, New Delhi, 1988. With an introduction by George E. Andrews. MR 947735

Dissections of the generating functions of

and

Dennis Stanton, Sign variations of the Macdonald identities, SIAM J. Math. Anal. 17 (1986), no. 6, 1454–1460. MR 860926, DOI 10.1137/0517103
Frederick M. Goodman and Kathleen M. O’Hara, On the Gaussian polynomials, $q$-series and partitions (Minneapolis, MN, 1988) IMA Vol. Math. Appl., vol. 18, Springer, New York, 1989, pp. 57–66. MR 1019842, DOI 10.1007/978-1-4684-0637-5_{5}

Proof of certain identities in combinatory analysis

20

Lasse Winquist, An elementary proof of $p(11m+6)\equiv 0\,(\textrm {mod}\ 11)$, J. Combinatorial Theory 6 (1969), 56–59. MR 236136, DOI 10.1016/S0021-9800(69)80105-5