CyclePartition from Permutation

The orbits, as a set partition of the positions.

Permutation → SetPartition

Removed once for giving every position the same label. The cause was the laziness rule in tableau.ts — folding over a list taken out of the accumulator instead of indexing a range. Written by index it is right first time.

Called as CombinatorialMap(x, "CyclePartition"). The head's page has its examples.

Live

FindStat Mp00151

to cycle type

Matched by value: our map agrees with FindStat's image on each of the 873 rows of size at most 6 we can read (0 unread; its table has 1200 rows).

SurjectionGraded mapas FindStat records them, checked by FindStat's contributors.

Let $\pi=c_1\dots c_r$ a permutation of size $n$ decomposed in its cyclic parts. The associated set partition of $[n]$ then is $S=S_1\cup\dots\cup S_r$ such that $S_i$ is the set of integers in the cycle $c_i$.

Object	Image
[1]	{{1}}
[1,2]	{{1},{2}}
[2,1]	{{1,2}}
[1,2,3]	{{1},{2},{3}}

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/Mp00151 has the rest.