Weighted Rogers–Ramanujan partitions and Dyson crank

Uncu, Ali Kemal

doi:10.1007/s11139-017-9903-8

Cited by 15 publications

(16 citation statements)

References 11 publications

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“…It is easy to check that with the trivial choice a = b = c = d = q, ω π (q, q, q, q) = q |π| , the 4parameter generating functions become the generating function for the number of partitions and number of partitions into distinct parts, respectively. The choice (a, b, c, d) = (q, q, 1, 1), proves Schmidt problem and yields another similar theorem of the sort for ordinary partitions: Theorem 1.2 (Theorems 1.3 and 6.3 [19]).…”

Section: Introductionmentioning

confidence: 71%

“…This result has been proven and re-discovered by many over the years. For example, P. Mork [17] gave a solution to Schmidt's problem, the second author gave an independent proof in 2018 [19] without the knowledge of Schmidt's problem, and recently Andrews-Paule [5]. Andrews-Paule paper is particularly important because it led many new researchers into this area and many new proofs of Date: May 20, 2022.…”

Section: Introductionmentioning

confidence: 99%

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On Finite Analogs of Schmidt's Problem and Its Variants

Berkovich¹,

Uncu²

2022

Preprint

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We refine Schmidt's problem and a partition identity related to 2-color partitions which we will refer to as Uncu-Andrews-Paule theorem. We will approach the problem using Boulet-Stanley weights and a formula on Rogers-Szegő polynomials by Berkovich-Warnaar, and present various Schmidt's problem alike theorems and their refinements. Our new Schmidt type results include the use of even-indexed parts' sums, alternating sum of parts, and hook lengths as well as the odd-indexed parts' sum which appears in the original Schmidt's problem. We also translate some of our Schmidt's problem alike relations to weighted partition counts with multiplicative weights in relation to Rogers-Ramanujan partitions.

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Section: Introductionmentioning

confidence: 71%

Section: Introductionmentioning

confidence: 99%

On Finite Analogs of Schmidt's Problem and Its Variants

Berkovich¹,

Uncu²

2022

Preprint

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“…In this section we denote by C(n) the number of partition of n with non-negative crank. Recently, Uncu [23] proved that the number of partitions into even number of distinct parts whose odd-indexed parts sum is n is equal to the number of partitions of n with non-negative crank. In this context he provided the following result.…”

Section: Truncated Identity Of Auluckmentioning

confidence: 99%

Rank partition functions and truncated theta identities

Merca¹

2020

Preprint

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In 1944, Freeman Dyson defined the concept of rank of an integer partition and introduced without definition the term of crank of an integer partition. A definition for the crank satisfying the properties hypothesized for it by Dyson was discovered in 1988 by G. E. Andrews and F. G. Garvan. In this paper, we introduce truncated forms for two theta identities involving the generating functions for partitions with non-negative rank and non-negative crank. As corollaries we derive new infinite families of linear inequalities for the partition function p(n). The number of Garden of Eden partitions are also considered in this context in order to provide other infinite families of linear inequalities for p(n).

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“…In this paper, we discuss a couple of examples of weighted partition identities connecting D and U, thus taking a step towards the eventual solution to Alladi's problem. The reader interested in weighted partition identities and their applications may also wish to examine [1], [4], [6], [7], and [22]. In particular, Theorem 1.2 is extended in [22].…”

Section: Introductionmentioning

confidence: 99%

“…The reader interested in weighted partition identities and their applications may also wish to examine [1], [4], [6], [7], and [22]. In particular, Theorem 1.2 is extended in [22]. In Section 2 we will introduce different representations of partitions which we are going to use later.…”

Section: Introductionmentioning

confidence: 99%

Variation on a theme of Nathan Fine. New weighted partition identities

Berkovich

Uncu

2017

Journal of Number Theory

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We utilize false theta function results of Nathan Fine to discover three new partition identities involving weights. These relations connect Göllnitz-Gordon type partitions and partitions with distinct odd parts, partitions into distinct parts and ordinary partitions, and partitions with distinct odd parts where the smallest positive integer that is not a part of the partition is odd and ordinary partitions subject to some initial conditions, respectively. Some of our weights involve new partition statistics, one is defined as the number of different odd parts of a partition larger than or equal to a given value and another one is defined as the number of different even parts larger than the first integer that is not a part of the partition.

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Weighted Rogers–Ramanujan partitions and Dyson crank

Cited by 15 publications

References 11 publications

On Finite Analogs of Schmidt's Problem and Its Variants

On Finite Analogs of Schmidt's Problem and Its Variants

Rank partition functions and truncated theta identities

Variation on a theme of Nathan Fine. New weighted partition identities

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