  For any finite connected poset $P$, Galashin introduced a simple convex
$(|P|-2)$-dimensional polytope $\mathscr{A}(P)$ called the poset associahedron.
Let $P$ be a poset with a proper autonomous subposet $S$ that is a chain of
size $n$. For $1\leq i \leq n$, let $P_i$ be the poset obtained from $P$ by
replacing $S$ by an antichain of size $i$. We show that the $h$-polynomial of
$\mathscr{A}(P)$ can be written in terms of the $h$-polynomials of
$\mathscr{A}(P_i)$ and type B Narayana polynomials. We then use the identity to
deduce several identities involving Narayana polynomials, Eulerian polynomials,
and stack-sorting preimages.
