Inequalities for the crank

Abstract

Garvan first defined certain "vector partitions" and assigned to each such partition a "rank." Denoting by N-V(r, m, n) the (weighted) count of the vector partitions of Ir with rank I module III, he gave a number of relations between the numbers N-v(r, m, mn + k) when m = 5, 7 and 11, 0 less than or equal to r, k < m. The true crank whose existence was conjectured by Dyson was discovered by Andrews and Garvan who also showed that N-V(r, m, n) = M(r, m, n) unless n = 1, where M(r, m, n) denotes the number of partitions of n whose cranks are congruent to r module m. In the case of module 11, a simpler form of Garvan's results have been found by Hirschhorn. In fact, the Hirschhorn result was derived using Winquist's identity, but the details were omitted. In this work, from the simpler form we deduce some new inequalities between the M(r, 11, 11n + k)'s and give the details of Hirschhorn's result. We also prove some conjectures of Garvan in the case of module 7. (C) 1998 Academic Press.