Let $S=\lbrace s_1,\ldots,s_n \rbrace \subset \lbrace1,\ldots,2n\rbrace$. Consider two operations on $S$: the complement $C(S)=\lbrace 1,\ldots,2n \rbrace \setminus S$ and a reflection* $R(S)=\lbrace2n+1−s_1,\ldots,2n+1−s_n\rbrace$. In general, $C(S) \ne R(S)$, but the sums of their elements are equal.
E.g., $S=\lbrace3,4,5\rbrace \subset \lbrace1,\ldots,6\rbrace$ has $C(S)=\lbrace1,2,6\rbrace$, sum 9, and $R(S)=\lbrace2,3,4\rbrace$, sum 9.
My question is not how to prove this (it's a nice proof appropriate for a discrete math course), rather the history of this result or at least a citation. For me, this arose in looking at applications of permutations to fair division of indivisible goods.
Footnote * This operation $R$ is also called the complement in Egge, Annals of Combinatorics, 2007; Marc van Leeuwen suggested calling it reflection to avoid confusion.
Caveat: I recently ran this question on math.stackexchange where, despite a bonus, it generated no answers. I'm still figuring out which site is better for certain questions.
