Combinatorial Proof of the Minimal Excludant Theorem · arXivDesk1908.06789Aug 19, 201915 pages; this version includes a combinatorial proof of the generalization
Combinatorial Proof of the Minimal Excludant Theorem
Cristina Ballantine, Mircea Merca
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
. Recently, Andrews and Newman obtained a new combinatorial interpretations for
σmex(n) . They showed, using generating functions, that
σmex(n) equals the number of partitions of
into distinct parts using two colors. In this paper, we provide a purely combinatorial proof of this result and new properties of the function
σmex(n) . We generalize this combinatorial interpretation to
σrmex(n) , the sum of least
-gaps in all partitions of
. The least
-gap of a partition
is the smallest positive integer that does not appear at least
times as a part of
.