On the $f$-vectors of $r$-multichain subdivisions

For a poset $P$ and an integer $r\geq 1$, let $P_r$ be a collection of all $r$-multichains in $P$. Corresponding to each strictly increasing map $\i:[r]\rightarrow [2r]$, there is an order $\preceq_{\i}$ on $P_r$. Let $\D(G_{\i}(P_r))$ be the clique complex of the graph $G_{\i}$ associated to $P_r$ and $\i$. In a recent paper \cite{NW}, it is shown that $\D(G_{\i}(P_r))$ 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 $\Delta$ to the $f$- and $h$-vectors of these subdivisions when $P$ is a poset of faces of $\D$. We study two important subdivisions Cheeger-M\"{u}ller-Schrader's subdivision and the $r$-colored barycentric subdivision which fall in our class of $r$-multichain subdivisions.