On Completing Latin Squares

This paper characterizes a particular scheme of partially filled Latin squares and when they can be completed to full Latin squares. In particular, given an \(n \times n\) array with the first \(s\) rows and the first \(d\) cells of row \(s+1\) filled with \(n\) distinct symbols in such a way that no symbol occurs more than once in any row or column, necessary and sufficient conditions are found for when this array can be completed to a full Latin square.