A Generalization of the Farkas and Kra

We prove generalizations of some partition theorems of Farkas and Kra. 1. The Farkas and Kra partition theorem and its generalization In a recent talk given at the University of Melbourne, Hershel Farkas asked for a(nother) proof of the following elegant partition theorem. Theorem 1.1 (Farkas and Kra [2],[3, Chapter 7]). Consider the positive integers such that multiples of 7 occur in two copies, say 7k and 7k. Let E(n) be the number of partitions of the even integer 2n into distinct even parts and let O(n) be the number of partitions of the odd integer 2n+ 1 into distinct odd parts. Then E(n) = O(n). For example, E(9) = O(9) = 9, with the following admissible partitions of 18 and 19: (18), (16, 2), (14, 4), (14, 4), (12, 6), (12, 4, 2), (10, 8), (10, 6, 2), (8, 6, 4)