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 $\mathbf{d}$-steps having one of $r$ colors in $[r]={1, 2, \dots, r}$. In this paper, we consider several statistics on the set $\mathcal{A}<em>{n,0}<sup>{(r)}$ of $r$-colored Dyck paths of length $2n$ with no two consecutive $\mathbf{d}$-steps having the same colors. Precisely, the paper studies the statistics number of points" at level $\ell$,number of $\mathbf{u}$-steps" at level $\ell+1$, number of peaks" at level $\ell+1$ andnumber of $\mathbf{udu}$-steps" on the set $\mathcal{A}</em>{n,0}<sup>{(r)}$. The counting formulas of the first three statistics are established by Riordan arrays related to $S(a,b; x)$, the weighted generating function of $(a,b)$-Schr\"{o}der paths. By a useful and surprising relations satisfied by $S(a,b; x)$, several identities related to these counting formulas are also described.