Generalized Motzkin Paths

Faculty Sponsor

Dr. Paul Drube


Arts and Sciences



ORCID Identifier(s)

Madeleine Naquin 0000-0003-2176-511X, Frank Seidl 0000-0001-7752-9103, Isaac DeJager 0000-0003-4925-9518

Presentation Type

Poster Presentation

Symposium Date

Summer 7-29-2019


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.

Biographical Information about Author(s)

Isaac DeJager is a rising junior at LeTourneau University in Longview, TX, where he is majoring in Mathematics and Computer Science and pursuing a minor in Physics. He grew up in San Jose, CA.

Madeleine Naquin is a rising senior at Spring Hill College in Mobile, AL, where she is majoring in Mathematics and minoring in Economics. She is from New Orleans, LA.

Frank Seidl is a rising sophomore at the University of Michigan in Ann Arbor, MI, where he is majoring in Mathematics and Computer Science. He is from Ann Arbor, MI.

DeJager, Naquin, and Seidl are all students participating in the Valparaiso Experience in Research for Undergraduate Mathematicians (VERUM).

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