InverseFoataHan from SignedPermutation

The inverse of Foata and Han's transformation: the last letter is peeled off and the blocks of the rest are rotated back.

SignedPermutation → SignedPermutationbijectioninverse

FindStat's Mp00194, sending the flag inversion number to the flag major index.

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

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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 Mp00194

Foata-Han inverse

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 inverse of 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/Mp00194 has the rest.