FoataMajorToInversions from IntegerComposition

Foata's bijection on the word of parts: each part is put on the end of the image so far, which is cut into blocks, and every block is rotated one place right.

IntegerComposition → IntegerCompositionbijectioninverse

FindStat's Mp00314. An equal part counts as the smaller; it sends the major index to the number of inversions.

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

Live

Explore

Each element of IntegerComposition 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 IntegerComposition. Its distribution over the images of IntegerComposition at each size is computed, and compared with the 44 statistics on IntegerComposition: those that share it are the ones the map sends it to, as far as the sizes computed show.

Statistic LargestPartFirstPartLastPartSmallestPartDistinctPartsLargestMultiplicitySmallestMultiplicityDistinctMultiplicitiesAscentsWeakDescentCountPeaksInversionsCompositionMajorIndexMajorIndexWeakMajorIndexAbsoluteVariationVariationAscentVariationStrongRecordCountWeakRecordCountStrongRecordPositionSumWeakRecordPositionSumLongestWeaklyIncreasingSubsequenceLongestDecreasingSubsequenceLongestStaircaseConstantRunCountLongestConstantRunShortestConstantRunBargraphPeakCountBargraphHalfPerimeterBargraphUpStepCountBargraphValleyHeightSumAsymmetryDegreePartsEqualToReversedPartRotationOrbitSizeStandardImmaculateTableauCountStandardRibbonTableauCountRibbonCornerCountSeaweedIndexDyckPathsAboveBouncePathCountInverseDescentParkingFunctionCountStandardCompositionTableauCountCompositionNakayamaGlobalDimensionDualImmaculateMonomialColumnSum Sizes 1 to

Also from

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

FindStat Mp00314

Foata bijection

Matched by value: our map agrees with FindStat's image on each of the 63 rows of size at most 6 we can read (0 unread; its table has 1023 rows).

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

The Foata bijection for compositions.

Object	Image
[1]	[1]
[2]	[2]
[3]	[3]
[4]	[4]

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/Mp00314 has the rest.