Enumerating several statistics of r-colored Dyck paths with no dd-steps having the same colors

Abstract

An r-colored Dyck path is a Dyck path with all d-steps having one of r colors in $\left[r\right]=\{1,2,\dots ,r\}$. In this paper, we consider several statistics on the set $A_{n,0}^{\left(r\right)}$ of r-colored Dyck paths of length 2n with no two consecutive d-steps having the same colors. Precisely, the paper studies the statistics “number of points” at level ℓ, “number of u-steps” at level $\ell +1$, “number of peaks” at level $\ell +1$ and “number of udu-steps” on the set $A_{n,0}^{\left(r\right)}$. The counting formulas of the first three statistics are established by Riordan arrays related to $S(a,b;x)$, the weighted generating function of $(a,b)$-Schröder paths. By a useful and surprising relations satisfied by $S(a,b;x)$, several identities related to these counting formulas are also described.