Existence of frame SOLS of type \(a^n b^1\) (Q1613445)

A frame self-orthogonal Latin square, \(\text{FSOLS}(h_1^{n_1}h_2^{n_2}\dots h_k^{n_k})\), is a holey Latin square orthogonal to its transpose, where \(h_1^{n_1}h_2^{n_2}\dots h_k^{n_k}\) denotes its type based on the hole set which partitions the underlying set of elements. The authors use a series of constructions, either known or generalized, to prove the existence of \(\text{FSOLS}(a^nb^1)\), when \(n\) is even and \(\text{FSOLS}(a^n 1^1)\) when \(n\) is odd.