On the f-vectors of r-multichain subdivisions

Abstract

For a poset P and an integer r≥1, let Pr be the collection of all r-multichains in P. Corresponding to each strictly increasing map ı:[r]→[2r], there is an order ⪯ı on Pr. Let Δ(Gı(Pr)) be the clique complex of the graph Gı associated to Pr and ı. In a recent paper [14], it is shown that Δ(Gı(Pr)) is a subdivision of P for a class of strictly increasing maps. In this paper, we show that all these subdivisions have the same f-vector. We give an explicit description of the transformation matrices from the f- and h-vectors of Δ to the f- and h-vectors of these subdivisions when P is a poset of faces of Δ. We study two important subdivisions, namely Cheeger-Müller-Schrader's subdivision and the r-colored barycentric subdivision which fall in our class of r-multichain subdivisions.