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

TL;DR

This paper introduces a new generalization of the minimal excludant (mex) called the r-chain mex, inspired by an analogue of Franklin's identity, and derives its generating function and combinatorial properties.

Contribution

It presents a novel generalization of the mex called the r-chain mex and establishes its generating function and combinatorial identities, extending previous results by Andrews and Newman.

Findings

Derived the generating function for the sum of r-chain mex over partitions.

Established a combinatorial identity generalizing Andrews and Newman's result.

Proved an analogue of Franklin's identity involving partitions with multiples of r.

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 $n$.