Inverting the General Order Sweep Map

TL;DR

This paper introduces a family of bijective Order sweep maps for general Dyck paths, extending previous algorithms for rational Dyck paths and unifying various sweep map variants within a comprehensive framework.

Contribution

It generalizes the inversion algorithm for the sweep map to a broader class of Dyck paths, establishing bijections and unifying multiple sweep map variants.

Findings

Developed a family of bijective Order sweep maps for general Dyck paths.

Extended the inversion algorithm from rational Dyck paths to broader classes.

Unified different sweep map variants within a single framework.

Abstract

Building upon the foundational work of Thomas and Williams on the modular sweep map, Garsia and Xin have developed a straightforward algorithm for the inversion of the sweep map on rational $(m,n)$-Dyck paths, where $(m,n)$ represents coprime pairs of integers. Our research reveals that their innovative approach readily generalizes to encompass a broader spectrum of Dyck paths. To this end, we introduce a family of Order sweep maps applicable to general Dyck paths, which are differentiated by their respective sweep orders at level $0$. We demonstrate that each of these Order sweep maps constitutes a bijective transformation. Our findings encapsulate the sweep maps for both general Dyck paths and their incomplete counterparts as specific instances within this more extensive framework.