Colored Motzkin Paths of Higher Order

TL;DR

This paper introduces a generalized class of colored Motzkin paths of higher order, explores their enumeration via Riordan arrays, and establishes bijections with various well-known combinatorial objects, broadening understanding of their structure.

Contribution

It generalizes higher-order Motzkin paths with coloring schemes, connects them to Riordan arrays, and links them to multiple combinatorial structures through bijections.

Findings

Enumeration via Riordan arrays for colored paths

Bijections with generalized Dyck paths and $k$-ary trees

Extension of combinatorial interpretations of Motzkin paths

Abstract

Motzkin paths of order-$\ell$ are a generalization of Motzkin paths that use steps $U=(1,1)$, $L=(1,0)$, and $D_i=(1,-i)$ for every positive integer $i \leq \ell$. We further generalize order-$\ell$ Motzkin paths by allowing for various coloring schemes on the edges of our paths. These $(\vec{\alpha},\vec{\beta})$-colored Motzkin paths may be enumerated via proper Riordan arrays, mimicking the techniques of Aigner in his treatment of Catalan-like numbers. After an investigation of their associated Riordan arrays, we develop bijections between $(\vec{\alpha},\vec{\beta})$-colored Motzkin paths and a variety of well-studied combinatorial objects. Specific coloring schemes $(\vec{\alpha},\vec{\beta})$ allow us to place $(\vec{\alpha},\vec{\beta})$-colored Motzkin paths in bijection with different subclasses of generalized $k$-Dyck paths, including $k$-Dyck paths that remain weakly above horizontal lines $y=-a$, $k$-Dyck paths whose peaks all have the same height modulo-$k$, and Fuss-Catalan generalizations of Fine paths. A general bijection is also developed between $(\vec{\alpha},\vec{\beta})$-colored Motzkin paths and certain subclasses of $k$-ary trees.