Discrete Tomography and plane partitions

A \textit{plane partition} is a $p\times q$ matrix $A=(a_{ij})$, where $1\leq i\leq p$ and $1\leq j\leq q$, with non-negative integer entries, and whose rows and columns are weakly decreasing. From a geometric point of view plane partitions are equivalent to \textit{pyramids}, subsets of the integer lattice $\mathbb{Z}^3$ which play an important role in Discrete Tomography. As a consequence, some typical problems concerning the tomography of discrete lattice sets can be rephrased and considered via plane partitions. In this paper we focus on some of them. In particular, we get a necessary and sufficient condition for additivity, a canonical procedure for checking the existence of (weakly) bad configurations, and an algorithm which constructs "minimal" pyramids with assigned projection of a bad configurations.