Elliptic Curves, Riordan arrays and Lattice Paths

2025/07/22 by Paul Barry, Barry, Paul
Mathematics · #11B37 #11B83 #14H52 #15B36 #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics #Primary 05A15 #Secondary 11G05

paper · pdf · doi:10.48550/arxiv.2507.16765

openalex publication_date 2025/07/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this note, we show that to each elliptic curve of the form y²-axy-y=x³-bx²-cx, we can associate a family of lattice paths whose step set is determined by the parameters of the elliptic curve. The enumeration of these lattice paths is by means of an associated Riordan array. The curves and the paths have associated Somos 4 sequences which are essentially the same. For the curves the link to Somos 4 sequences is a classical result, via the elliptic divisibility sequence. For the paths the link is via a Hankel transform.

Citations

• Notes on Riordan arrays and lattice paths
• Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials
• Hankel Determinant Solution for Elliptic Sequence
• Integrality and the Laurent phenomenon for Somos 4 sequences
• The On-Line Encyclopedia of Integer Sequences

Related

• Notes on Riordan arrays and lattice paths
• Generating functions of lattice paths
• Riordan Paths and Derangements
• Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials
• Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences