A new generalization of the minimal excludant arising from an analogue of Franklin's identity

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 $\sigma_{rc} \textup{mex}(n)$, the sum of $r$-chain mex taken over all partitions of $n$, thereby deducing a combinatorial identity for $\sigma_{rc} \textup{mex}(n)$, which neatly generalizes the result of Andrews and Newman for $\sigma \textup{mex}(n)$, the sum of mex over all partitions of $n$.