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Keywords:
symmetric group; simplicial complex; $f$- and $h$-vector (triangle); barycentric subdivision of a simplicial complex
symmetric group; simplicial complex; $f$- and $h$-vector (triangle); barycentric subdivision of a simplicial complex
Summary:
For a simplicial complex $\Delta $ we study the behavior of its $f$- and $h$-triangle under the action of barycentric subdivision. In particular we describe the $f$- and $h$-triangle of its barycentric subdivision $\mathop {\rm sd}(\Delta )$. The same has been done for $f$- and $h$-vector of $\mathop {\rm sd}(\Delta )$ by F. Brenti, V. Welker (2008). As a consequence we show that if the entries of the $h$-triangle of $\Delta $ are nonnegative, then the entries of the $h$-triangle of $\mathop {\rm sd}(\Delta )$ are also nonnegative. We conclude with a few properties of the $h$-triangle of $\mathop {\rm sd}(\Delta )$.
References:
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[2] Brenti, F., Welker, V.: $f$-vectors of barycentric subdivisions. Math. Z. 259 849-865 (2008). DOI 10.1007/s00209-007-0251-z | MR 2403744 | Zbl 1158.52013
[3] Miller, E., Sturmfels, B.: Combinatorial Commutative Algebra. Graduate Texts in Mathematics 227 Springer, New York (2005). MR 2110098 | Zbl 1090.13001
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