Given a partition \(\lambda\), we define \(r_k(\lambda)\) to be the number of ways of placing \(k\) rooks on the Young diagram of \(\lambda\) so that no two rooks are in the same row or column. Two partitions, \(\lambda\) and \(\mu\), are said to be rook equivalent if \(r_k(\lambda) = r_k(\mu)\) for all \(k\). A partition \(\lambda\) is a \(t\)-core if none of the hook numbers in its Young diagram are multiples of \(t\). The hook number of a cell in a Young diagram is one more than the number of cells to the right in that row or below in that column. The authors prove that for \(t \leq 4\), two \(t\)-core partitions are rook equivalent if and only if they are conjugate. They give a simple proof that for all \(t \geq 5\), there exist non-conjugate pairs of rook equivalent \(t\)-cores, and conjecture that for each \(t \) there is an integer \(N(t)\) such for all \(n \geq N(t)\), there is a pair of non-conjugate \(t\)-cores of \(n\) that are rook equivalent. For \(t\) = 3 and 4, the authors also explore the number of partitions that are rook equivalent to each \(t\)-core. This problem is related to the calculation of the class number of imaginary quadratic fields through the fact that the class number of the field of discriminant \(-32n-20\) is twice the number of 4-cores of \(n\).