Hankel Determinants of convoluted Catalan numbers and nonintersecting lattice paths: A bijective proof of Cigler's Conjecture

Abstract

In recent preprints, Cigler considered certain Hankel determinants of convoluted Catalan numbers and conjectured identities for these determinants. In this note, we shall give a bijective proof of Cigler's Conjecture by interpreting determinants as generating functions of nonintersecting lattice paths: this proof employs the reflection principle, the Lindström-Gessel-Viennot-method and a certain construction involving reflections and overlays of nonintersecting lattice paths. Shortly after this bijective proof was presented here, Cigler provided a shorter proof based on earlier results.

arXiv commentCorrected some (minor) errors and included references to the (much shorter) proof Cigler himself provided shortly after the first version of this note