Complement from SetPartition

Each element i replaced by n + 1 - i: the set partition seen from the other end.

SetPartition → SetPartitioninvolution

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

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Each element of SetPartition at a size, and its image, applied in your browser; sizes past 7 are not offered.

Size

Properties

Sizes 1 to

Transport a statistic

Pick a statistic on SetPartition. Its distribution over the images of SetPartition at each size is computed, and compared with the 111 statistics on SetPartition: those that share it are the ones the map sends it to, as far as the sizes computed show.

Statistic BlocksLargestBlockSmallestBlockBlockSizeSpanSingletonBlocksBlocksAtLeastTwoBlocksSizeTwoLastBlockSizeCrossingsNestingsCrossingNestingTotalSetPartitionRankDimensionIndexSumOfBlockMinimaSumOfBlockMaximaAntisingletonsSingletonsAndAntisingletonsBlocksAndAntisingletonsSuccessionsMaximalBlockDifferenceFirstBlockCardinalityLargestElementInFirstBlockInternalPointsOverlappingBlockPairsSameBlockPairsSameBlockTriplesDistinctBlockPairsDimensionExponentLargestOpenerSmallestCloserTerminalClosersBlocksWithOddMinimumBlocksWithEvenMinimumRotationOrbitSizeIntertwiningNumberRightOpenerSmallerRightOpenerBiggerLeftOpenerSmallerRightCloserSmallerRightCloserBiggerLeftCloserSmallerLeftCloserBiggerBlockMajorIndexFlattenedMajorIndexHuangRhoadesMajorIndexFirstAtomicComponentBlockCountDepthIndexFlattenedAscentTopsAfterSmallerInterlacingNumberFirstBlockSecondOrSmallestNonsingletonMaximumMahonianWordExcedancesMahonianWordDescentsMahonianWordInversionsMahonianWordMajorIndexMahonianWordZIndexOccurrencesOf12Over3OccurrencesOf13Over2OccurrencesOf1Over23OccurrencesOf1Over2Over3OccurrencesOf13Over24OccurrencesOf12Over34SingletonBeforeCloserPairsOpenerBeforeCloserPairsCloserBeforeSingletonPairsCloserBeforeOpenerPairsSingletonBeforeLargerPairsSmallerBeforeCloserPairsCloserBeforeLargerPairsOccurrencesOf1Over2Over3Openers2Closers3OccurrencesOf13Over2Openers1Closers2OccurrencesOf13Over2Openers1Closers3Arc13OccurrencesOf1Over2Over3Openers3Closers12OccurrencesOf1Over2Over3Openers1Closers3OccurrencesOf13Over2Closers2Arc13OccurrencesOf1Over23Openers2OccurrencesOf1Over2Over3Openers1OccurrencesOf1Over2Over3Openers13Closers2OccurrencesOf1Over23Closers1Arc23OccurrencesOf1Over23Openers2Closers1Arc23OccurrencesOf1Over2Over3Closers2OccurrencesOf1Over2Over3Closers1OccurrencesOf1Over2Over3Openers12OccurrencesOf13Over2Openers12Arc13OccurrencesOf1Over23Openers1OccurrencesOf1Over2Over3Openers3Closers1OccurrencesOf1Over23Openers2Arc23OccurrencesOf1Over23Openers12Closers3Arc23OccurrencesOf1Over23Arc23OccurrencesOf13Over2Openers1Arc13OccurrencesOf1Over23Openers12Arc23OccurrencesOf13Over2Openers1OccurrencesOf1Over2Over3Openers23OccurrencesOf1Over2Over3Openers3Closers2OccurrencesOf1Over23Closers3Arc23OccurrencesOf1Over23Closers13Arc23OccurrencesOf1Over23Openers2Closers3Arc23OccurrencesOf1Over2Over3Openers12Closers3OccurrencesOf1Over23Openers12OccurrencesOf13Over2Closers2OccurrencesOf1Over23Closers1OccurrencesOf1Over23Openers1Arc23OccurrencesOf13Over2Openers2Closers3Arc13OccurrencesOf1Over23Openers1Closers3Arc23OccurrencesOf1Over2Over3Closers13CrossingNumberNestingNumberSetPartitionDimensionMinimalArcLengthMaximalArcLengthUnsplittableFactorCountTopologicallyConnectedComponents Sizes 1 to

Also from

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

FindStat Mp00112

complement

Matched by value: our map agrees with FindStat's image on each of the 278 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 set partition obtained by replacing $i$ with $n+1-i$.

Object	Image
{{1}}	{{1}}
{{1,2}}	{{1,2}}
{{1,2,3}}	{{1,2,3}}
{{1},{2}}	{{1},{2}}

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/Mp00112 has the rest.