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We define nice partitions of the multicomplex associated with a Stanley ideal. As the main result we show that if the monomial ideal $I$ is a CM Stanley ideal, then $I^p$ is a Stanley ideal as well, where $I^p$ is the polarization of $I$.

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 $\mathrm{sd}\left(\Delta \right)$.