Latin hypercubes realizing integer partitions

Abstract

For an integer partition $h_1+\cdots +h_n=N$, a 2-realization of this partition is a latin square of order $N$ with disjoint subsquares of orders $h_1,\dots ,h_n$. The existence of 2-realizations is a partially solved problem posed by Fuchs. In this paper, we extend Fuchs' problem to $m$-ary quasigroups, or, equivalently, latin hypercubes. We construct latin cubes for some partitions with at most two distinct parts and highlight how the new problem is related to the original.

17 pages, 16 figures