Complement from SignedPermutation

Each entry ±i replaced by ±(n + 1 − i), its sign kept.

SignedPermutation → SignedPermutationinvolution

FindStat's Mp00188.

Called as CombinatorialMap(x, "Complement"). 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, SetPartition, SetComposition, IntegerComposition, BinaryWord.

FindStat Mp00188

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 mapInvolutionas FindStat records them, checked by FindStat's contributors.

The 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/Mp00188 has the rest.