# Reciprocity for multi-parameter Ehrhart polynomials

In McMullen's 1977 paper "Valuations and Euler-type relations on certain classes of convex polytopes" (https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms/s3-35.1.113), he shows that for $\mathcal{Q}_1,\ldots,\mathcal{Q}_m$ lattice polytopes in $\mathbb{R}^n$, the quantity $$L_{\mathcal{Q}_1,\ldots,\mathcal{Q}_m}(k_1,\ldots,k_m) := \#((k_1\mathcal{Q}_1+\cdots+k_m\mathcal{Q}_m)\cap\mathbb{Z}^n)$$ is given by a polynomial in $k_1,\ldots,k_m$ for $k_1,\ldots,k_m\in\mathbb{Z}_{\geq0}$. This is a "multi-parameter" Ehrhart polynomial. For a usual Ehrhart polynomial $L_{\mathcal{P}}(k)$ of a lattice polytope $\mathcal{P}$ in $\mathbb{R}^n$, we can interpret $L_\mathcal{P}(-k)$ for $k\in\mathbb{Z}_{\geq0}$ via Ehrhart reciprocity: $L_\mathcal{P}(-k)=(-1)^{\mathrm{dim}(\mathcal{P})}\cdot \# (\,\mathrm{int}(k\mathcal{P})\cap\mathbb{Z}^n)$.

Question: Is there any combinatorial interpretation of $L_{\mathcal{Q}_1,\ldots,\mathcal{Q}_m}(k_1,\ldots,k_m)$ for $k_1,\ldots,k_m\in\mathbb{Z}$?

In that same paper McMullen does give a general "reciprocity" result for arbitrary polytope valuations as an alternating sum over faces, but here I am asking about a very specific valuation where maybe we can do better (like we can in the case of the single variable Ehrhart polynomial). On the other hand, some simple examples show that the sign of $L_{\mathcal{Q}_1,\ldots,\mathcal{Q}_m}(k_1,\ldots,k_m)$ is not easily predicted, so my guess is that there is no nice combinatorial interpretation. But this feels like the kind of thing people may have looked at.

• As a somewhat related question, one could ask if there is some connection between $L_{\mathcal{Q}_1,\dots,\mathcal{Q}_m}(k_1,\dots,k_m)$ and $\#((k_1\mathcal{P}_1+\cdots+k_m\mathcal{P}_m)\cap \mathbb{Z}^n)$, where each $\mathcal{P_i}$ is either $\mathcal{Q}_i$ or $\mathrm{int}{\mathcal{Q}_i}$. – Richard Stanley Mar 29 '18 at 16:10