constructible. In the rest of this paper, a simplicial cell $d$-ball which is constructible (respectively shellable) will be called a *constructible $d$-ball* (respectively *shellable $d$-ball*).

We need the following well-known properties of constructibility and shellability. See e.g., [@Bj2 Theorem 11.4 and Section 12].

-   Let $P$ be a simplicial cell sphere. If $Q \subsetneq P$ is a shellable simplicial poset with $\dim Q=\dim P$, then $Q$ is a simplicial cell ball.

-   Boolean algebras and $\Delta_d(k)$ with $k <d$ are shellable balls.

-   Let $P$ and $Q$ be constructible $d$-balls, and let $I \subsetneq \partial P$ and $J \subsetneq \partial Q$ be order ideals with $I \cong J$. If $I$ is a constructible $(d-1)$-ball then $P {_{I}\cup_J} Q$ is a constructible $d$-ball.

For a finite set $X$, we write $\# X$ for the cardinality of $X$. The next well-known lemma, which immediately follows from Lemma \[3.2\] and the fact that a Boolean algebra has the $h$-vector $(1,0,\dots,0)$, is useful to compute $h$-vectors of shellable simplicial posets.

\[shellablehvector\] Let $P$ be a $(d-1)$-dimensional shellable simplicial poset with a shelling $\sigma_1,\sigma_2,\dots,\sigma_r$. Then, for $i=1,2,\dots,d$, one has $$h_i(P)=\# \big\{ \ell \in \{2,3,\dots,r\}: 
\langle \sigma_1,\sigma_2,\dots,\sigma_{\ell-1} \rangle \cap \langle \sigma_\ell \rangle
\cong \Delta_d(i)
\big\}.$$

Now 