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A survey of multipartitions: Congruences and identities. (English) Zbl 1183.11063

Alladi, Krishnaswami (ed.), Surveys in number theory. New York, NY: Springer (ISBN 978-0-387-78509-7/hbk). Developments in Mathematics 17, 1-19 (2008).
Summary: The concept of a multipartition of a number, which has proved so useful in the study of Lie algebras, is studied for its own intrinsic interest. Following up on the work of Atkin, we present an infinite family of congruences for \(P_k (n)\), the number of \(k\)-component multipartitions of \(n\). We also examine the enigmatic tripentagonal number theorem and show that it implies a theorem about tripartitions. Building on this latter observation, we examine a variety of multipartition identities connecting them with mock theta functions and the Rogers-Ramanujan identities.
For the entire collection see [Zbl 1147.11004].

MSC:

11P81 Elementary theory of partitions
11P83 Partitions; congruences and congruential restrictions
05A30 \(q\)-calculus and related topics
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[1] Andrews, G. E., An analytic generalization of the Rogers-Ramanujan identities for odd moduli, Proc. Nat. Acad. Sci. USA, 71, 4082-4085 (1974) · Zbl 0289.10010 · doi:10.1073/pnas.71.10.4082
[2] G. E. Andrews, Problems and prospects for basic hypergeometric functions, from The Theory and Applications of Special Functions (R. Askey, ed.), pp. 191-224, Academic Press, New York, 1975. · Zbl 0342.33001
[3] G. E. Andrews, The Theory of Partitions, Encycl. of Math. and Its Appl. (G.-C. Rota, ed.), Vol. 2, Addison-Wesley, Reading, 1975 (Reprinted: Cambridge University Press, Cambridge, 1998).
[4] Andrews, G. E., Multiple q-series identities, Houston Math. J., 7, 11-22 (1981) · Zbl 0468.05009
[5] Andrews, G. E., Multiple series Rogers-Ramanujan identities, Pac. J. Math., 114, 267-283 (1984) · Zbl 0547.10012
[6] Andrews, G. E., Umbral calculus, Bailey chains and pentagonal number theorems, J. Comb. Th., Ser. A, 91, 464-475 (2000) · Zbl 0985.11048 · doi:10.1006/jcta.2000.3111
[7] Andrews, G. E.; Garvan, F. G., Dyson’s crank of a partition, Bull. Amer. Math. Soc., 18, 167-171 (1988) · Zbl 0646.10008 · doi:10.1090/S0273-0979-1988-15637-6
[8] G. E. Andrews and R. Roy, Ramanujan’s method in q-series congruences, Elec. J. Comb., 4(2): R2, 7pp., 1997. · Zbl 0884.05012
[9] Atkin, A. O. L., Proof of a conjecture of Ramanujan, Glasgow Math. J., 8, 14-32 (1967) · Zbl 0163.04302 · doi:10.1017/S0017089500000045
[10] Atkin, A. O. L., Ramanujan congruences for p_-k (n), Canad. J. Math., 20, 67-78 (1968) · Zbl 0164.35101
[11] A. Berkovich, The tripentagonal number theorem and related identities, Int. J. Number Theory, (to appear). · Zbl 1192.33001
[12] Bouwknegt, P., Multipartitions, generalized Durfee squares and affine Lie algebra characters, J. Austral. Math. Soc., 72, 395-408 (2002) · Zbl 1019.05005
[13] Broueacute;, M.; Malle, G., Zyklotomische Heckealgebren, Asteacute;risque, 212, 119-189 (1993) · Zbl 0835.20064
[14] Cheema, M. S.; Haskell, C. T., Multirestricted and rowed partitions, Duke Math. J., 34, 443-451 (1967) · Zbl 0173.01804 · doi:10.1215/S0012-7094-67-03450-3
[15] Dyson, F. J., Some guesses in the theory of partitions, Eureka (Cambridge), 8, 10-15 (1944)
[16] Fayers, M., Weights of multipartitions and representations of Ariki-Koike algebras, Adv. in Math., 206, 112-144 (2006) · Zbl 1111.20009 · doi:10.1016/j.aim.2005.07.017
[17] Garvan, F. G., New combinatorial interpretations of Ramanujan’s partition congruences mod 5, 7 and 11, Trans. Amer. Math. Soc., 305, 47-77 (1988) · Zbl 0641.10009 · doi:10.2307/2001040
[18] F. G. Garvan, Ranks and cranks for bipartitions \mod 5, (in preparation).
[19] Gasper, G.; Rahman, N., Basic Hypergeometric Series (1990), Cambridge: Cambridge University Press, Cambridge · Zbl 0695.33001
[20] Gupta, H., Selected Topics in Number Theory (1980), Turnbridge Wells: Abacus Press, Turnbridge Wells · Zbl 0425.10001
[21] Gupta, H.; Gwyther, C. E.; Miller, J. C. P., Tables of Partitions, Royal Society Math. Tables (1958), Cambridge: Cambridge University Press, Cambridge · Zbl 0079.06202
[22] Hardy, G. H.; Wright, E. M., An Introduction to the Theory of Numbers (1960), London: Oxford University Press, London · Zbl 0086.25803
[23] Mahlburg, K., Partition congruences and the Andrews-Garvan-Dyson crank, Proc. Nat. Acad. Sci., U.S.A, 102, 15373-15376 (2005) · Zbl 1155.11350 · doi:10.1073/pnas.0506702102
[24] Paule, P., On identities of the Rogers-Ramanujan type, J. Math. Anal. and Appl., 107, 255-284 (1985) · Zbl 0582.10008 · doi:10.1016/0022-247X(85)90368-3
[25] S. Ramanujan, The Lost Notebook and Other Unpublished Papers, intro. by G. E. Andrews · Zbl 0639.01023
[26] L. J. Rogers, On two theorems of combinatory analysis and allied identities, Proc. London Math. Soc. (2), 16 (1917), 315-336. · JFM 46.0109.01
[27] Slater, L. J., Further identities of the Rogers-Ramanujan type, Proc. London Math. Soc., 54, 2, 147-167 (1952) · Zbl 0046.27204 · doi:10.1112/plms/s2-54.2.147
[28] Sylvester, J. J., A constructive theory of partitions arranged in three acts, an interact and an exodion, Amer. J. Math., 5, 251-330 (1882) · JFM 15.0129.02 · doi:10.2307/2369545
[29] Watson, G. N., The mock theta functions (2), Proc. London Math. Soc., Ser., 2, 42, 274-304 (1937) · JFM 62.1228.04 · doi:10.1112/plms/s2-42.1.274
[30] Watson, G. N., A note on Lerch’s functions, Quart. J. Math., Oxford Series, 8, 44-47 (1937) · JFM 63.0337.05
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