Counting Young Tableaux When Rows Are Cosets

Let \(T(m,n)\) denote the number of \(m \times n\) rectangular standard Young tableaux with the property that the difference of any two rows has all entries equal. Let \(T(n) = \sum\limits_{d|n} T(d,n/d)\). We find recurrence relations satisfied by the numbers \(T(m,n)\) and \(\hat{T}(n)\), compute their generating functions, and express them explicitly in some special cases.