Hello!

I'm working on a problem that has boiled down to combinatorial matrix theory (well, really, an array). Basically, one is given a skew-symmetric, n by n Latin Square, which uses the entries 1, 2, \dots, n. The first row (and first column) has the ordered entries 1, 2, \dots, n, and the ith row/column has the entries i, i+1, i+2, \dots, i+n-1 taken mod n. The problem involves taking this Latin Square, choosing a random entry and changing it (and its skew-symmetric equivalent) to some other entry from the set \lbrace 1, 2, \dots, n \rbrace. We then permit, in only whatever row/column this transformation has taken place, that there may be two identical elements, but that the array must otherwise satisfy the properties of a Latin Square. Is there a basic theorem that states that it is possible to achieve this by reorganizing the other elements in the Latin Square?

For instance, in row/column one, I change the "2" entry to a "1", and we say that row/column 1 may contain two entries of "1", but that row/column two may not (which already contains a "1" entry), and we therefore must perform some series of transpositions on the other entries in the Latin Square to achieve that it is it is otherwise still a Latin Square, if not for the case of there being two 1's in row/column one, which we allow.