The fact that there are eight different sets of four integers between 1 and 8 that sum to 18 invites the creation of a sort of magic square where each set occurs once as a row or column. There are even a few ways of doing this so that the diagonals also sum to 18. This seems like the sort of idea that is obvious enough that it would have to have been looked at before, but I have no idea how to find out where it may have been previously discussed.
Long shot, and not the sort of answer you're looking for, but maybe this fact is related to some of the bijections between bipartitions of a set of size 8 and 2-dimensional subspaces of (F_2)^4 that are used to construct the Golay code, Mathieu groups, etc. I know that in other parts of those constructions weight enumerator things come up where i has weight i, i.e. you might care about sums of subsets.
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