Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-29T03:31:55.716Z Has data issue: false hasContentIssue false
  • English
  • Français

More cranks and t-cores

Published online by Cambridge University Press:  17 April 2009

F. G. Garvan
F. G. Garvan
Affiliation:
Department of Mathematics, University of Florida, Gainesville, FL 32611, United States of America, e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Dedicated to George Szekeres on the occasion of his 90th Birthday

In 1990, new statistics on partitions (called cransk) were found which combinatorially prove Ramanujan's congruences for the partition function modulo 5, 7, 11 and 25. The methods are extended to find cranks for Ramanujan's partition congruence modulo 49. A more explicit form of the crank is given for the modulo 25 congruence.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 63 , Issue 3 , June 2001 , pp. 379 - 391
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Andrews, G.E., The theory of partitions, Encyclopedia of Mathematics and its Applications 2 (Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976).Google Scholar
[2]Andrews, G.E. and Garvan, F.G., ‘Dyson's crank of a partition’, Bull. Amer. Math. Soc. 18 (1988), 167–171.CrossRefGoogle Scholar
[3]Atkin, A.O.L., ‘Proof of a conjecture of Ramanujan’, Glasgow Math. J. 8 (1967), 14–32.CrossRefGoogle Scholar
[4]Atkin, A.O.L. and Swinnerton-Dyer, H.P.E., ‘Some properties of partitions’, Proc. London Math. Soc. (3) 4 (1954), 84–106.CrossRefGoogle Scholar
[5]Dyson, F., ‘Some guesses in the theory of partitions’, Eureka (Cambridge) 8 (1944), 10–15.Google Scholar
[6]Garvan, F.G., Kim, D. and Stanton, D., ‘Cranks and t-cores’, Invent. Math. 101 (1990), 1–17.CrossRefGoogle Scholar
[7]Kerber, J.G., The representation theory of the symmetric group (Addison-Wesley, Reading, MA, 1981).Google Scholar
[8]Reti, Z., Five problems in combinatorial number theory, Ph.D. thesis (University of Florida, 1994).Google Scholar
[9]Watson, G.N., ‘Ramanujans Vermutung über Zerfällungsanzahlen’, J. Reine. Angew. Math. 179 (1938), 97–128.CrossRefGoogle Scholar