2006/03/06 by Alexander Bérkovich, Alexander Berkovich, Berkovich, Alexander +2
Engineering · Mathematics · #Algebraic structures and combinatorial models #Finite Group Theory Research #graph theory and CDMA systems #math.CO #math.NT #msc:05A20 #msc:11F27
paper · pdf · doi:10.48550/arxiv.math/0603150
12 pages
arxiv created 2007/09/15 · arxiv updated 2009/12/01
A q-series with nonnegative power series coefficients is called positive. The partition statistics BG-rank is defined as an alternating sum of parities of parts of a partition. It is known that the generating function for the number of partitions of n that are 7-cores with given BG-rank can be written as certain sum of multi-theta functions. We give explicit representations for these generating functions in terms of sums of positive eta-quotients and derive inequalities for the their coefficients. New identities for the generating function of unrestricted 7-cores and inequalities for their coefficients are also obtained. Our proofs utilize Ramanujan's theory of modular equations.