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Bijections of Motzkin Paths Using Shifted Riordan Decompositions

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Abstract

This paper concerns bijections between Motzkin and Łukasiewicz paths arising from Riordan array decompositions. Bijections have been shown between Motzkin paths and Łukasiewicz paths with constant weights (Hennessy in A study of riordan srrays with applications to continued fractions, orthogonal polynomials and lattice paths. Ph.D. thesis, Waterford Institute of Technology, 2011). We introduce a shifting matrix technique to induce a bijection between Motzkin paths and Łukasiewicz paths with non constant weighted steps. We will show a generating function proof and a construction of these bijections.

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Notes

  1. Tridiagonal forms of production matrices have previously been referred to as Stieltjes matrices [14]. For consistency, we will refer to these matrices as production matrices throughout this article.

  2. A combinatorial proof of this identity was one of ten open questions proposed by Louis Shapiro at the Riordan array conference 2016

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  1. Waterford Institute of Technology, Waterford, Republic of Ireland

    Aoife Hennessy

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Hennessy, A. Bijections of Motzkin Paths Using Shifted Riordan Decompositions. Graphs and Combinatorics 35, 169–187 (2019). https://doi.org/10.1007/s00373-018-1982-9

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  • DOI: https://doi.org/10.1007/s00373-018-1982-9

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