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 $n$.