Dyson’s crank of a partition

Andrews, George E.; Garvan, Frank G.

doi:10.1090/s0273-0979-1988-15637-6

Cited by 323 publications

(341 citation statements)

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“…Atkin and Swinnerton-Dyer [4] proved Dyson's rank conjecture for 5 and 7. Andrews and Garvan [2] proved Dyson's crank conjecture by finding a crank which proves not only Ramanujan's conjecture for 11 but also for 5 and 7. Later, Garvan, Kim and Stanton [6] found new cranks which gave new interpretations of Ramanujan's congruences mod 5, 7, 11, and 25.…”

Section: Introductionmentioning

confidence: 82%

“…They gave explicit bijections between the equinumerous classes. In the present paper we extend the 380 F.G. Garvan [2] methods of [6] and give a crank which is a combinatorial refinement of the a = 1 case of (1.3), namely (1.5) p(49n + 47) = 0 (mod 49).…”

Section: Introductionmentioning

confidence: 95%

“…The map O : P5-core(5n + 4) > P5-core(5n + 4) available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700019481 [6] is defined by 2 no rii 4 n 2 2 n 3 3 3 n 0 6nx 4 n 2 2 n 3 2 n 0 3 n i 3 n 2 n 3 1 6n 0 2 n i 3n 2 is a bijection. See [6, p.8].…”

Section: Cranks For T-cores and Partitionsmentioning

confidence: 99%

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More cranks andt-cores

Garvan¹

2001

Bull. Austral. Math. Soc.

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110

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Section: Introductionmentioning

confidence: 82%

Section: Introductionmentioning

confidence: 95%

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More cranks andt-cores

Garvan¹

2001

Bull. Austral. Math. Soc.

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110

226

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“…The first author established the following remarkable congruences spt(5n + 4) ≡ 0 (mod 5) spt(7n + 5) ≡ 0 (mod 7) spt(13n + 6) ≡ 0 (mod 13)…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

On the Distribution of the spt-Crank

2013

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Andrews, Garvan and Liang introduced the spt-crank for vector partitions. We conjecture that for any n the sequence {N S (m, n)} m is unimodal, where N S (m, n) is the number of S-partitions of size n with crank m weight by the spt-crank. We relate this conjecture to a distributional result concerning the usual rank and crank of unrestricted partitions. This leads to a heuristic that suggests the conjecture is true and allows us to asymptotically establish the conjecture. Additionally, we give an asymptotic study for the distribution of the spt-crank statistic. Finally, we give some speculations about a definition for the spt-crank in terms of "marked" partitions. A "marked" partition is an unrestricted integer partition where each part is marked with a multiplicity number. It remains an interesting and apparently challenging problem to interpret the spt-crank in terms of ordinary integer partitions.

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“…In the early 1980s, Frank Garvan (5) wrote his Pennsylvania State Ph.D. thesis precisely on the formulas of Ramanujan relative to the soon-to-be-unearthed crank. In 1987, Garvan and Andrews (6) were able to find and describe the crank that had been hiding in Ramanujan's work for 68 years.…”

mentioning

confidence: 99%