Three correspondences between
nonnegative integer arrays and plane partitions, due to Burge, Knuth, and Hillman
and Grassl, are investigated. A variety of parallel and complementary properties are
derived for the first two. In particular, the images of an array under the
correspondences are characterized in terms of certain sets of “paths” in the arrays.
The correspondences are also related to each other through the action of the dihedral
group on rectangular arrays. Finally, the Hillman-Grassl correspondence is similarly
characterized in terms of sets of paths and is shown to be an extension of Burge’s
correspondence.
