论文信息 - On Some Relations between the Rank and the Crank

On Some Relations between the Rank and the Crank

Abstract Garvan noted that some “curious” relations hold between the numbers N(r, m; n) and M(r, m; n) when m = 5. Alternative versions of these relations are given, together with some similar relations for the case m = 7. Here, N(r, m; n) and M(r, m; n) denote the number of partitions of n whose ranks (as defined by Dyson), respectively cranks (as defined by Andrews and Garvan), are congruent to r modulo n.

Richard Lewis | Richard P. Lewis

[1] George E. Andrews,et al. Dyson's crank of a partition , 1988 .

[2] Frank G. Garvan,et al. New combinatorial interpretations of Ramanujan’s partition congruences mod 5,7 and 11 , 1988 .

[3] Frank G. Garvan,et al. The crank of partitions mod 8,9 and 10 , 1990 .

[4] Peter Swinnerton-Dyer,et al. Some Properties of Partitions , 1954 .

[5] Freeman J. Dyson,et al. Mappings and symmetries of partitions , 1989, J. Comb. Theory, Ser. A.