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 $\mathrm{sd}\left(\Delta \right)$. The same has been done for $f$- and $h$-vector of $\mathrm{sd}\left(\Delta \right)$ 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 $\mathrm{sd}\left(\Delta \right)$ are also nonnegative. We conclude with a few properties of the $h$-triangle of $\mathrm{sd}\left(\Delta \right)$.

Order complex is an important object associated to a partially ordered set. Following a suggestion from V. A. Vassiliev (1994), we investigate an order complex associated to the partially ordered set of nontrivial ideals in a commutative ring with identity. We determine the homotopy type of the geometric realization for the order complex associated to a general commutative ring with identity. We show that this complex is contractible except for semilocal rings with trivial Jacobson radical when...