[Abstract] F.G. Garvan ; The crank of partitions mod 8, 9 and 10 , Trans Amer. Math. Soc., 322 (1990), 79-94.


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