ArcRepresentation from SetPartition
Each position linked to the next in its block, or to itself when last.
The statistics frontier calls this the arc representation: within each block, consecutive elements (b1,b2), (b2,b3), .... Encoding it as an endofunction rather than a bare list of pairs keeps it a typed carrier — Crossings, Nestings and CrossingNestingTotal (@enumeratio/statistics) read the arcs off this without needing a carrier of their own.
Called as CombinatorialMap(x, "ArcRepresentation"). The head's page has its examples.
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Each element of SetPartition at a size, and its image, applied in your browser; sizes past 7 are not offered.
Properties
Transport a statistic
Pick a statistic on Endofunction. Its distribution over the images of SetPartition at each size is computed, and compared with the 111 statistics on SetPartition: those that share it are the ones the map sends it to, as far as the sizes computed show.
No statistic is declared on Endofunction.