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.