Sweeping up Zeta

TL;DR

This paper proves that modular sweep maps are bijective by constructing their inverses via equitable partitions, establishing the bijectivity of general sweep maps, and confirming the zeta map on rational Dyck paths as a bijection.

Contribution

It introduces a novel proof of bijectivity for modular and general sweep maps using equitable partitions and lattice theory, including the first proof for the zeta map on rational Dyck paths.

Findings

Modular sweep maps are bijective.

General sweep maps are bijective.

Zeta map on rational Dyck paths is a bijection.

Abstract

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.