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

For any finite connected poset P, Galashin introduced a simple convex $(|P|-2)$-dimensional polytope 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≤ i ≤ n, let Pᵢ be the poset obtained from P by replacing S by an antichain of size i. We show that the h-polynomial of A(P) can be written in terms of the h-polynomials of A(Pᵢ) and type B Narayana polynomials. We then use the identity to deduce several identities involving Narayana polynomials, Eulerian polynomials, and stack-sorting preimages.