On the rank and the crank modulo and
Authors:
Richard Lewis and Nicolas SantaGadea
Journal:
Trans. Amer. Math. Soc. 341 (1994), 449465
MSC:
Primary 11P83; Secondary 05A17
MathSciNet review:
1136545
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Abstract 
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Additional Information
Abstract: In this paper we prove some identities, conjectured by Lewis, for the rank and crank of partitions concerning the modulo and . These identities are similar to Dyson's identities for the rank modulo and which give a combinatorial interpretation to Ramanujan's partition congruences. For this, we use multisection of series and some of the results that Watson established for the third order mock theta functions.
 [A]
George
E. Andrews, Applications of basic hypergeometric functions,
SIAM Rev. 16 (1974), 441–484. MR 0352557
(50 #5044)
 [AG]
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 (89b:11079), http://dx.doi.org/10.1090/S027309791988156376
 [AH]
G. E. Andrews and D. Hickerson, Ramanujan's "lost" notebook. VVI: The sixth order mock theta functions, preprint.
 [AS]
A.
O. L. Atkin and P.
SwinnertonDyer, Some properties of partitions, Proc. London
Math. Soc. (3) 4 (1954), 84–106. MR 0060535
(15,685d)
 [D1]
F. J. Dyson, Some guesses in the theory of partitions, Eureka (Cambridge) 8 (1944), 1015.
 [D2]
Freeman
J. Dyson, Mappings and symmetries of partitions, J. Combin.
Theory Ser. A 51 (1989), no. 2, 169–180. MR 1001259
(90f:05009), http://dx.doi.org/10.1016/00973165(89)900435
 [G1]
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
(89b:11081), http://dx.doi.org/10.1090/S00029947198809201468
 [G2]
, Combinatorial interpretations of Ramanujan's partitions congruences, Ramanujan Revisted: Proc. of the Centenary Conference (Univ. of Illinois at UrbanaChampaign, June 15, 1987), Academic Press, San Diego, 1988.
 [G3]
, The crank of partitions , and , Trans. Amer. Math. Soc. 322 (1990), 803821.
 [GKS]
Frank
Garvan, Dongsu
Kim, and Dennis
Stanton, Cranks and 𝑡cores, Invent. Math.
101 (1990), no. 1, 1–17. MR 1055707
(91h:11106), http://dx.doi.org/10.1007/BF01231493
 [H]
Dean
Hickerson, A proof of the mock theta conjectures, Invent.
Math. 94 (1988), no. 3, 639–660. MR 969247
(90f:11028a), http://dx.doi.org/10.1007/BF01394279
 [J]
M.
Jackson, On some formulae in partition theory, and bilateral basic
hypergeometric series, J. London Math. Soc. 24
(1949), 233–237. MR 0032048
(11,246b)
 [L1]
Richard
Lewis, On some relations between the rank and the crank, J.
Combin. Theory Ser. A 59 (1992), no. 1,
104–110. MR 1141325
(92k:11113), http://dx.doi.org/10.1016/00973165(92)90101Y
 [L2]
Richard
Lewis, On the rank and the crank modulo
4, Proc. Amer. Math. Soc.
112 (1991), no. 4,
925–933. MR 1057957
(91j:11088), http://dx.doi.org/10.1090/S00029939199110579572
 [Ra]
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.
 [Ri]
John
Riordan, Combinatorial identities, John Wiley & Sons,
Inc., New YorkLondonSydney, 1968. MR 0231725
(38 #53)
 [S]
N. SantaGadea, On the rank and the crank moduli , and , Ph.D. thesis, Penn State Univ., 1990.
 [W]
G.
N. Watson, The final problem: an account of the mock theta
functions, Ramanujan: essays and surveys, Hist. Math., vol. 22,
Amer. Math. Soc., Providence, RI, 2001, pp. 325–347. MR
1862757
 [A]
 G. E. Andrews, Applications of basic hypergeometric functions, SIAM Rev. 16 (1974), 441484. MR 0352557 (50:5044)
 [AG]
 G. E. Andrews and F. G. Garvan, Dyson's crank of a partition, Bull. Amer. Math. Soc. (N.S.) 18 (1988), 167171. MR 929094 (89b:11079)
 [AH]
 G. E. Andrews and D. Hickerson, Ramanujan's "lost" notebook. VVI: The sixth order mock theta functions, preprint.
 [AS]
 A. O. L. Atkin and P. SwinnertonDyer, Some properties of partitions, Proc. London Math. Soc. (3) 4 (1954), 84106. MR 0060535 (15:685d)
 [D1]
 F. J. Dyson, Some guesses in the theory of partitions, Eureka (Cambridge) 8 (1944), 1015.
 [D2]
 , Mappings and symmetries of partitions, J. Combin. Theory Ser. A 51 (1989), 169180. MR 1001259 (90f:05009)
 [G1]
 F. G. Garvan, New combinatorial interpretations of Ramanujan's congruences , and , Trans. Amer. Math. Soc. 305 (1988), 4777. MR 920146 (89b:11081)
 [G2]
 , Combinatorial interpretations of Ramanujan's partitions congruences, Ramanujan Revisted: Proc. of the Centenary Conference (Univ. of Illinois at UrbanaChampaign, June 15, 1987), Academic Press, San Diego, 1988.
 [G3]
 , The crank of partitions , and , Trans. Amer. Math. Soc. 322 (1990), 803821.
 [GKS]
 F. G. Garvan, D. Kim, and D. Stanton, Cranks and cores, Invent. Math. 101 (1990), 117. MR 1055707 (91h:11106)
 [H]
 D. Hickerson, A proof of the mock theta conjectures, Invent. Math. 94 (1988), 639660. MR 969247 (90f:11028a)
 [J]
 M. Jackson, On some formulae in partition theory, and bilateral basic hypergeometric series, J. London Math. Soc. 24 (1949), 233237. MR 0032048 (11:246b)
 [L1]
 R. Lewis, On some relations between the rank and the crank, J. Combin. Theory Ser. A 59 (1992), 104110. MR 1141325 (92k:11113)
 [L2]
 , On the rank and the crank modulo , Proc. Amer. Math. Soc. 112 (1991), 925933. MR 1057957 (91j:11088)
 [Ra]
 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.
 [Ri]
 J. Riordan, Combinatorial identities, Wiley, New York, 1967. MR 0231725 (38:53)
 [S]
 N. SantaGadea, On the rank and the crank moduli , and , Ph.D. thesis, Penn State Univ., 1990.
 [W]
 G. N. Watson, The final problem: an account of the mock theta functions, J. London Math. Soc. 11 (1936), 5580. MR 1862757
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199411365450
PII:
S 00029947(1994)11365450
Keywords:
Partitions,
rank,
crank,
mock theta functions
Article copyright:
© Copyright 1994
American Mathematical Society
