Two examples of mixed lattice point enumerators

Let ${\cal P}_1,\dots,{\cal P}_k$ be convex polytopes in $\mathbb{R}^n$ with integer vertices, and let $t_1,\dots,t_k$ be nonnegative integers. Define the ``Minkowski sum'' $$ t_1{\cal P}_1 +\cdots+t_k{\cal P}_k=\{t_1v_1+\cdots+t_kv_k\,:\, v_i\in{\cal P}_i\}. $$ The \emph{mixed lattice point enumerator} of $P_1,\dots,P_k$ is the function of $t_1,\dots,t_k$ defined by $$ F(t_1,\dots,t_k) = \#\left( (t_1{\cal P}_1 +\cdots+t_k{\cal P}_k )\cap\mathbb{Z}^n\right). $$ It was first shown by McMullen that $F(t_1,\dots,t_k)\in \mathbb{Q}[t_1,\dots,t_k]$. The function $F(t_1,\dots,t_k)$ contains a lot of interesting information about the ${\cal P}_i$'s, such as their volumes, mixed volumes, and Ehrhart polynomials. We will discuss two recently discovered examples of mixed lattice point enumerators. The first (in joint work with A. Postnikov) is closely connected with Kostant's partition function for the root system $A_n$. The second (with J. Pitman) is closely connected with parking functions.