Resolution of a conjecture of Andrews and Lewis involving cranks of partitions

Andrews and Lewis have conjectured that the sign of the number of partitions of $n$ with crank congruent to 0 mod 3, minus the number of partitions of $n$ with crank congruent to 1 mod 3, is determined by the congruence class of $n$ mod 3 apart from a finite number of specific exceptions. We prove this by using the ``circle method" to approximate the value of this difference to great enough accuracy to determine its sign for all sufficiently large $n$.