New identities for 7-cores with prescribed BG-rank

Let ππ be a partition. BG-rank(π)(π) is defined as an alternating sum of parities of parts of ππ [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⩾0at,j(n)qn, where at,j(n)at,j(n) denotes the number of t-cores of n   with BG-rank=jBG-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 function∏j⩾1(1-q7j)7(1-qj).These formulas enable us to establish a number of striking inequalities for a7,j(n)a7,j(n) with j=-1,0,1,2j=-1,0,1,2 and a7(n)a7(n), such asa7(2n+2)⩾2a7(n),a7(4n+6)⩾10a7(n).Here a7(n)a7(n) denotes a number of unrestricted 7-cores of n. Our techniques are elementary and require creative imagination only.