Abstract: For any fixed integer $k\ge2$, we define a statistic on partitions called the $k$-rank. The definition involves the decomposition into successive Durfee squares. Dyson's rank corresponds to the $2$-rank. Generating function identities are given. The sign of the $k$-rank is reversed by an involution which we call $k$-conjugation. We prove the following partition theorem: the number of self-$2k$-conjugate partitions of $n$ is equal to the number of partitions of $n$ with no parts divisible by $2k$ and the parts congruent to $k$ (mod $2k$) are distinct. This generalizes the well-known result: the number of self-conjugate partitions of $n$ is equal to the number of partitions into distinct odd parts.