  Dyson's rank function and the Andrews--Garvan crank function famously give
combinatorial witnesses for Ramanujan's partition function congruences modulo
5, 7, and 11. While these functions can be used to show that the corresponding
sets of partitions split into 5, 7, or 11 equally sized sets, one may ask how
to make the resulting bijections between partitions organized by rank or crank
combinatorially explicit. Stanton recently made conjectures which aim to
uncover a deeper combinatorial structure along these lines, where it turns out
that minor modifications of the rank and crank are required. Here, we prove two
of these conjectures. We also provide abstract criteria for quotients of
polynomials by certain cyclotomic polynomials to have non-negative coefficients
based on unimodality and symmetry. Furthermore, we extend Stanton's conjecture
to an infinite family of cranks. This suggests further applications to other
combinatorial objects. We also discuss numerical evidence for our conjectures,
connections with other analytic conjectures such as the distribution of
partition ranks.
