  Let \pi be a partition. In [2] we defined BG-rank(\pi) as an alternating sum
of parities of parts. This statistic was employed to generalize and refine the
famous Ramanujan modulo 5 partition congruence. Let p_j(n)(a_{t,j}(n)) denote a
number of partitions (t-cores) of n with BG-rank=j. Here, we provide an elegant
combinatorial proof that 5|p_j(5n+4) by showing that the residue of the 5-core
crank mod 5 divides the partitions enumerated by p_j(5n+4) into five equal
classes. This proof uses the orbit construction in [2] and new identity for
BG-rank. In addition, we find eta-quotient representation for the generating
functions for coefficients a_{t,floor((t+1)/4)}(n), a_{t,-floor((t-1)/4)}(n)
when t is an odd, positive integer. Finally, we derive explicit formulas for
the coefficients a_{5,j}(n) with j=0,1,-1.

Combinatorics;Number Theory