Partitions with rounded occurrences and attached parts

Lovejoy, Jeremy

doi:10.1007/978-1-4614-7858-4_19

Cited by 4 publications

(6 citation statements)

References 18 publications

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“…Here we give an example of the above bijection. Let γ = (1, 1, 2, 2, 2, 3, 3, 3,4,4,5,5,5,5,6,6,7,7,7,8,8,8,9,9,9,10,10,11,11,12,12,14,14,15,17,17,17) in Q 10,9,8,6,6;1 (311 which is the Gordon marking representation of an overpartition in Q 9,8,7,5,5;1 (254). It can be checked that the above process is reversible.…”

Section: Examiningmentioning

confidence: 99%

“…Here we give an example of the above bijection. Let γ = (1, 1, 2, 2, 2, 3,3,3,4,4,5,5,5,5,6,6,7,7,7,8,8,8,9,9,9,10,10,11,11,12,12,14,14,15,17,17,17)…”

Section: The Gordon Marking Of An Overpartitionmentioning

confidence: 99%

“…Furthermore, overpartitions possess many analogous properties of ordinary partitions, see Lovejoy [13,15]. For example, various overpartition analogues of the Rogers-Ramanujan-Gordon theorem have been obtained by Corteel and Lovejoy [9], Corteel, Lovejoy and Mallet [10] and Lovejoy [13,14,16,17].…”

Section: Introductionmentioning

confidence: 99%

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The Rogers-Ramanujan-Gordon theorem for overpartitions

Chen

Sang

Shi

2013

Proceedings of the London Mathematical Society

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Let B k,i (n) be the number of partitions of n with certain difference condition and let A k,i (n) be the number of partitions of n with certain congruence condition. The Rogers-Ramanujan-Gordon theorem states that B k,i (n) = A k,i (n). Lovejoy obtained an overpartition analogue of the Rogers-Ramanujan-Gordon theorem for the cases i = 1 and i = k. We find an overpartition analogue of the Rogers-Ramanujan-Gordon theorem in the general case. Let D k,i (n) be the number of overpartitions of n satisfying certain difference condition and C k,i (n) be the number of overpartitions of n whose non-overlined parts satisfy certain congruence condition. We show that C k,i (n) = D k,i (n). By using a function introduced by Andrews, we obtain a recurrence relation that implies that the generating function of D k,i (n) equals the generating function of C k,i (n). By introducing the Gordon marking of an overpartition, we find a generating function formula for D k,i (n) that can be considered an overpartition analogue of an identity of Andrews for ordinary partitions.

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Section: Examiningmentioning

confidence: 99%

“…Here we give an example of the above bijection. Let γ = (1, 1, 2, 2, 2, 3,3,3,4,4,5,5,5,5,6,6,7,7,7,8,8,8,9,9,9,10,10,11,11,12,12,14,14,15,17,17,17)…”

Section: The Gordon Marking Of An Overpartitionmentioning

confidence: 99%

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The Rogers-Ramanujan-Gordon theorem for overpartitions

Chen

Sang

Shi

2013

Proceedings of the London Mathematical Society

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“…The conjecture for j = 1 has been recently proved by Kim [27]. The overpartition analogues of classical partition theorems have been received great attention, see for example Chen, Sang and Shi [13][14][15], Choi, Kim and Lovejoy [16], Corteel and Lovejoy [17], Corteel, Lovejoy and Mallet [18], Corteel and Mallet [19], Dousse [20,21], Goyal [24], He, Ji, Wang and Zhao [25], He, Wang and Zhao [26], Kurşungöz [31], Lovejoy [32,33,[35][36][37], Lovejoy and Mallet [38], Padmavathamma and Raghavendra [39], and Sang and Shi [41]. The main objective of this paper is to give an overpartition analogue of Bressoud's conjectured combinatorial theorem, which provides overpartition analogues of many classical partition theorems including Euler's partition theorem, the Rogers-Ramanujan-Gordon theorem and the Andrews-Göllnitz-Gordon theorem.…”

Section: Introductionmentioning

confidence: 99%

Overpartitions and Bressoud's conjecture, I

He¹,

Ji²,

Zhao³

2019

Preprint

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In a memoir in 1980, Bressoud obtained an analytic generalization of the Rogers-Ramanujan-Gordon identities based on Andrews' generalization of Watson's q-analogue of Whipple's theorem. Let j = 0 or 1, Bressoud also defined two partition functions A j and B j depending on multiple parameters as combinatorial counterparts of his identity, where the function A j can be viewed as the generating function of partitions with certain congruence conditions and the function B j can be viewed as the generating function of partitions with certain difference conditions. Bressoud conjectured A j = B j which specializes to most of the well-known theorems in the theory of partitions, including the Rogers-Ramanujan-Gordon identities and the Andrews-Göllnitz-Gordon identities. Special cases of his conjecture have been subsequently proved by Bressoud, Andrews, Kim and Yee. Recently, Kim proved that Bressoud's conjecture is true for the general case when j = 1.In this paper, we introduce a new partition function B j which could be viewed as an overpartition analogue of the partition function B j introduced by Bressoud. By constructing a bijection, we showed that there is a relationship between B 1 and B 0 and a relationship between B 0 and B 1 . Based on the relationship between B 1 and B 0 and Bressoud's theorems on the Rogers-Ramanujan-Gordon identities and the Göllnitz-Gordon identities, we obtain the overpartition analogue of the Rogers-Ramanujan-Gordon identities due to Chen, Sang and Shi and a new overpartition analogue of the Andrews-Göllnitz-Gordon identities. On the other hand, by using the relation between B 0 and B 1 and Bressoud's conjecture for j = 1 proved by Kim, we obtain an overpartition analogue of Bressoud's conjecture for j = 0, which provides overpartition analogues of many classical partition theorems including Euler's partition theorem. The generating function of the overpartition analogue of Bressoud's conjecture for j = 0 is also obtained with the aid of Bailey pairs.

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“…In recent years, there are many overpartition analogues of classical partition theorems. For example, Corteel and Mallet [13], Corteel, Lovejoy and Mallet [14], Lovejoy [23][24][25][26] found many overpartition analogues of the Rogers-Ramanujan-Gordon theorem. Recently, Chen, Sang and Shi [11] obtained overpartition analogue of Rogers-Ramanujan-Gordon theorem in the general case.…”

Section: Introductionmentioning

confidence: 99%

The Bressoud-Göllnitz-Gordon Theorem for Overpartitions of Even Moduli

He¹,

Wang²,

Zhao³

2017

Taiwanese J. Math.

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We give an overpartition analogue of Bressoud's combinatorial generalization of the Göllnitz-Gordon theorem for even moduli in general case. Let O k,i (n) be the number of overpartitions of n whose parts satisfy certain difference condition and P k,i (n) be the number of overpartitions of n whose non-overlined parts satisfy certain congruence condition. We show that O k,i (n) = P k,i (n) for 1 ≤ i < k.

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Partitions with rounded occurrences and attached parts

Cited by 4 publications

References 18 publications

The Rogers-Ramanujan-Gordon theorem for overpartitions

The Rogers-Ramanujan-Gordon theorem for overpartitions

Overpartitions and Bressoud's conjecture, I

The Bressoud-Göllnitz-Gordon Theorem for Overpartitions of Even Moduli

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