On the combinatorics of r-chain minimal and maximal excludants

Abstract

The minimal excludant (mex) of a partition was introduced by Grabner and Knopfmacher under the name `least gap' and was revived by a couple of papers due to Andrews and Newman. It has been widely studied in recent years together with the complementary partition statistic maximal excludant (maex), first introduced by Chern. Among such recent works, the first and second authors along with Maji introduced and studied the $r$-chain minimal excludants (r-chain mex) which led to a new generalization of Euler's classical partition theorem and the sum-of-mex identity of Andrews and Newman. In this paper, we first give combinatorial proofs for these two results on r-chain mex. Then we introduce the r-chain maximal excludants (r-chain maex) and establish the associated identity for the sum of r-chain maex over all partitions, both analytically and combinatorially.