Foata from Permutation

Foata's fundamental bijection: cycles rotated to their max, ordered by increasing max, parentheses erased.

Permutation → Permutation

The first fundamental transformation — it sends a permutation with k cycles to one with k left-to-right maxima. (The catalog's title also names maj → inv, which is the SECOND fundamental transformation's property; this map is the first.)

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

Live

FindStat Mp00087

inverse first fundamental transformation

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). FindStat's title says inverse first fundamental transformation, while its code is the forward fundamental_transformation(); the match is by value.

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

Let $\sigma = (i_{11}\cdots i_{1k_1})\cdots(i_{\ell 1}\cdots i_{\ell k_\ell})$ be a permutation given by cycle notation such that every cycle starts with its maximal entry, and all cycles are ordered increasingly by these maximal entries.

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

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/Mp00087 has the rest.