Let $M$ be a closed, triangulated manifold of dimension $m$ and $K(M)$ be its triangulation. Let $f_i$ denote the number of $i$-simplices of $K(M)$. As proved by Klee the face numbers satisfy the following *Dehn-Sommerville* relations
$$ f_k = \sum_{i=k}^m (-1)^{i+m} \binom{i+1}{k+1} f_i,$$
for $k = 0, 1, \dots, m$.

The above formula holds true for more general spaces called the *semi-Eulerian complexes* (i.e., these are simplicial complexes such that the Euler characteristic of the link of every non-empty face is equal to that of an appropriate-dimensional sphere).

Recall that the $h$-vector of a simplicial complex is defined as $h_i = \sum_{j=0}^i (-1)^{i-j} \binom{m-j}{m-i} f_{j-1}$ for $0\leq i\leq m+1$. The Dehn-Sommerville relations have a particularly beautiful expression in terms of these $h$-vectors: $$h_{m+1-i} - h_i = (-1)^i\binom{m}{i}(\chi(M) - \chi(S^m)) $$ for $0\leq i\leq m+1$.

My question: Do these relations hold if one were to have a $\Delta$-complex (i.e., semi-simplicial complex) instead of simplicial triangulation of the manifold (one might have to assume the structure to be regular, i.e., attaching maps for homeomorphisms) ?

Is there an analogue of semi-Eulerian complex for $\Delta$-complexes ?