Reductions of additive sets, sets of uniqueness and pyramids (Q1367042)

If \([m]= \{1,\dots, m\}\), \(m\) a positive integer, consider the three-dimensional box \(B= [p]\times [q]\times [r]\), with \(S\) a subset of \(B\). The idea of a reduction of a set is introduced and an equivalence relation is defined on \(B\). Two subsets of \(B\) are box equivalent if both can be reduced to a third set; every set in this class can be reduced to it. A set \(S\) in \(B\) is additive if and only if any reduction of \(S\) is additive. A special class of subsets called pyramids is introduced and it is pointed out that there is a one-to-one correspondence between these and plane partitions, already known to P. A. MacMahon.