Combinatorial Proof of the Minimal Excludant Theorem

2019/08/19 by Ballantine, Cristina, Merca, Mircea · 1 citation
#05A19 #11A63 #11P81 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1908.06789

Abstract

The minimal excludant of a partition λ, \rmmex(λ), is the smallest positive integer that is not a part of λ. For a positive integer n, σ \rmmex(n) denotes the sum of the minimal excludants of all partitions of n. Recently, Andrews and Newman obtained a new combinatorial interpretations for σ \rmmex(n). They showed, using generating functions, that σ \rmmex(n) equals the number of partitions of n into distinct parts using two colors. In this paper, we provide a purely combinatorial proof of this result and new properties of the function σ \rmmex(n). We generalize this combinatorial interpretation to σ_r \rmmex(n), the sum of least r-gaps in all partitions of n. The least r-gap of a partition λ is the smallest positive integer that does not appear at least r times as a part of λ.

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