[@Ko] studied $h$-vectors of simplicial cell balls and gave a characterization of $h$-vectors of simplicial cell balls up to dimension $6$. In this paper, we extend the results of Kolins and give a complete characterization of $h$-vectors of simplicial cell balls.

Let $P$ be a simplicial poset. We say that an element $\sigma \in P$ has *rank $i$*, denoted ${\ensuremath{\mathrm{rank}}\hspace{1pt}}\sigma =i$, if $[\hat 0,\sigma]$ is a Boolean algebra of rank $i$. The *dimension* of $P$ is $$\dim P= \max \{{\ensuremath{\mathrm{rank}}\hspace{1pt}}\sigma : \sigma \in P\}-1.$$ Let $d=\dim P +1$ and let $f_i=f_i(P)$ be the number of elements $\sigma \in P$ having rank $i+1$ for $i=-1,0,\dots,d-1$. Thus $f_i$ is the number of $i$-dimensional cells of $\Gamma (P)$. The vector $f(P)=(f_{-1},f_0,\dots,f_{d-1})$ is called the *$f$-vector (face vector) of $P$*. We define the *$h$-vector $h(P)=(h_0(P),h_1(P),\dots,h_d(P)) \in { \ensuremath{\mathbb{Z}}}^{d+1}$ of $P$* by the relation $$\sum_{i=0}^d f_{i-1}  (1-t)^{d-i} = \sum_{i=0}^d h_i(P) t^{d-i}.$$ Then knowing $f(P)$ is equivalent to knowing $h(P)$.

On face vectors of simplicial cell spheres, Stanley [@St] and Masuda [@Ma] proved the following result, which characterizes all possible $h$-vectors of simplicial cell spheres.

\[1.1\] Let ${\mathbf{h}}=(h_0,h_1,\dots,h_d) \in { \ensuremath{\mathbb{Z}}}^{d+1}$. Then ${\mathbf{h}}$ is the $h$-vector of a simplicial cell $(d-1)$-sphere if and only if it 