KrewerasComplement from SignedPermutation

π⁻¹c with the long cycle c = (1 … n −1 … −n): position i holds c(π⁻¹(i)).

SignedPermutation → SignedPermutationbijection

FindStat's Mp00168. A bijection of order 2n.

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

Live

Explore

Each element of SignedPermutation at a size, and its image, applied in your browser; sizes past 6 are not offered.

Size

Properties

Sizes 1 to

Transport a statistic

Pick a statistic on SignedPermutation. Its distribution over the images of SignedPermutation at each size is computed, and compared with the 18 statistics on SignedPermutation: those that share it are the ones the map sends it to, as far as the sizes computed show.

Statistic NegativeEntryCountPositiveEntryCountTypeBInversionCountNegativeSumPairCountFlagMajorIndexFlagDescentFlagExcedanceFlagWeakExcedancePositiveBeforeLargerNegativePairCountIncreasingPositivePairCountPositiveBeforeNegativeCountRightDescentsSignedPermutationOrderSignedReflectionLengthFlagDenertIndexSignedSortingIndexSignedAlignmentCountStasinskiVollLength Sizes 1 to

Also from

The same name out of other carriers, each its own implementation: Permutation.

FindStat Mp00168

Kreweras complement

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

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

The Kreweras complement of a signed permutation.

Object	Image
[1]	[-1]
[-1]	[1]
[1,2]	[2,-1]
[2,1]	[-1,2]

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/Mp00168 has the rest.