NoncrossingPartition from DyckPath

Biane's noncrossing set partition: the up steps whose matching downs lie in one run of downs form a block.

DyckPath → SetPartition

FindStat's Mp00138. Injective: it reaches exactly the noncrossing partitions.

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

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Each element of DyckPath 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 DyckPath at each size is computed, and compared with the 171 statistics on DyckPath: 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 Mp00138

to noncrossing partition

Matched by value: our map agrees with FindStat's image on each of the 196 rows of size at most 6 we can read (0 unread; its table has 1200 rows).

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

Biane's map to noncrossing set partitions.

Object	Image
[1,0]	{{1}}
[1,0,1,0]	{{1},{2}}
[1,1,0,0]	{{1,2}}
[1,0,1,0,1,0]	{{1},{2},{3}}

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/Mp00138 has the rest.