Enumerating several statistics of r-Colored Dyck paths with no dd-steps having the same colors
Yidong Sun, Jinyi Wang, Xinyu Wang

TL;DR
This paper investigates various combinatorial statistics of r-colored Dyck paths with no consecutive same-colored down steps, providing explicit formulas and identities using Riordan arrays and Schröder path generating functions.
Contribution
It introduces new counting formulas for specific statistics on r-colored Dyck paths with no consecutive same-colored down steps, connecting them to Riordan arrays and Schröder path generating functions.
Findings
Derived formulas for the number of points at level ℓ
Established counts for u-steps and peaks at certain levels
Discovered identities related to the generating functions of these paths
Abstract
An -colored Dyck path is a Dyck path with all -steps having one of colors in . In this paper, we consider several statistics on the set of -colored Dyck paths of length with no two consecutive -steps having the same colors. Precisely, the paper studies the statistics ``number of points" at level , ``number of -steps" at level , ``number of peaks" at level and ``number of -steps" on the set . The counting formulas of the first three statistics are established by Riordan arrays related to , the weighted generating function of -Schr\"{o}der paths. By a useful and surprising relations satisfied by , several identities related to these counting formulas are also described.
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