New identities for 7-cores with prescribed BG-rank
| dc.citation.epage | 5259 | en_US |
| dc.citation.issueNumber | 22 | en_US |
| dc.citation.spage | 5246 | en_US |
| dc.citation.volumeNumber | 308 | en_US |
| dc.contributor.author | Berkovich, A. | en_US |
| dc.contributor.author | Yesilyurt, H. | en_US |
| dc.date.accessioned | 2016-02-08T10:06:58Z | |
| dc.date.available | 2016-02-08T10:06:58Z | |
| dc.date.issued | 2008 | en_US |
| dc.department | Department of Mathematics | en_US |
| dc.description.abstract | Let π be a partition. BG-rank(π) is defined as an alternating sum of parities of parts of π [A. Berkovich, F.G. Garvan, On the Andrews-Stanley refinement of Ramanujan's partition congruence modulo 5 and generalizations, Trans. Amer. Math. Soc. 358 (2006) 703-726. [1]]. Berkovich and Garvan [The BG-rank of a partition and its applications, Adv. in Appl. Math., to appear in 〈http://arxiv.org/abs/math/0602362〉] found theta series representations for the t-core generating functions ∑n ≥ 0 at, j (n) qn, where at, j (n) denotes the number of t-cores of n with BG-rank = j. In addition, they found positive eta-quotient representations for odd t-core generating functions with extreme values of BG-rank. In this paper we discuss representations of this type for all 7-cores with prescribed BG-rank. We make an essential use of the Ramanujan modular equations of degree seven [B.C. Berndt, Ramanujan's Notebooks, Part III, Springer, New York, 1991] to prove a variety of new formulas for the 7-core generating functionunder(∏, j ≥ 1) frac((1 - q7 j)7, (1 - qj)) .These formulas enable us to establish a number of striking inequalities for a7, j (n) with j = - 1, 0, 1, 2 and a7 (n), such asa7 (2 n + 2) ≥ 2 a7 (n), a7 (4 n + 6) ≥ 10 a7 (n) . Here a7 (n) denotes a number of unrestricted 7-cores of n. Our techniques are elementary and require creative imagination only. © 2007 Elsevier B.V. All rights reserved. | en_US |
| dc.identifier.doi | 10.1016/j.disc.2007.09.044 | en_US |
| dc.identifier.eissn | 1872-681X | |
| dc.identifier.issn | 0012-365X | |
| dc.identifier.uri | http://hdl.handle.net/11693/22956 | |
| dc.language.iso | English | en_US |
| dc.publisher | Elsevier BV * North-Holland | en_US |
| dc.relation.isversionof | http://dx.doi.org/10.1016/j.disc.2007.09.044 | en_US |
| dc.source.title | Discrete Mathematics | en_US |
| dc.subject | 7 - cores | en_US |
| dc.subject | BG - rank | en_US |
| dc.subject | Modular equations | en_US |
| dc.subject | Partition inequalities | en_US |
| dc.subject | Positive eta - quotients | en_US |
| dc.subject | 7 - cores | en_US |
| dc.subject | BG - rank | en_US |
| dc.subject | Modular equations | en_US |
| dc.subject | Partition inequalities | en_US |
| dc.subject | Positive eta-quotients | en_US |
| dc.subject | Function evaluation | en_US |
| dc.title | New identities for 7-cores with prescribed BG-rank | en_US |
| dc.type | Article | en_US |
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