On the Sweep Map for k-Dyck Paths

2018/11/19 by Xin, Guoce, Zhang, Yingrui · 2 citations
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1811.07475

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± 1. They introduced an intermediate family T_n^k of certain standard Young tableau. Then inverting the sweep map is done by a simple walking algorithm on a T∈ T_n^k. We find their idea naturally extends for k^±-Dyck paths, and also for k-Dyck paths (reducing to k-Dyck paths for the equal parameter case). The intermediate object becomes a similar type of tableau in T_k of different column lengths. This approach is independent of the Thomas-Williams algorithm for inverting the general modular sweep map.

Cited by

• Symmetry of the refined q,t-Catalan polynomials for k-Dyck paths
• Enumerating several statistics of r-Colored Dyck paths with no dd-steps having the same colors

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