KrewerasComplement

π⁻¹c with the long cycle c = (1 2 ... n), FindStat's order: position i holds c(π⁻¹(i)).

KrewerasComplement(Permutation)w⁻¹c with the long cycle c = (1 2 ... n), FindStat's order: position i holds c(w⁻¹(i)).

Typed signatures

Type	Package	Applies
`((permutation) -> permutation) & ((signed_permutation) -> signed_permutation)`	`combinatorics`	always

Domain: Combinatorial maps

Details

Takes a Permutation and returns a Permutation — a typed map, so a wrong carrier is a type error rather than a wrong answer.
Total, a bijection of S_n; it restricts to the classical Kreweras complement of the non-crossing permutations (those below c in absolute order), mapping NC(n) onto itself. K∘K is conjugation by c.
Also takes a SignedPermutation, returning a SignedPermutation: π⁻¹c with the long cycle c = (1 … n −1 … −n): position i holds c(π⁻¹(i)).
FindStat's Mp00168. A bijection of order 2n.

Examples

Basic #

the one-line word $231$

\mathrm{KrewerasComplement}({\operatorname{Permutation}}([2, 3, 1]))

Out

{\operatorname{Permutation}}([1, 2, 3])

the signed permutation $2\,\bar 3\,1$

\mathrm{KrewerasComplement}({\operatorname{SignedPermutation}}([2, -3, 1]))

Out

{\operatorname{SignedPermutation}}([-1, 2, -3])

About

Examples	2
Crosswalk	Wikipedia Wikidata FindStat ×2