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 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.
Highlights
The sweep map is a mysterious simple sorting algorithm that is invertible
Based on the idea of Thomas and Williams, Garsia and Xin [5] gave a geometric construction for inverting the sweep map on (m, n)-rational Dyck paths
Problem: Is there a linear algorithm to invert the sweep map, at least for a more general class of paths? We find positive answers for k+, k−, and k-Dyck paths by extending Garsia-Xin’s idea
Summary
The sweep map is a mysterious simple sorting algorithm that is invertible. The best way to introduce the sweep map is by using rational Dyck paths, because it already raises complicated enough problems and it has natural generalizations. Based on the idea of Thomas and Williams, Garsia and Xin [5] gave a geometric construction for inverting the sweep map on (m, n)-rational Dyck paths. These algorithms are nice iteration algorithms, but are not linear: the number of iterations is measured by the sum of the ranks of D. By using a completely different approach, Garsia and Xin find a O(m + n) algorithm for inverting sweep map on (m, n)-Dyck paths in the Fuss case m = kn ± 1
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