Congruences for Andrews' Smallest Parts Partition Function and New Congruences for Dyson's Rank

By Frank G. Garvan. 2007

89 citations recorded by the source corpus.

Let spt(n) denote the total number of appearances of smallest parts in the partitions of n. Recently, Andrews showed how spt(n) is related to the second rank moment, and proved some surprising Ramanujan-type congruences mod 5, 7 and 13. We prove a generalization of these congruences using known relations between rank and crank moments. We obtain explicit Ramanujan-type congruences for spt(n) mod p for p = 11, 17, 19, 29, 31 and 37. Recently, Bringmann and Ono proved that Dyson's rank function has infinitely many Ramanujan-type congruences. Their proof is non-constructive and utilizes the theory of weak Maass forms. We construct two explicit nontrivial examples mod 11 using elementary congruences between rank moments and half-integer weight Hecke eigenforms.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/22c4ab49201f5263e1c8d05b0b4b12a9d11bc622

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Citations in the corpus

Higher Order SPT-Functions (2010, methodology): “Further congruence properties of Andrews’ spt-function were found by the author [17], Folsom and Ono [14] and Ono [21].”
Asymptotics for rank and crank moments (2009, background): “Also, part (1) of his conjecture does not appear in [20], but was mentioned in private communication to the first two authors.”
Asymptotic inequalities for positive crank and rank moments (2012, background): “Atkin and Garvan described algebraic and arithmetic relationships between crank and rank moments, but the nature of their exact numerical values was first explored by Garvan, who conjectured a strong inequality between the two families of moments in [24]; a related set of conjectures was later given”
Taylor coefficients of mock-Jacobi forms and moments of partition statistics (2014, background): “These observations have been the catalyst for many works studying the relations between the rank and crank statistics (see [7, 9, 21, 22]).”
Rank and crank moments for overpartitions (2008, background): “kN(m,n), (1.1) and Mk(n) := X m∈Z mkM(m,n). (1.2) 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]. A key role in several of these studies is played by the fact that the crank mo”