A new shellability proof of an identity of Dixon
| Autor: | Davidson, Ruth, O'Keefe, Augustine, Parry, Daniel |
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| Rok vydání: | 2015 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: Changes to introduction, discussion, future work, and bibliography |
| Databáze: | arXiv |
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