An identity involving $h$-polynomials of poset associahedra and type B Narayana polynomials

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