Higher order SPT functions for overpartitions, overpartitions with smallest part even, and partitions with smallest part even and without repeated odd parts

Jennings-Shaffer, Chris

doi:10.1016/j.jnt.2014.10.011

Cited by 17 publications

(28 citation statements)

References 26 publications

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“…Various results on the D-rank and M 2 -rank of overpartitions can be found in [3,4,15,22,[24][25][26][35][36][37][38]. In 2014, Chan and Mao [16] proposed the following monotonicity conjecture on N (m, n) and N2(m, n): Conjecture 1.3 (Chan and Mao [16]).…”

Section: Introductionmentioning

confidence: 99%

Monotonicity properties for ranks of overpartitions

Xiong

Zang

2020

Journal of Number Theory

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The rank of partitions play an important role in the combinatorial interpretations of several Ramanujan's famous congruence formulas. In 2005 and 2008, the D-rank and M 2 -rank of an overpartition were introduced by Lovejoy, respectively. Let N(m, n) and N2(m, n) denote the number of overpartitions of n with D-rank m and M 2 -rank m, respectively. In 2014, Chan and Mao proposed a conjecture on monotonicity properties of N (m, n) and N2(m, n). In this paper, we prove the Chan-Mao monotonicity conjecture. To be specific, we show that for any integer m and nonnegative integer n, N2(m, n) ≤ N2(m, n + 1); and for (m, n) = (0, 4) with n = |m| + 2, we have N(m, n) ≤ N (m, n + 1). Furthermore, when m increases, we prove that N (m, n) ≥ N(m + 2, n) and N2(m, n) ≥ N2(m + 2, n) for any m, n ≥ 0, which is an analogue of Chan and Mao's result for partitions.

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Section: Introductionmentioning

confidence: 99%

Monotonicity properties for ranks of overpartitions

Xiong

Zang

2020

Journal of Number Theory

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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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“…The smallest parts function spt(n), counting the total number of appearances of the smallest parts in all partitions of n, was introduced by Andrews [1], and the function has received great attention since its introduction. For example, see [3,4,13,15,18,22,23,24].…”

Section: Introductionmentioning

confidence: 99%

Generating functions and congruences for some partition functions related to mock theta functions

Baruah

Begum

2019

Int. J. Number Theory

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Higher order SPT functions for overpartitions, overpartitions with smallest part even, and partitions with smallest part even and without repeated odd parts

Abstract: openAccessArticle: Falsecover date: 2015-04-01pii: S0022-314X(14)00360-6Harvest Date: 2016-01-06 13:08:10issueName:Page Range: 285-285href scidir: http://www.sciencedirect.com/science/article/pii/S0022314X14003606pubType

Cited by 17 publications

References 26 publications

Monotonicity properties for ranks of overpartitions

Monotonicity properties for ranks of overpartitions

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

Generating functions and congruences for some partition functions related to mock theta functions

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