  Using techniques introduced by H. Thomas and N. Williams in "Cyclic Symmetry
of the Scaled Simplex," we prove that modular sweep maps are bijective. We
construct the inverse of the modular sweep map by passing through an
intermediary set of equitable partitions; motivated by an analogy to stable
marriages, we prove that the set of equitable partitions for a fixed word forms
a distributive lattice when ordered componentwise. We conclude that the general
sweep maps defined by D. Armstrong, N. Loehr, and G. Warrington in "Sweep Maps:
A Continuous Family of Sorting Algorithms" are bijective. As a special case of
particular interest, this gives the first proof that the zeta map on rational
Dyck paths is a bijection.
