Inverting the General Order Sweep Map
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 -Dyck paths, where 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 . 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.
Cite
@article{arxiv.2307.15357,
title = {Inverting the General Order Sweep Map},
author = {Ying Wang and Guoce Xin and Yingrui Zhang},
journal= {arXiv preprint arXiv:2307.15357},
year = {2024}
}