CallanSwitch from SetPartition

Callan's involution: switches the first closer with the lesser of the least closer and the second element of the block of 1.

SetPartition → SetPartition

FindStat's Mp00249, a joint symmetry of St000971 and St001784.

Called as CombinatorialMap(x, "CallanSwitch"). 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

FindStat Mp00249

Callan switch

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.

Switch the first closer and the minimum of the smallest closer and the second element of the block containing 1 in a set partition.

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

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/Mp00249 has the rest.