FoataHan from SignedPermutation

Foata and Han's fundamental transformation: each letter in turn is put on the end of the image so far, which is cut into blocks and rotated.

SignedPermutation → SignedPermutationbijectioninverse

FindStat's Mp00190, a bijection that sends the flag major index to the flag inversion number. Inverse: InverseFoataHan.

Called as CombinatorialMap(x, "FoataHan"). 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

FindStat Mp00190

Foata-Han

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 Foata-Han bijection for signed permutations.

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

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/Mp00190 has the rest.