Finney has used orthogonal partitions in the context of the search for higher order (coarser) partitions of given Latin squares. Hedayat and Seiden use the term $F$-square to denote higher order partitions that are orthogonal to both rows and columns. This note is a short expository treatment of orthogonal partitions in general and is based on the identification of a partition with a vector subspace of Euclidean $N$-space $R^N$. This identification is not new as it is part of the usual vector space approach to analysis of variance. This approach puts the concept of orthogonal partitions in a simple light unencumbered by the language of design of experiments. Another advantage is that certain published bounds on the maximum number of orthogonal partitions of specified type are immediate from the dimensionality restriction imposed by $R^N$. In addition, some counting problems are identified which are of possible interest to researchers in design of experiments and combinatorics.
"A Note on Orthogonal Partitions and Some Well-Known Structures in Design of Experiments." Ann. Statist. 5 (3) 565 - 570, May, 1977. https://doi.org/10.1214/aos/1176343857