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

Garsia and Xin gave a linear algorithm for inverting the sweep map for Fuss rational Dyck paths in $D_{m,n}$ where $m=knpm 1$. 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 $Tin mathcal{T}_n^k$. 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.