Generalized Motzkin Paths
Madeleine Naquin, Frank Seidl, Isaac DeJager · ValpoScholar (Valparaiso University) · 2019
Motzkin paths are integer lattice paths that use steps U=(1,1), L=(1,0), D=(1,-1) and stay weakly above the line y=0. We generalize Motzkin paths to allow for down steps with multiple slopes, and then allow for various coloring schemes on the edges of the resulting paths. These higher-order, colored Motzkin paths provide a general setting where specific coloring schemes yield sets that are in bijection with many well-studied combinatorial objects. We developed bijections between various classes of higher-order, colored Motzkin paths and generalized l-ary paths, generalized Fine paths, and certain subclasses of l-ary trees. All of this utilizes the language of proper Riordan arrays, and we close by proving a series of results about the Riordan arrays whose entries enumerate our sets of generalized Motzkin paths.