Abstract:
Euler's classical identity states that the number of partitions of an integer into odd parts and distinct parts are equinumerous. Franklin gave a generalization by considering partitions with exactly j different multiples of r, for a positive integer r. We prove an analogue of Franklin's identity by studying the number of partitions with j multiples of r in total and in the process, discover a natural generalization of the minimal excludant (mex) which we call the r-chain mex. Further, we derive the generating function for , the sum of r-chain mex taken over all partitions of n, thereby deducing a combinatorial identity for , which neatly generalizes the result of Andrews and Newman for , the sum of mex over all partitions of n.