  For any $n > 0$ and $0 \leq m < n$, let $P_{n,m}$ be the poset of projective
equivalence classes of $\{-,0,+\}$-vectors of length $n$ with sign variation
bounded by $m$, ordered by reverse inclusion of the positions of zeros. Let
$\Delta_{n,m}$ be the order complex of $P_{n,m}$. A previous result from the
third author shows that $\Delta_{n,m}$ is Cohen-Macaulay over $\mathbb{Q}$
whenever $m$ is even or $m = n-1$. Hence, it follows that the $h$-vector of
$\Delta_{n,m}$ consists of nonnegative entries. Our main result states that
$\Delta_{n,m}$ is partitionable and we give an interpretation of the $h$-vector
when $m$ is even or $m = n-1$. When $m = n-1$ the entries of the $h$-vector
turn out to be the new Eulerian numbers of type $D$ studied by Borowiec and M\l
otkowski in [{\em Electron. J. Combin.}, 23(1):Paper 1.38, 13, 2016]. We then
combine our main result with Klee's generalized Dehn-Sommerville relations to
give a geometric proof of some facts about these Eulerian numbers of type $D$.

Combinatorics