PerfectMatchings
A name from the enumeratio catalogue with no kernel of its own yet; this page carries its crosswalk only.
Catalogued in the enumeratio database, with crosswalk rows in oeis, sage, wikipedia; not yet written up here.
✓ checked against that system · ! checked, with disagreements (hover a chip)Wikipedia
Perfect matchingMathWorld1-FactorWikidataQ99567093OEISA001147✓SageMathsage.combinat.perfect_matching.PerfectMatchings(k)PerfectMatchings(...)Catalogued in the enumeratio database, with crosswalk rows in oeis, sage, wikipedia; not yet written up here.Typed signatures
| Type | Package | Applies |
|---|---|---|
(integer<0..>) -> indexed_collection<set_partition> | combinatorics | always |
Statistics
On SetPartition, called as CombinatorialStat(x, "Name").
- AntisingletonsSt000248Elements i with i + 1 in the same block, counting n + 1 as 1.
- BlocksSt000105The number of blocks.
- BlocksAndAntisingletonsSt000250Blocks plus antisingletons.
- BlocksAtLeastTwoSt000251Blocks containing at least two elements.
- BlockSizeSpanLargest block size minus smallest.
- BlocksSizeTwoBlocks containing exactly two elements.
- BlocksWithEvenMinimumSt001153Blocks whose smallest element is even.
- BlocksWithOddMinimumSt001151Blocks whose smallest element is odd.
- CrossingNestingTotalCrossings plus nestings.
- CrossingsSt000232Pairs of arcs a < b < c < d with a~c and b~d.
- DimensionExponentSt000572The sum over blocks of span minus (size - 1).
- DimensionIndexSt000229The sum over blocks of largest minus smallest element, plus one.
- DistinctBlockPairsSt000564Pairs of elements in different blocks.
- FirstBlockCardinalitySt000504The size of the block containing 1.
- InternalPointsSt000562Elements with an earlier and a later element whose blocks open after theirs does.
- LargestBlockSt001062The size of the largest block.
- LargestElementInFirstBlockSt000505The largest element of the block containing 1.
- LargestOpenerSt000839The largest of the blocks' smallest elements.
- LastBlockSizeThe size of the final block.
- MaximalBlockDifferenceSt000503The largest difference between two elements of one block.
- NestingsSt000233Pairs of arcs a < b < c < d with a~d and b~c.
- OverlappingBlockPairsSt000563Pairs of blocks A, B with min A < min B < max A < max B.
- SameBlockPairsSt000558Pairs of elements in one block.
- SameBlockTriplesSt000561Triples of elements in one block.
- SetPartitionRankSt000211The number of elements minus the number of blocks.
- SingletonBlocksSt000247Blocks containing exactly one element.
- SingletonsAndAntisingletonsSt000249Singleton blocks plus antisingletons.
- SmallestBlockSt001075The size of the smallest block.
- SmallestCloserSt000971The smallest of the blocks' largest elements.
- SuccessionsSt000502Elements i with i + 1 in the same block.
- SumOfBlockMaximaSt000231The sum of the largest elements of the blocks.
- SumOfBlockMinimaSt000230The sum of the smallest elements of the blocks.
- TerminalClosersSt001050Closers e with every larger element also a closer.
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.
- RestrictedGrowthStringSetPartition → RestrictedGrowthStringinverseA set partition's restricted growth string: each position labelled with its block's index, from 0.
To SetPartition
- CyclePartitionPermutation → SetPartitionSurjectionGraded mapMp00151The orbits, as a set partition of the positions.
- SetPartitionRestrictedGrowthString → SetPartitioninverseThe set partition a restricted growth string labels: block j holds the positions labelled j.
24 more in FindStat, not implemented here
- Mp00079SetPartition → IntegerPartitionSurjectionGraded mapshape
- Mp00080SetPartition → PermutationInjectionGraded mapto permutation
- Mp00091SetPartition → SetPartitionBijectionGraded maprotate increasing
- Mp00092PerfectMatching → SetPartitionInjectionGraded mapto set partition
- Mp00112SetPartition → SetPartitionBijectionGraded mapInvolutioncomplement
- Mp00115SetPartition → SetPartitionBijectionGraded mapInvolutionKasraoui-Zeng
- Mp00128SetPartition → CompositionSurjectionGraded mapto composition
- Mp00138DyckPath → SetPartitionInjectionGraded mapto noncrossing partition
- Mp00164SetPartition → SetPartitionBijectionGraded mapInvolutionChen Deng Du Stanley Yan
- Mp00171SetPartition → SetPartitionBijectionGraded mapintertwining number to dual major index
- Mp00174SetPartition → SetPartitionBijectionGraded mapdual major index to intertwining number
- Mp00176SetPartition → SetPartitionBijectionGraded maprotate decreasing
- Mp00215SetPartition → SetPartitionBijectionGraded mapWachs-White
- Mp00216SetPartition → SetPartitionBijectionGraded mapinverse Wachs-White
- Mp00217SetPartition → SetPartitionBijectionGraded mapWachs-White-rho
- Mp00218SetPartition → SetPartitionBijectionGraded mapinverse Wachs-White-rho
- Mp00219SetPartition → SetPartitionBijectionGraded mapinverse Yip
- Mp00220SetPartition → SetPartitionBijectionGraded mapYip
- Mp00221SetPartition → SetPartitionBijectionGraded mapInvolutionconjugate
- Mp00240Permutation → SetPartitionSurjectionGraded mapweak exceedance partition
- Mp00249SetPartition → SetPartitionBijectionGraded mapInvolutionCallan switch
- Mp00258SetPartition → StandardTableauSurjectionStandard tableau associated to a set partition
- Mp00284StandardTableau → SetPartitionInjectionGraded maprows
- Mp00285SetComposition → SetPartitionSurjectionGraded mapto set partition