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 $R.$ We show that the generalized Hilbert-Kunz function $HK(s)=ell({mathcal R}(mathfrak n)/(mathfrak n, mathfrak n t)^{[s]})$ is given by a polynomial for all large $s.$ We calculate it in many examples and also provide a Macaulay2 code for computing $HK(s).$