# Narayana polynomials as numerators of Ehrhart series rational functions?

The Narayana polynomials (OEIS 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 (cf. also Computing the Continuous Discretely 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?

• As an example of where Ehrhart power series arise, see "Stringy Chern classes of singular toric varieties and their applications" by Batyrev and Schaller arxiv.org/abs/1607.04135 – Tom Copeland Nov 5 '17 at 16:54

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.

• Thanks. Could you provide a reference? Also, it would be nice to have this info in the OEIS along with a ref. I could enter this link and paste this info there in the comment section if you don't wish to do so yourself. – Tom Copeland Jul 1 '17 at 17:29
• The linear extensions $\sigma$ of the direct product of a 2-element chain and $n$-element chain are in bijection with ballot sequences $\beta$ of length $2n$, and the descent set of $\sigma$ is the same as that of $\beta$. Moreover, it is well-known (though I don't know a precise reference) that the Narayana polynomial enumerates ballot sequences by number of descents. The conclusion follows from standard facts about order polytopes and $P$-partitions. I can provide further details if necessary. I don't know whether this result has already appeared somewhere, – Richard Stanley Jul 2 '17 at 11:41
• "Two Poset Polytopes" by Richard P. Stanley dedekind.mit.edu/~rstan/pubs/pubfiles/66.pdf – Tom Copeland Nov 19 '19 at 13:28

As another example, the polytopes

$$\mathrm{conv}\{0, e_i - e_j \mid 1 \leq j < i \leq n \} \subset \mathbb{R}^n$$

also have Ehrhart $h^*$-polynomials which are the Narayana polynomials. This is Example 6 of Benjamin Braun's survey article Unimodality Problems in Ehrhart Theory.

• Thanks. Would be nice to have depictions of the low dimensional polytopes. – Tom Copeland Aug 3 '17 at 19:08