Counting Young Tableaux When Rows Are Cosets

Abstract

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 $T(n)=\sum _{d|n}T(d,n/d)$. 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.