InverseKrewerasComplement

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

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

Typed signatures

Type	Package	Applies
`((dyck_path) -> dyck_path) & ((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.
FindStat's Mp00089; the inverse of KrewerasComplement.
Also takes a SignedPermutation, returning a SignedPermutation: cπ⁻¹ with the long cycle c = (1 … n −1 … −n): position i holds π⁻¹(c(i)).
FindStat's Mp00167, the inverse of KrewerasComplement.
Also takes a DyckPath, returning a DyckPath: The inverse Kreweras complement of the path read as a noncrossing partition: the blocks are the downs after each up step.
FindStat's Mp00327. A bijection of each size; Mp00089 is the same map on noncrossing permutations.

Examples

Basic #

the one-line word $231$

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

Out

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

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

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

Out

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

the Dyck path $UUDDUD$

\mathrm{InverseKrewerasComplement}({\operatorname{DyckPath}}([1, 1, 0, 0, 1, 0]))

Out

{\operatorname{DyckPath}}([1, 0, 1, 1, 0, 0])

About

Examples	3
Crosswalk	FindStat ×3