Séminaire Lotharingien de Combinatoire, 84B.23 (2020), 12 pp.

Cristina Ballantine and Mircea Merca

The Minimal Excludant and Colored Partitions

Abstract. The minimal excludant of a partition λ, mex(λ), is the smallest positive integer that is not a part of λ. For a positive integer n, $ σmex(n) denotes the sum of the minimal excludants of all partitions of n. Recently, Andrews and Newman obtained a new combinatorial interpretation for σmex(n). They showed, using generating functions, that σmex(n) equals the number of partitions of n into distinct parts using two colors. We give a purely combinatorial proof of this result and derive its generalization to the sum of least r-gaps. We introduce several new identities connecting the function σmex(n) to the number of partitions with colored parts satisfying certain congruences.

Received: November 20, 2019. Accepted: February 20, 2020. Final version: April 30, 2020.

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