Dyson’s crank of a partition

Abstract

largest part minus the number of parts. He let 7V(ra, t, n) denote the number of partitions of n of rank congruent to m modulo £, and he conjectured (1.1) 7V(m, 5,5n + 4) = ±p(5n + 4), 0 < m < 4; (1.2) JV(ra,7,7n + 5) = ±p(7n + 5), 0 < m < 6,