Cranks and t-cores
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Publication:2640646
DOI10.1007/BF01231493zbMath0721.11039MaRDI QIDQ2640646
Frank G. Garvan, Dongsu Kim, Dennis W. Stanton
Publication date: 1990
Published in: Inventiones Mathematicae (Search for Journal in Brave)
Full work available at URL: https://eudml.org/doc/143794
Combinatorial aspects of partitions of integers (05A17) Partitions; congruences and congruential restrictions (11P83)
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Cites Work
- Unnamed Item
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- Unnamed Item
- Unnamed Item
- Unnamed Item
- On a system of modular functions connected with the Ramanujan identities
- The Crank of Partitions Mod 8, 9 and 10
- Some identities involving the partition function
- Sieved Partition Functions and q-Binomial Coefficients
- Some combinatorial results involving shifted Young diagrams
- New Combinatorial Interpretations of Ramanujan's Partition Congruences Mod 5,7 and 11
- Dyson’s crank of a partition
- Frobenius symbols for partitions and degrees of spin characters.
- Applications of Basic Hypergeometric Functions
- Proof of a conjecture of Ramanujan
- Ramanujans Vermutung über Zerfällungszahlen.
- A NOTE ON TWO OF RAMANUJAN'S FORMULAE
- Some Properties of Partitions