Determinants of (generalised) Catalan numbers

TL;DR

This paper explains recent determinant evaluations involving Catalan numbers using lattice path combinatorics and a simple determinant lemma, leading to natural extensions and generalisations.

Contribution

It introduces a combinatorial explanation for determinant evaluations of Catalan numbers using the Lindström-Gessel-Viennot theorem and a determinant lemma, enabling broader generalisations.

Findings

Provides a combinatorial interpretation of determinant evaluations

Extends the evaluations to more general cases

Connects lattice path theory with algebraic determinant identities

Abstract

We show that recent determinant evaluations involving Catalan numbers and generalisations thereof have most convenient explanations by combining the Lindstr\"om-Gessel-Viennot theorem on non-intersecting lattice paths with a simple determinant lemma from [Manuscripta Math. 69 (1990), 173-202]. This approach leads also naturally to extensions and generalisations.