The Narayana polynomials (OEIS [A001263](https://oeis.org/A001263)) are the h-polynomials of the associahedra (the Stasheff polytopes) and their dual simplicial polytopes (cf. the Fomin and Reading ref in the OEIS entry). 


Are these polynomials related to the [Ehrhart series](https://en.m.wikipedia.org/wiki/Ehrhart_polynomial) (cf. also [Computing the Continuous Discretely](http://math.sfsu.edu/beck/ccd.html) by Beck and Robins) of any families of polytopes? 


Are there any properties of the Narayana polynomials that preclude them from being the numerator polynomials associated to the Ehrhart series rational functions of any family of polytopes?



Narayana polynomials as numerators of Ehrhart series rational functions?
The $n$th Narayana polynomial is the numerator of the generating function for the Ehrhart polynomial of the order polytope and the chain polytope of the product of a 2-element chain and an $n$-element chain. 


