Partition congruences and the Andrews-Garvan-Dyson crank.

Abstract:

:In 1944, Freeman Dyson conjectured the existence of a "crank" function for partitions that would provide a combinatorial proof of Ramanujan's congruence modulo 11. Forty years later, Andrews and Garvan successfully found such a function and proved the celebrated result that the crank simultaneously "explains" the three Ramanujan congruences modulo 5, 7, and 11. This note announces the proof of a conjecture of Ono, which essentially asserts that the elusive crank satisfies exactly the same types of general congruences as the partition function.

authors

Mahlburg K

doi

10.1073/pnas.0506702102

keywords:

subject

Has Abstract

pub_date

2005-10-25 00:00:00

pages

15373-6

issue

43

eissn

0027-8424

issn

1091-6490

pii

0506702102

journal_volume

102

pub_type

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