On the Sweep Map for $\vec{k}$-Dyck Paths

Abstract

Garsia and Xin gave a linear algorithm for inverting the sweep map for Fuss rational Dyck paths in $D_{m,n}$ where $m=kn\pm 1$. They introduced an intermediate family ${\mathcal{T}}_n^k$ of certain standard Young tableau. Then inverting the sweep map is done by a simple walking algorithm on a $T\in {\mathcal{T}}_n^k$. We find their idea naturally extends for ${\mathbf{k}}^{\pm }$-Dyck paths, and also for $\mathbf{k}$-Dyck paths (reducing to $k$-Dyck paths for the equal parameter case). The intermediate object becomes a similar type of tableau in ${\mathcal{T}}_{\mathbf{k}}$ of different column lengths. This approach is independent of the Thomas-Williams algorithm for inverting the general modular sweep map.