Relations Between the Ranks and Cranks of Partitions

By A. Atkin, Frank G. Garvan. 2002

147 citations recorded by the source corpus.

New identities and congruences involving the ranks and cranks of partitions are proved. The proof depends on a new partial differential equation connecting their generating functions.

Subfields: Algebra and Number Theory, Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/e4dd348f658f58131b15db8dda542815368701da

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In Algebra and Number Theory

In Applied Mathematics

Citations in the corpus

Zagier-type dualities and lifting maps for harmonic Maass–Jacobi forms (2010, methodology): “Bringmann [3] and Bringmann, Garvan, and Mahlburg [4] examined quasiharmonic Maass forms, i.e., linear combinations of derivatives of harmonic Maass forms, which are also closely related to derivatives of Jacobi forms.”
The number of smallest parts in the partitions of n (2008, background): “In [6], Atkin and Garvan introduce the moments of ranks”
Automorphic Properties of Generating Functions for Generalized Rank Moments and Durfee Symbols (2008, methodology): “Atkin and Garvan [4] initiated the study of rank moments, the kth rank moment Nk(n) being defined by Nk(n) := ∑ m∈Z mkN(m,n).”
Unearthing the visions of a master: harmonic Maass forms and number theory (2008, background): “Several years ago, Atkin and Garvan [30] discovered a striking partial differential equation which related the rank and crank generating functions.”
The spt-crank for overpartitions (2013, background): “These rank and crank moments were introduced by Atkin and the first author in [8].”
Rank and crank moments for overpartitions (2008, methodology): “2) Since their introduction by Atkin and Garvan [4], the rank and crank moments and their linear combinations have been the subject of a number of works [2, 3, 5, 6, 16, 17].”
Inequalities between ranks and cranks (2009, background): “Both the crank and rank moments vanish when k is odd due to the symmetries of the statistics [6].”
Congruences for Andrews' Smallest Parts Partition Function and New Congruences for Dyson's Rank (2007, methodology): “2n 1 : In [ 8 ] we obtained the following identity”
Partition Statistics and Quasiharmonic Maass Forms (2009, background): “In this section we recall certain relations between rank and crank moments and consider twisted generalisations. For details we refer the reader to [ 2 ].”
Higher Order SPT-Functions (2010, result): “Many of the congruences follow from known results for rank and crank moments [6].”
Partitions, Durfee symbols, and the Atkin–Garvan moments of ranks (2007, background): “In a paper relating cranks and ranks [10], Atkin and Garvan found that the moments of ranks and cranks were important in the study of further partition congruences.”