Abstract:
Recently new combinatorial interpretations of Ramanujan's
partition congruences modulo 5, 7 and 11 were found. These were
in terms of the crank. A refinement of the congruence modulo 5 is
proved. The number of partitions of 5n+4 with even crank is congruent
to 0 modulo 5. The residue of the even crank modulo 10 divides these
partitions into five equal classes. Other relations for the crank modulo
8, 9 and 10 are also proved. The dissections of certain generating
functions associated with these results are calculated. All of the results are
proved by elementary methods