On the sweep map for \(\vec{k}\)-Dyck paths (Q2325763)

Summary: \textit{A. Garsia} and \textit{G. Xin} [Electron. J. Comb. 24, No. 1, Research Paper P1.64, 9 p. (2017; Zbl 1358.05022)] 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 tableaux. 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.