Centered polygon numbers, heptagons and nonagons, and the Robbins numbers

TL;DR

This paper investigates determinantal formulas for Robbins numbers using advanced combinatorial techniques, linking them to polygonal numbers and conjecturing new Hankel transform relations involving Fibonacci and Catalan numbers.

Contribution

It introduces new determinantal representations of Robbins numbers and explores their connections to polygonal numbers, Fibonacci, and Catalan numbers, including conjectures on Hankel transforms.

Findings

Determinantal descriptions of Robbins numbers using Hankel and symmetric matrices

Connections established between Robbins numbers and centered polygonal numbers, heptagons, and nonagons

Conjectured Hankel transform determinants involving Fibonacci and Catalan numbers

Abstract

In this note, we explore certain determinantal descriptions of the Robbins numbers. Techniques used for this include continued fractions, Riordan arrays and series inversion. Proven and conjectured representations involve the determinants of both Hankel and symmetric matrices. In specific cases, links are drawn to centered polygonal numbers, and to heptagons and nonagons. We conjecture a Hankel transform determinant for the Robbins numbers related to the Fibonacci and the Catalan numbers.