notatio

SetPartition ​

The singular-inhabitant constructor for a partition of a set, as its blocks.

WikipediaPartition of a setNoncrossing partitionvia NonCrossingPartitionsPerfect matchingvia PerfectMatchingsMathWorldSetPartitionWikidataQ381060Britannicatopic/partition-of-a-setOEISA000110via SetPartitionsA000108via NonCrossingMatchingsA001147via PerfectMatchingsA008277via SetPartitionsIntoKBlocksWolfram LanguageBellBvia SetPartitionsStirlingS2via SetPartitionsIntoKBlocksSageMathSymPymultiset_partitionsvia SetPartitionsmathlib4Finpartition (Finset.univ : Finset (Fin n))via SetPartitionsNat.bellvia SetPartitionsNat.stirlingSecondvia SetPartitionsIntoKBlocks
SetPartition(list)The singular-inhabitant constructor for a partition of a set, as its blocks.
Typed signatures
TypePackageApplies
((list<list<integer>>) -> set_partition) & ((restricted_growth_string) -> set_partition)combinatoricsalways

Domain: Collections·Drawn as Graphics

Details
  • Ordered by refinement: π is below σ when every block of π lies inside a block of σ. A lattice, so Min is the meet (common refinement) and Max the join (finest common coarsening), with Compare giving NaN for incomparable partitions.

Statistics

On SetPartition, called as CombinatorialStat(x, "Name").

Maps

Into and out of SetPartition, called as CombinatorialMap(x, "Name"). Properties are the laws we state and what FindStat records.

From SetPartition

  • ArcRepresentationSetPartition → EndofunctionEach position linked to the next in its block, or to itself when last.
  • AssociatedStandardTableauSetPartition → StandardTableauSurjectionMp00258The standard tableau whose j-th column holds the j-th smallest elements of the blocks, in increasing order.
  • BlockSizeCompositionSetPartition → IntegerCompositionsurjectionSurjectionGraded mapMp00128The block sizes of a set partition, its blocks taken in order of their least elements.
  • BlockSizePartitionSetPartition → IntegerPartitionsurjectionSurjectionGraded mapMp00079The partition of n a set partition's block sizes form, largest first.
  • CallanSwitchSetPartition → SetPartitionBijectionGraded mapInvolutionMp00249Callan's involution: switches the first closer with the lesser of the least closer and the second element of the block of 1.
  • ComplementSetPartition → SetPartitioninvolutionBijectionGraded mapInvolutionMp00112Each element i replaced by n + 1 - i: the set partition seen from the other end.
  • DecreasingRotationSetPartition → SetPartitionbijectioninverseBijectionGraded mapMp00176Every element lowered by one, cyclically: 1 becomes the largest.
  • DualMajorIndexToIntertwiningSetPartition → SetPartitionbijectioninverseBijectionGraded mapMp00174The inverse of IntertwiningToDualMajorIndex: read the growth word as an intertwining word and rebuild the blocks.
  • IncreasingCyclePermutationSetPartition → PermutationInjectionGraded mapMp00080The permutation whose cycles are the blocks, each cycle increasing: every element goes to the next larger in its block, the largest to the least.
  • IncreasingRotationSetPartition → SetPartitionbijectioninverseBijectionGraded mapMp00091Every element raised by one, cyclically: the largest becomes 1.
  • IntertwiningToDualMajorIndexSetPartition → SetPartitionbijectioninverseBijectionGraded mapMp00171The set partition whose growth word is the intertwining word: a bijection taking the intertwining number to the dual major index.
  • InverseWachsWhiteSetPartition → SetPartitionbijectioninverseBijectionGraded mapMp00216The set partition joined by the arcs of the rook placement that Wachs and White's bijection gamma assigns to the set partition.
  • InverseWachsWhiteRhoSetPartition → SetPartitionbijectioninverseBijectionGraded mapMp00218The set partition joined by the arcs of the rook placement that Wachs and White's bijection rho assigns to the set partition.
  • InverseYipSetPartition → SetPartitionbijectioninverseBijectionGraded mapMp00219The set partition joined by the arcs of the rook placement that Yip's bijection psi assigns to the set partition.
  • KasraouiZengSetPartition → SetPartitionBijectionGraded mapInvolutionMp00115The Kasraoui-Zeng involution: exchanges the number of crossings with the number of nestings.
  • RestrictedGrowthStringSetPartition → RestrictedGrowthStringinverseA set partition's restricted growth string: each position labelled with its block's index, from 0.
  • WachsWhiteSetPartition → SetPartitionbijectioninverseBijectionGraded mapMp00215The set partition whose arcs, read as a rook placement, Wachs and White's bijection gamma carries to a set partition.
  • WachsWhiteRhoSetPartition → SetPartitionbijectioninverseBijectionGraded mapMp00217The set partition whose arcs, read as a rook placement, Wachs and White's bijection rho carries to a set partition.
  • YipSetPartition → SetPartitionbijectioninverseBijectionGraded mapMp00220The set partition whose arcs, read as a rook placement, Yip's bijection psi carries to a set partition.

To SetPartition

  • CyclePartitionPermutation → SetPartitionsurjectionSurjectionGraded mapMp00151The orbits, as a set partition of the positions.
  • NoncrossingPartitionDyckPath → SetPartitionInjectionGraded mapMp00138Biane's noncrossing set partition: the up steps whose matching downs lie in one run of downs form a block.
  • RowSetPartitionStandardTableau → SetPartitionInjectionGraded mapMp00284The set partition whose blocks are the rows of the tableau.
  • SetPartitionRestrictedGrowthString → SetPartitioninverseThe set partition a restricted growth string labels: block j holds the positions labelled j.
  • UnderlyingSetPartitionSetComposition → SetPartitionsurjectionSurjectionGraded mapMp00285The set partition with the same blocks, the order of the blocks forgotten.
  • WeakExceedancePartitionPermutation → SetPartitionSurjectionGraded mapMp00240The coarsest set partition containing every arc from i to π(i) with i ≤ π(i).
3 more in FindStat, not implemented here
  • Mp00092PerfectMatching → SetPartitionInjectionGraded mapto set partition
  • Mp00164SetPartition → SetPartitionBijectionGraded mapInvolutionChen Deng Du Stanley Yan
  • Mp00221SetPartition → SetPartitionBijectionGraded mapInvolutionconjugate

Examples

Basic #

From its restricted growth string: block j holds the positions labelled j

live ↗
InSetPartition⁡(RestrictedGrowthString⁡([0,1,0,2]))\operatorname{SetPartition}({\operatorname{RestrictedGrowthString}}([0, 1, 0, 2]))OutSetPartition⁡([[1,3],[2],[4]]){\operatorname{SetPartition}}([[1, 3], [2], [4]])
About
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