Geodesic
$f$-vectors of subdivided simplicial complexes (extended abstract)
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010) (2010).

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We take a geometric point of view on the recent result by Brenti and Welker, who showed that the roots of the $f$-polynomials of successive barycentric subdivisions of a finite simplicial complex $X$ converge to fixed values depending only on the dimension of $X$. We show that these numbers are roots of a certain polynomial whose coefficients can be computed explicitly. We observe and prove an interesting symmetry of these roots about the real number $-2$. This symmetry can be seen via a nice realization of barycentric subdivision as a simple map on formal power series. We then examine how such a symmetry extends to more general types of subdivisions. The generalization is formulated in terms of an operator on the (formal) ring on the set of simplices of the complex. 
@article{DMTCS_2010_special_259_a17,
     author = {Delucchi, Emanuele and Pixton, Aaron and Sabalka, Lucas},
     title = {$f$-vectors of subdivided simplicial complexes (extended abstract)},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010)},
     year = {2010},
     doi = {10.46298/dmtcs.2822},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2822/}
}
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Delucchi, Emanuele; Pixton, Aaron; Sabalka, Lucas. $f$-vectors of subdivided simplicial complexes (extended abstract). Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010) (2010). doi : 10.46298/dmtcs.2822. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2822/

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