Asymptotic formulae for partition ranks

Dousse, Jehanne; Mertens, Michael H.

doi:10.4064/aa168-1-5

Cited by 21 publications

(22 citation statements)

References 11 publications

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“…Similar to the present work, these statistics have gained a lot of attention because of their connections to exotic modular objects, such as quasimodular forms and mock modular forms [23,24,67,73]. Significant results about the distribution of these statistics appear notably in the works of Bringmann and Mahlburg and their coauthors [15,21,30,31,69].…”

Section: A History Of Partition Statisticssupporting

confidence: 67%

Integer partitions, probabilities and quantum modular forms

Ngo

Rhoades

2017

Res Math Sci

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What is the probability that the smallest part of a random integer partition of N is odd? What is the expected value of the smallest part of a random integer partition of N? It is straightforward to see that the answers to these questions are both 1, to leading order. This paper shows that the precise asymptotic expansion of each answer is dictated by special values of an arithmetic L-function. Alternatively, the asymptotics are dictated by the asymptotic expansions of quantum modular forms. A quantum modular form is a function on the rational numbers which has pseudo-modular properties and nice asymptotic expansions near each root of unity. This paper contains five examples involving some of the most famous quantum modular forms of Don Zagier. Additionally, this paper contains new generating function identities for the partition questions relevant to this work and three general circle method asymptotics which may be of independent interest. Mathematics Subject Classification: 05A16, 05A17, 11P81, 11P82, 11P84, 60C05 BackgroundLet P(N ) denote the set of all integer partitions of N and write p(N ) = P(N ) for the number of integer partitions of N . The study of the analytic approximation of partition statistics started with Hardy-Ramanujan asymptotic formula [49], further strengthened by Rademacher's exact formula [72]. These formulas give for any positive integer R the asymptoticwhere K =

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Section: A History Of Partition Statisticssupporting

confidence: 67%

Integer partitions, probabilities and quantum modular forms

Ngo

Rhoades

2017

Res Math Sci

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“…In [10], Dyson conjectured an asymptotic formula for the crank statistic for integer partitions: Bringmann and Dousse [4] proved that (1.1) holds for all |m| ≤ ( √ n log n)/(π √ 6). In [8], Dousse and Mertens proved the same result for N(m, n). For more results on the asymptotics of the rank and crank statistic for integer partitions, see [6,7,13,14].…”

Section: Introductionmentioning

confidence: 56%

On the Distribution of the Rank Statistic for Strongly Concave Compositions

Zhou

2019

Bull. Aust. Math. Soc.

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“…More recently, Dousse and M. Mertens [10] showed that the same asymptotic formula holds for N (m, n); i.e., the distribution for cranks and ranks are asymptotically similar when m is relatively smaller than n (more precisely, for |m| ≤ √ n log n π √ 6…”

Section: Introductionmentioning

confidence: 76%

On the asymptotic distribution of cranks and ranks of cubic partitions

Kim

Nam

2016

Journal of Mathematical Analysis and Applications

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Cubic partitions are a special kind of bi-partitions whose name is inspired by a connection between a cubic continued fraction and an arithmetic property of this bi-partition. There are two partition statistics, namely rank and crank, for cubic partitions, which explain cubic partition congruences combinatorially. We obtain asymptotics for the number of cubic partitions of rank (resp. crank) m which reveal the distribution of the rank (resp. crank) among cubic partitions. As applications, we derive asymptotic inequalities between cubic partition rank and crank functions, and we remark upon the similarities and the differences between these and the corresponding inequalities for ordinary partitions.

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Asymptotic formulae for partition ranks

Abstract: Using an extension of Wright's version of the circle method, we obtain asymptotic formulae for partition ranks similar to formulae for partition cranks which where conjectured by F. Dyson and recently proved by the first author and K. Bringmann.

Cited by 21 publications

References 11 publications

Integer partitions, probabilities and quantum modular forms

Integer partitions, probabilities and quantum modular forms

On the Distribution of the Rank Statistic for Strongly Concave Compositions

On the asymptotic distribution of cranks and ranks of cubic partitions

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