A new shellability proof of an identity of Dixon

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-Poincaré relation.

Changes to introduction, discussion, future work, and bibliography

References in corpus (1)

• A non-partitionable Cohen-Macaulay simplicial complex