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arXiv · 1811.07475

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

Guoce Xin·Yingrui Zhang
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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.

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Guoce Xin, Yingrui Zhang. 2018-11-19. On the Sweep Map for $\vec{k}$-Dyck Paths. https://arxiv.org/abs/1811.07475

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