2020/12/29 by Isaac DeJager, DeJager, Isaac, Madeleine Naquin +5 · 1 citation
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Geometric and Algebraic Topology #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2012.14947
openalex publication_date 2020/12/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Motzkin paths of order-ℓ are a generalization of Motzkin paths that use steps U=(1,1), L=(1,0), and Di=(1,-i) for every positive integer i ≤ ℓ. We further generalize order-ℓ Motzkin paths by allowing for various coloring schemes on the edges of our paths. These (α,β)-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 (α,β)-colored Motzkin paths and a variety of well-studied combinatorial objects. Specific coloring schemes (α,β) allow us to place (α,β)-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 (α,β)-colored Motzkin paths and certain subclasses of k-ary trees.