Enumeration of the faces of complexes and normalizations of distributive lattices

For an arbitrary face system $\Phi\subseteq 2^{[m]}$ of the power set of the set $[m]=\{1,\dots,m\}$, we consider the vector descriptions $f(\Phi;m),h(\Phi;m)\in Q^{m+1}$ and the generating functions
$$ F_{\Phi;m}(y-1)=\sum_{l=0}^mf_l(\Phi;m)(y-1)^{m-l} = H_{\Phi;m}(y)=\sum_{l=0}^mh_{l}(\Phi;m)y^{m-l}, $$
where $f_l(\Phi;m)=|\{A\in\Phi\colon |A|=l\}|$, $0\leq l\leq m$. The corresponding valuations on the Boolean lattice of all subsets of the power set $2^{[m]}$ are defined.
For a partition of a face system $\Phi\subseteq{2}^{[m]}$ into Boolean intervals such that the partition consists of $p_{i,j}$ intervals $[A,B]$ with $|A|=j$ and $|B-A|=i$,
$$ h_l(\Phi;m)=(-1)^l\sum_{i=0}^{m-l}\sum_{j=0}^l (-1)^j p_{i,j} \binom{m-i-j}{l-j}. $$

For a pair of mutually dual face systems $\Phi,\Phi^*\subseteq2^{[m]}$, where $\Phi^*=\{[m]-A\colon A\in{2}^{[m]}, A\notin\Phi\}$,
$$ h_l(\Phi;m)+(-1)^l\sum_{j=l}^m \binom jlh_j(\Phi^*;m)=0, \qquad 1\leq l\leq m. $$