Let p(n) denote the number of partitions of n. Ramanujan discovered and later proved that p(5n+4) =0 (mod 5) (1).

The first combinatorial interpretation of (1) is due to Dyson, who introduced new (then) integral partition statistic rank:= (largest part of partition) - (number of parts of partition) and conjectured that rank divides partitions enumerated by p(5n+4) into 5 equal classes. Garvan, Kim and Stanton provided the first combinatorial proof of (1).

In my talk I outline the combinatorial proof of the new refinement of (1), which emerged in my recent joint work with Prof. Frank Garvan. This refinement uses new partition statistic, termed bg-rank. Remarkably, bg-rank enables one to discover new congruences mod 5 for partitions of n not = 4 (mod 5).