A new shellability proof of an identity of Dixon

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 th...

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Bibliographic Details
Published in:arXiv.org 2016-05
Main Authors: Davidson, Ruth, O'Keefe, Augustine, Parry, Daniel
Format: Article
Language:English
Subjects:
Combinatorial analysis
Integers
Online Access:Get full text
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Summary: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.
ISSN:2331-8422