GENERALIZATIONS OF DYSON RANK AND NON-ROGERS-RAMANUJAN PARTITIONS

For any fixed integer k greater-than-or-equal-to 2, we define a statis tic on partitions called the k-rank. The definition involves the decom position into successive Durfee squares. Dyson's rank corresponds to t he 2-rank. Generating function identities are given. The sign of the k -rank is reversed by an involution which we call k-conjugation. We pro ve the following partition theorem: the number of self-2k-conjugate pa rtitions 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. Th is generalizes the well-known result: the number of self-conjugate par titions of n is equal to the number of partitions into distinct odd pa rts.