Congruences for the Andrews
            <i>spt</i>
            function

Ono, Ken

doi:10.1073/pnas.1015339107

Cited by 30 publications

(54 citation statements)

References 14 publications

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“…This fact has provided the starting point for a number of recent studies on congruences for spt.n/ (see, for example, works of Bringmann [6], the first author with Bringmann and Lovejoy [2], Garvan [19][20][21] and Ono [28]). The situation is analogous in some ways -and different in others -from the better-known situation of congruences for the ordinary partition function p.n/.…”

Section: Introductionmentioning

confidence: 92%

“…In analogy with (1.5), we establish systematic Hecke relations among the elements of the grid. Using these, we are able to replace the congruences (1.4) with equalities (see Section 3 for the statements of these results, which imply the congruences of [2] and [28]). In Section 4 we introduce a grid associated to other smallest parts functions which were considered in [2], and we obtain analogous results.…”

Section: Introductionmentioning

confidence: 94%

“…n 2k 1 q n 1 q n (where .s/ is the Riemann zeta function), and the modular j function (this is a normalized version of the function which is called M in [2] and [28]). After work of Bringmann [6], it is known that M.z/ is a mock modular form; in particular we have…”

Section: Preliminariesmentioning

confidence: 99%

“…The congruences for spt.n/ modulo prime powers`m in [2] (which extend the congruences modulo`of [28]) descend from congruences for translates of a certain mock modular form M.z/ under various Hecke operators T .`2 m / (see below for details). For example, Theorem 1.2 loc.…”

Section: Introductionmentioning

confidence: 98%

See 3 more Smart Citations

Mock modular grids and Hecke relations for mock modular forms

Ahlgren

Kim

2012

Forum Mathematicum

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We construct and study several infinite grids of mock modular forms. The "rows" of each grid encode an infinite family of forms, while the "columns" encode a second infinite family. Each of these grids contains a generating function of interest as its first entry (for example, the "smallest parts" function of Andrews, or a mock theta function of Ramanujan). We show that the Hecke operators transform these grids in systematic ways, and from this we deduce many corollaries for the generating functions under consideration. For example, we recover the systematic families of congruences for the smallest parts function which have recently been discovered by various authors, and we obtain families of Hecke relations in the grid associated to the mock theta function f .q/.

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Section: Introductionmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 94%

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

See 2 more Smart Citations

Mock modular grids and Hecke relations for mock modular forms

Ahlgren

Kim

2012

Forum Mathematicum

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“…Much attention has been drawn to the investigation of the spt-function, in particular, the sptcrank of an S-partition, see, for example, Andrews, Dyson and Rhoades [5], Andrews, Garvan and Liang [6,7], Folsom and Ono [13], Garvan [14] and Ono [19].…”

Section: Introductionmentioning

confidence: 99%

The spt-crank for ordinary partitions

Chen

Zang

2014

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. The spt-function spt(n) was introduced by Andrews as the weighted counting of partitions of n with respect to the number of occurrences of the smallest part. Andrews, Garvan and Liang defined the spt-crank of an S-partition which leads to combinatorial interpretations of the congruences of spt(n) mod 5 and 7. Let N S (m, n) denote the net number of S-partitions of n with spt-crank m. Andrews, Garvan and Liang showed that N S (m, n) is nonnegative for all integers m and positive integers n, and they asked the question of finding a combinatorial interpretation of N S (m, n). In this paper, we introduce the structure of doubly marked partitions and define the spt-crank of a doubly marked partition. We show that N S (m, n) can be interpreted as the number of doubly marked partitions of n with spt-crank m. Moreover, we establish a bijection between marked partitions of n and doubly marked partitions of n. A marked partition is defined by Andrews, Dyson and Rhoades as a partition with exactly one of the smallest parts marked. They consider it a challenge to find a definition of the spt-crank of a marked partition so that the set of marked partitions of 5n + 4 and 7n + 5 can be divided into five and seven equinumerous classes. The definition of spt-crank for doubly marked partitions and the bijection between the marked partitions and doubly marked partitions leads to a solution to the problem of Andrews, Dyson and Rhoades.

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Congruences for Andrews’ spt-Function Modulo 32760 and Extension of Atkin’s Hecke-Type Partition Congruences

Garvan

2013

Springer Proceedings in Mathematics &Amp; Statistics

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New congruences are found for Andrews' smallest parts partition function spt(n). The generating function for spt(n) is related to the holomorphic part α(24z) of a certain weak Maass form M(z) of weight 3 2 . We show that a normalized form of the generating function for spt(n) is an eigenform modulo 72 for the Hecke operators T (ℓ 2 ) for primes ℓ > 3, and an eigenform modulo p for p = 5, 7 or 13 provided that (ℓ, 6p) = 1. The result for the modulus 3 was observed earlier by the author and considered by Ono and Folsom. Similar congruences for higher powers of p (namely 5 6 , 7 4 and 13 2 ) occur for the coefficients of the function α(z). Analogous results for the partition function were found by Atkin in 1966. Our results depend on the recent result of Ono that M ℓ (z/24) is a weakly holomorphic modular form of weight 3 2 for the full modular group where M ℓ (z) = M(z)|T (ℓ 2 ) − 3 ℓ (1 + ℓ)M(z).

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Congruences for the Andrews spt function

Cited by 30 publications

References 14 publications

Mock modular grids and Hecke relations for mock modular forms

Mock modular grids and Hecke relations for mock modular forms

The spt-crank for ordinary partitions

Congruences for Andrews’ spt-Function Modulo 32760 and Extension of Atkin’s Hecke-Type Partition Congruences

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