On some relations between the rank and the crank (Q1185885)

Let \(N(r,m;n)\) denote the number of partitions of \(n\) with rank (as defined by Dyson) congruent to \(r\pmod m\). Let \(M(r,m;n)\) denote the number of partitions of \(n\) with crank (as defined by Andrews and Garvan) congruent to \(r\pmod m\). In this paper a number of linear relations among the functions \(N(r,5;5n+k)\) and \(M(r,5;5n+k)\) are established for various values of \(r\) and \(k\), as well as several relations among \(N(r,7;7n+k)\) and \(M(r,7;7n+k)\). These include various relations established previously by Garvan plus some additional ones. Garvan had also proved certain linear relations among the \(M(r,m;n)\) with \(m=8,9,10\) and 11. The author presents as conjectures a further set of linear relations with \(m=8\) and 9. In a note added in the proof the author remarks that since writing this paper, he and Santa Gadea have proved these conjectures.