Colored Motzkin Paths of Higher Order

Motzkin paths of order 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 <= l. We further generalize order-l Motzkin paths by allowing for various coloring schemes on the edges of our paths. These ((alpha)over-right-arrow, (beta)over-right-arrow)-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 ((alpha)over-right-arrow, (beta)over-right-arrow)-colored Motzkin paths and a variety of well-studied combinatorial objects. Specific coloring schemes ((alpha)over-right-arrow, (beta)over-right-arrow) allow us to place ((alpha)over-right-arrow, (beta)over-right-arrow)-colored Motzkin paths in bijection with different sub-classes 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 ((alpha)over-right-arrow, (beta)over-right-arrow)-colored Motzkin paths and certain sub-classes of k-ary trees.