The odd moments of ranks and cranks

Andrews, George E.; Chan, Ssc; Kim, Byungchan

doi:10.1016/j.jcta.2012.07.001

Cited by 51 publications

(54 citation statements)

References 12 publications

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“…The smallest parts function spt(n), counting the total number of appearances of the smallest parts in all partitions of n, has received great attention since it was introduced in [9]. For generalizations and analogues of spt(n), we refer the reader to [12,18,20,23,27,28].…”

Section: Introductionmentioning

confidence: 99%

Partitions associated with the Ramanujan/Watson mock theta functions ω(q), ν(q)and ϕ(q)

2015

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The generating function of partitions with repeated (resp. distinct) parts such that each odd part is less than twice the smallest part is shown to be the third order mock theta function ω(q) (resp. ν(−q)). Similar results for partitions with the corresponding restriction on each even part are also obtained, one of which involves the third order mock theta function φ(q). Congruences for the smallest parts partition function(s) associated to such partitions are obtained. Two analogues of the partition-theoretic interpretation of Euler's pentagonal number theorem are also obtained.

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Section: Introductionmentioning

confidence: 99%

Partitions associated with the Ramanujan/Watson mock theta functions ω(q), ν(q)and ϕ(q)

2015

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“…Studying how their distributions differ is one of the main themes in the theory of partitions. To this end, Atkin and Garvan [4] introduced moments of partition ranks and cranks and Andrews et al [1] introduced the positive moments. Define M(m, n) (respectively N(m, n)) to be the number of partitions of n with crank (respectively rank) m. We define the partial moments…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…Andrews et al [1] proved that ospt(n) = M t 1,0 (n) − N t 1,0 (n) counts certain kinds of strings along the partitions of n. In the light of the significance of the ospt(n) function and Theorem 1.1, it is natural to study ospt t r (n) = M t 1,r (n) − N t 1,r (n), which also illuminates how the tail distributions of cranks and ranks differ.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

A Remark on Tail Distributions of Partition Rank and Crank

Kim

2015

Bull. Aust. Math. Soc.

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“…The ospt function is the difference of "first" moments of the crank and rank distributions, see [17]. From (2.1) we have…”

Section: Generating Functions For N S (M N)mentioning

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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The odd moments of ranks and cranks

Cited by 51 publications

References 12 publications

Partitions associated with the Ramanujan/Watson mock theta functions ω(q), ν(q)and ϕ(q)

Partitions associated with the Ramanujan/Watson mock theta functions ω(q), ν(q)and ϕ(q)

A Remark on Tail Distributions of Partition Rank and Crank

On the Distribution of the spt-Crank

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