If you generate the distance matrix of distances between all set partitions over a 7-element set and then make a contour-plot showing only level 5, this is what you get.

Don't let yourself be fooled by the labels on the axes. This is a plot over discrete space. I simply haven't figured out how to fit 877 labels (each consisting of 7 numerals) on the axes. If I did, you'd see (1,1,1,1,1,1,1) instead of 0.0 and (1,2,3,4,5,6,7) instead of 1.0. Yes, and everything in "between".

The set partitions are ordered lexicographical. E.g. (1,2,1) directly precedes (1,2,2).

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