Berkovich, A.Yesilyurt, H.2016-02-082016-02-0820080012-365Xhttp://hdl.handle.net/11693/22956Let 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.English7 - coresBG - rankModular equationsPartition inequalitiesPositive eta - quotients7 - coresBG - rankModular equationsPartition inequalitiesPositive eta-quotientsFunction evaluationArticle10.1016/j.disc.2007.09.0441872-681X