New identities for 7-cores with prescribed BG-rank

Let be a partition. BG-rank() is defined as an alternating sum of parities of parts of [1]. In [2], Berkovich and Garvan found theta series representations for the t-core generating functions Pn0 at,j(n)qn. Here, at,j(n) denotes a 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 7 [3] to prove a variety of new formulas for 7-core generating function Y j1 (1 - q7j)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 as a7(2n + 2) 2a7(n), a7(4n + 6) 10a7(n). Here a7(n) denotes a number of unrestricted 7-cores of n. Our techniques are elementary and require creative imagination only. `Behind every inequality there lies an i...

• Type: Journal
• Year: 2008
• Where: DM
• Authors: Alexander Berkovich, Hamza Yesilyurt
• Keywords: 7-core Generating Function, DM 2008, Ramanujan Modular Equations, Series Representations