A new shellability proof of an identity of Dixon

TL;DR

This paper presents a topological combinatorics proof of Dixon's classical identity by constructing shellable simplicial complexes and analyzing their Betti numbers, offering a new perspective on the identity.

Contribution

It introduces a novel shellability-based proof of Dixon's identity through the construction and analysis of infinite families of simplicial complexes.

Findings

Constructed an infinite family of shellable simplicial complexes

Computed Betti numbers using shelling and generating functions

Re-established Dixon's identity via topological methods

Abstract

We give a new proof of an old identity of Dixon (1865-1936) that uses tools from topological combinatorics. Dixon's identity is re-established by constructing an infinite family of non-pure simplicial complexes $\Delta(n)$, indexed by the positive integers, such that the alternating sum of the numbers of faces of $\Delta(n)$ of each dimension is the left-hand side of the identity. We show that $\Delta(n)$ is shellable for all $n$. Then, using the fact that a shellable simplicial complex is homotopy equivalent to a wedge of spheres, we compute the Betti numbers of $\Delta(n)$ by counting (via a generating function) the number of facets of $\Delta(n)$ of each dimension that attach along their entire boundary in the shelling order. In other words, Dixon's identity is re-established by using the Euler-Poincar\'{e} relation.