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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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