Inventiones mathematicae

, Volume 101, Issue 1, pp 1–17 | Cite as

Cranks andt-cores

  • Frank Garvan
  • Dongsu Kim
  • Dennis Stanton


New statistics on partitions (calledcranks) are defined which combinatorially prove Ramanujan's congruences for the partition function modulo 5, 7, 11, and 25. Explicit bijections are given for the equinumerous crank classes. The cranks are closely related to thet-core of a partition. Usingq-series, some explicit formulas are given for the number of partitions which aret-cores. Some related questions for self-conjugate and distinct partitions are discussed.


Partition Function Explicit Formula Related Question Distinct Partition Function Modulo 
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Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Frank Garvan
    • 1
  • Dongsu Kim
    • 2
  • Dennis Stanton
  1. 1.School of Mathematics, Physics, Computing and ElectronicsMacquarie UniversitySydneyAustralia
  2. 2.School of MathematicsUniversity of MinnesotaMinneapolisUSA

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