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Washington State University

Combinatorics, Linear Algebra and Number Theory (CLaN) Seminar

Department of Mathematics

Neill Hall 5W

February 22, Monday, 4:10 - 5:00 PM

Ruth Davison

Department of Mathematics

UIUC

Combinatorics, Linear Algebra and Number Theory (CLaN) Seminar

Department of Mathematics

Neill Hall 5W

February 22, Monday, 4:10 - 5:00 PM

Ruth Davison

Department of Mathematics

UIUC

Title: A new shellability proof of an old 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 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. No background in topological combinatorics will be assumed for this talk. This is joint work with Augustine O'Keefe and Daniel Parry.

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 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. No background in topological combinatorics will be assumed for this talk. This is joint work with Augustine O'Keefe and Daniel Parry.