Inequalities for the Crank

Garvan first defined certain “vector partitions” and assigned to each such partition a “rank.” Denoting byNV(r, m, n) the (weighted) count of the vector partitions ofnwith rankrmodulom, he gave a number of relations between the numbersNV(r, m, mn+k) whenm=5, 7 and 11, 0⩽r,k<m. The true crank whose existence was conjectured by Dyson was discovered by Andrews and Garvan who also showed thatNV(r, m, n)=M(r, m, n) unlessn=1, whereM(r, m, n) denotes the number of partitions ofnwhose cranks are congruent tormodulem. In the case of module 11, a simpler form of Garvanʹs results have been found by Hirschhorn. In fact, the Hirschhorn result was derived using Winquistʹs identity, but the details were omitted. In this work, from the simpler form we deduce some new inequalities between theM(r, 11, 11n+k)ʹs and give the details of Hirschhornʹs result. We also prove some conjectures of Garvan in the case of module 7.