Coefficients of Ehrhart polynomials, in the binomial-coefficient basis

Question

Let $P$ be the convex hull of a finite set of points in $\mathbb Z^d$, and $p(n) = \#\{nP \cap \mathbb Z^d\}$ be its Ehrhart polynomial, which is also the Hilbert polynomial of the corresponding projective toric variety.

If we write it as $\sum_{k=0}^d c_k {n \choose k}$, then since $p$ is $\mathbb Z$-valued, the $c_k$ are also in $\mathbb Z$; one can compute $c_k$ as $(\Delta^k p)(0)$ where $(\Delta q)(m) = q(m)-q(m-1)$.

Is there a succinct description of the coefficients $c_k$ in terms of the original $P$?

Answer 1

One possible answer: if you write the Ehrhart polynomial as $\sum_{k=0}^{d} c_k\binom{n-1}{k}$, then the coefficients $c_k$ are called the $f^{\ast}$-vector by Felix Breuer in the paper Ehrhart f*-coefficients of polytopal complexes are non-negative integers. He gives a combinatorial interpretation in terms of counting 'atomic lattice points' by 'height'. The relevant material is in Section 3. Of course, if you want to use $\binom{n}{k}$, then I do not know if there is any obvious combinatorial interpretation, aside from the fact that the resulting coefficients can be easily computed from the $f^{\ast}$-vector using the Pascal relation for binomial coefficients.