The BG-rank of a partition and its applications

Let π   denote a partition into parts λ1⩾λ2⩾λ3⩾⋯. In a 2006 paper we defined BG-rank(π)(π) asBG-rank(π)=∑j⩾1(−1)j+11−(−1)λj2. This statistic was employed to generalize and refine the famous Ramanujan modulo 5 partition congruence. Let pj(n)pj(n) denote the number of partitions of n   with BG-rank=jBG-rank=j. Here, we provide a combinatorial proof thatpj(5n+4)≡0(mod5),j∈Z, by showing that the residue of the 5-core crank mod 5 divides the partitions enumerated by pj(5n+4)pj(5n+4) into five equal classes. This proof uses the orbit construction from our previous paper and a new identity for the BG-rank. Let at,j(n)at,j(n) denote the number of t-cores of n   with BG-rank=jBG-rank=j. We find eta-quotient representations for∑n⩾0at,⌊t+14⌋(n)qnand∑n⩾0at,−⌊t−14⌋(n)qn, when t   is an odd, positive integer. Finally, we derive explicit formulas for the coefficients a5,j(n)a5,j(n), j=0,±1j=0,±1.