Cranks and t-cores

Frank Garvan; Dongsu Kim; Dennis Stanton

doi:10.1007/BF01231493

Cranks and t-cores

Frank Garvan
, Dongsu Kim
, Dennis Stanton

School of Mathematics

Research output: Contribution to journal › Article › peer-review

267 Scopus citations

Abstract

New statistics on partitions (called cranks) 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 the t-core of a partition. Using q-series, some explicit formulas are given for the number of partitions which are t-cores. Some related questions for self-conjugate and distinct partitions are discussed.

Original language	English (US)
Pages (from-to)	1-17
Number of pages	17
Journal	Inventiones Mathematicae
Volume	101
Issue number	1
DOIs	https://doi.org/10.1007/BF01231493
State	Published - Dec 1990

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10.1007/BF01231493

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abstract = "New statistics on partitions (called cranks) 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 the t-core of a partition. Using q-series, some explicit formulas are given for the number of partitions which are t-cores. Some related questions for self-conjugate and distinct partitions are discussed.",

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AU - Kim, Dongsu

AU - Stanton, Dennis

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N2 - New statistics on partitions (called cranks) 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 the t-core of a partition. Using q-series, some explicit formulas are given for the number of partitions which are t-cores. Some related questions for self-conjugate and distinct partitions are discussed.

AB - New statistics on partitions (called cranks) 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 the t-core of a partition. Using q-series, some explicit formulas are given for the number of partitions which are t-cores. Some related questions for self-conjugate and distinct partitions are discussed.

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Cranks and t-cores

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