Promotion of Lattice Paths by Riordan Arrays

Aoife Hennessy (), Kieran Murphy, Narciso Gonzaga and Paul Barry
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Aoife Hennessy: Department of Computing and Mathematics, School of Science and Computing, Waterford Campus, South East Technological University, X91 K0EK Waterford, Ireland
Kieran Murphy: Department of Computing and Mathematics, School of Science and Computing, Waterford Campus, South East Technological University, X91 K0EK Waterford, Ireland
Narciso Gonzaga: Department of Computing and Mathematics, School of Science and Computing, Waterford Campus, South East Technological University, X91 K0EK Waterford, Ireland
Paul Barry: Department of Computing and Mathematics, School of Science and Computing, Waterford Campus, South East Technological University, X91 K0EK Waterford, Ireland

Mathematics, 2025, vol. 13, issue 18, 1-24

Abstract: This paper investigates the use of Riordan arrays in the enumeration and transformation of lattice paths through a combinatorial framework of promotion. We demonstrate how Dyck paths can be promoted to generalised Motzkin and Schröder paths via two key transformations: the Binomial and Chebyshev transforms, each associated with specific Riordan arrays. These promotions yield classical integer sequences and continued fraction representations that enumerate weighted lattice paths. The framework is further extended to analyse grand paths, which are permitted to cross below the x-axis. We develop constructive bijections establishing explicit correspondences between promoted path families. The promotion framework offers new insights into known integer sequences and enables a unified approach to the generalisation and classification of lattice paths.

Keywords: Riordan arrays; lattice paths; enumerative combinatorics (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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