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.
@article{10_37236_8346,
author = {Guoce Xin and Yingrui Zhang},
title = {On the sweep map for {\(\vec{k}\)-Dyck} paths},
journal = {The electronic journal of combinatorics},
year = {2019},
volume = {26},
number = {3},
doi = {10.37236/8346},
zbl = {1420.05018},
url = {http://geodesic.mathdoc.fr/articles/10.37236/8346/}
}
TY - JOUR
AU - Guoce Xin
AU - Yingrui Zhang
TI - On the sweep map for \(\vec{k}\)-Dyck paths
JO - The electronic journal of combinatorics
PY - 2019
VL - 26
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/8346/
DO - 10.37236/8346
ID - 10_37236_8346
ER -
%0 Journal Article
%A Guoce Xin
%A Yingrui Zhang
%T On the sweep map for \(\vec{k}\)-Dyck paths
%J The electronic journal of combinatorics
%D 2019
%V 26
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/8346/
%R 10.37236/8346
%F 10_37236_8346
Guoce Xin; Yingrui Zhang. On the sweep map for \(\vec{k}\)-Dyck paths. The electronic journal of combinatorics, Tome 26 (2019) no. 3. doi: 10.37236/8346
