Generalized Hilbert-Kunz function of the Rees algebra of the face ring of a simplicial complex

Let $R$ be the face ring of a simplicial complex of dimension $d-1$ and $\mathcal{R}(\mathfrak{n})$ be the Rees algebra of the maximal homogeneous ideal $\mathfrak{n}$ of $R\mathrm{.}$ We show that the generalized Hilbert-Kunz function $HK(s)=\mathrm{\ell }(\mathcal{R}(\mathfrak{n})\mathrm{/}(\mathfrak{n},\mathfrak{n}t{)}^{[s]})$ is given by a polynomial for all large $s\mathrm{.}$ We calculate it in many examples and also provide a Macaulay2 code for computing $HK(s)\mathrm{.}$