Some asymptotics for cranks

Chan, O-Yeat

doi:10.4064/aa120-2-1

Cited by 15 publications

(14 citation statements)

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“…As Chan's method in [11] can be applied to find asymptotics for general eta-quotients, we only summarize the computation and refer to his results whenever it is necessary. By Cauchy's integral formula, for a circle C of radius r = e −2πρ = e −2π /N 2 for a positive N to be determined, we have…”

Section: Proof Of Theorem 13mentioning

confidence: 99%

“…partition statistic, which he named the 'crank', that would similarly explain Ramanujan's third partition congruence p(11n + 6) ≡ 0 (mod 11). His observations on the rank were first proved by Atkin and Swinnerton-Dyer [7] in 1954 and the crank was found by Andrews and Garvan [5] in 1988.…”

mentioning

confidence: 91%

“…We apply O.Y. Chan's modification [11] of the HardyRamanujan circle method to obtain an asymptotic formula for M e (n) − M o (n):…”

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confidence: 99%

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Partitions weighted by the parity of the crank

Choi

Kang

Lovejoy

2009

Journal of Combinatorial Theory, Series A

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The 'crank' is a partition statistic which originally arose to give combinatorial interpretations for Ramanujan's famous partition congruences. In this paper, we establish an asymptotic formula and a family of Ramanujan type congruences satisfied by the number of partitions of n with even crank M e (n) minus the number of partitions of n with odd crank M o (n). We also discuss the combinatorial implications of q-series identities involving M e (n) − M o (n). Finally, we determine the exact values of M e (n) − M o (n) in the case of partitions into distinct parts. These values are at most two, and zero for infinitely many n.

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Section: Proof Of Theorem 13mentioning

confidence: 99%

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confidence: 91%

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Partitions weighted by the parity of the crank

Choi

Kang

Lovejoy

2009

Journal of Combinatorial Theory, Series A

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“…Remarkably, the generating function (2.7) for cranks can be found in Ramanujan's lost notebook [19, p. 179] in the form 8) and so from (2.7),…”

Section: Dyson Ranks and Cranksmentioning

confidence: 99%

“…Using the Hardy-Ramanujan circle method, O.-Y. Chan [8] proved that the conjectured monotonicities did indeed hold. Upon checking the values of λ n below the thresholds of monotonicity, Chan then completed the proof that all ten tables of values of n are complete (with the exception of one missing value noted earlier).…”

Section: Dissections Of the Crankmentioning

confidence: 99%