KrewerasComplement from Permutation

w⁻¹c, for w below the long cycle c = (1 2 ... n) in absolute order.

Permutation → Permutation

Only defined on the non-crossing permutations (the embedding of NC(n) in S_n by absolute order beneath c) — everywhere else this declines rather than answering for a word it was never defined on. K∘K is conjugation by c, and K is a bijection of NC(n) onto itself.

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

Live

FindStat Mp00088

Kreweras complement

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

Sends the permutation $\pi \in \mathfrak{S}_n$ to the permutation $\pi^{-1}c$ where $c = (1,\ldots,n)$ is the long cycle.

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

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/Mp00088 has the rest.