SetPartitions

The partitions of the set $\{1, …, n\}$ into non-empty blocks, a lazy indexed family.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaPartition of a setMathWorldSetPartitionWikidataQ381060Britannicatopic/partition-of-a-setOEISA000110✓A203646✓A203645✓Wolfram LanguageBellB✓SageMathSetPartitions(n)✓SymPymultiset_partitionsmathlib4Finpartition (Finset.univ : Finset (Fin n))Nat.bell

SetPartitions(n)every way to split

\{1, …, n\}

into disjoint non-empty blocks.

CombinatoricaSetPartitions[If[ListQ[n], n, Range[n]]]

SetPartitions(n, k)the set partitions of

\{1, …, n\}

into exactly

k

blocks.

CombinatoricaKSetPartitions[Range[n], k]

SetPartitions(collection)the set partitions of any finite collection.

CombinatoricaSetPartitions[If[ListQ[collection], collection, Range[collection]]]

Typed signatures

Type	Package	Applies
`(collection<any> \| integer, integer?) -> indexed_collection<set_partition> \| list<list<list<any>> \| set_partition>`	`combinatorics`	always

Domain: Collections

Details

A lazy indexed collection; the count is the Bell number $B_n$ — $B_4 = 15$ . See BellNumber.
The partitions into exactly $k$ blocks number the Stirling numbers of the second kind $S(n, k)$ ; summing over $k$ gives $B_n$ . See Stirling.
Each element is the block list; the $set$ - $partition$ glyph draws it from its restricted-growth string.

Definition

Epsil, as the implementation runs it.

count(_n)

_tables[_n * (_n + 1) + 0 + 1]

unrank(_n, _r)

Apply(
       (uend: list<integer>) => Map(Function(Filter(
                                                     Range(1, _n, 1),
                                                     Function(Drop(uend, 2)[p] ==
                                                              j, p),
                                                   ), j), Range(
                                                                0,
                                                                uend[2] - 1,
                                                                1,
                                                              )),
       Fold(
             Function(Apply((ur: integer) => Apply((uo: integer) => Apply(
                                                                           (ua: integer) => Join(
                                                                                                  [ur % ua, uo],
                                                                                                  Drop(us, 2),
                                                                                                  [(ur -
                                                                                                    ur % ua) /
                                                                                                   ua],
                                                                                                ) if ur <
                                                                                                    uo * ua else Join([ur -
                                                                                                                uo * ua, uo + 1], Drop(us, 2), [uo]),
                                                                           _tables[(_n - ui) *
                                                                                   (_n + 1) + uo + 1],
                                                                         ), us[2]), us[1]), us, ui),
             [_r, 0],
             Range(1, _n, 1),
           ),
     )

rank(_n, _x)

Apply(
       (rgs0: list<integer>) => Fold(
                                      Function(Apply((rr: integer) => Apply((ro: integer) => Apply(
                                                                                                    (ra: integer) => [rr +
                                                                                                                      rv * ra, ro, rs[3] + 1] if rv < ro else [rr +
                                                                                                                                                        ro * ra, ro + 1, rs[3] + 1],
                                                                                                    _tables[(_n - rs[3]) *
                                                                                                            (_n + 1) + ro + 1],
                                                                                                  ), rs[2]), rs[1]), rs, rv),
                                      [0, 0, 1],
                                      rgs0,
                                    )[1],
       Apply((mins: list<integer>) => Map(
                                           Function(Fold(Function(
                                                                   Count(Filter(
                                                                                 mins,
                                                                                 Function(z < mins[j], z),
                                                                               )) if Contains(_x[j], x) else lbl,
                                                                   lbl,
                                                                   j,
                                                                 ), 0, Range(
                                                                             1,
                                                                             Length(_x),
                                                                             1,
                                                                           )), x),
                                           Range(1, _n, 1),
                                         ), Map(Function(Min(blk), blk), _x)),
     )

valid(_n, _x)

Apply((flat: list<integer>) => Length(flat) == _n &&
                               Fold(Function(
                                              ok_blk && Length(blk) >= 1,
                                              ok_blk,
                                              blk,
                                            ), True, _x) && Fold(
                                                                 Function(ok_x &&
                                                                          Contains(flat, x), ok_x, x),
                                                                 True,
                                                                 Range(1, _n, 1),
                                                               ), Flatten(_x))

tables(_n)

Fold(
      Function(Join(b_t, Fold(
                               Function(ReplaceAt(
                                                   b_row,
                                                   b_c + 1,
                                                   b_c * b_t[(b_s - 1) *
                                                             (_n + 1) + b_c + 1] +
                                                   (b_t[(b_s - 1) * (_n + 1) +
                                                        b_c + 1 + 1] if b_c < _n else 0),
                                                 ), b_row, b_c),
                               Map(Function(0, b_z), Range(0, _n + 1 - 1, 1)),
                               Range(0, _n + 1 - 1, 1),
                             )), b_t, b_s),
      Fold(
            Function(ReplaceAt(b_row, b_c + 1, 1), b_row, b_c),
            Map(Function(0, b_z), Range(0, _n + 1 - 1, 1)),
            Range(0, _n + 1 - 1, 1),
          ),
      Range(1, _n + 1 - 1, 1),
    )

Statistics

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

Antisingletons St000248Elements i with i + 1 in the same block, counting n + 1 as 1.
BlockMajorIndex St000565The sum over blocks i of i times the elements of block i above the least element of block i + 1.
Blocks St000105The number of blocks.
BlocksAndAntisingletons St000250Blocks plus antisingletons.
BlocksAtLeastTwo St000251Blocks containing at least two elements.
BlockSizeSpanLargest block size minus smallest.
BlocksSizeTwoBlocks containing exactly two elements.
BlocksWithEvenMinimum St001153Blocks whose smallest element is even.
BlocksWithOddMinimum St001151Blocks whose smallest element is odd.
CloserBeforeLargerPairs St000577Pairs a < b in different blocks with a the greatest element of its block.
CloserBeforeOpenerPairs St000576Pairs a < b in different blocks with a the greatest and b the least element of its block.
CloserBeforeSingletonPairs St000575Pairs a < b in different blocks with a the greatest element of its block and b a singleton block.
CrossingNestingTotalCrossings plus nestings.
CrossingNumber St000253The largest number of arcs that mutually cross.
Crossings St000232Pairs of arcs a < b < c < d with a~c and b~d.
DepthIndex St001094The sum of n - i over the arcs i, less the depth of each element, plus the depth of each arc.
DimensionExponent St000572The sum over blocks of span minus (size - 1).
DimensionIndex St000229The sum over blocks of largest minus smallest element, plus one.
DistinctBlockPairs St000564Pairs of elements in different blocks.
FirstAtomicComponentBlockCount St000695The least l for which the first l blocks hold exactly 1 .. m.
FirstBlockCardinality St000504The size of the block containing 1.
FirstBlockSecondOrSmallestNonsingletonMaximum St001784The smaller of the second element of the block holding 1 and the smallest closer; 1 when {1} is a block.
FlattenedAscentTopsAfterSmaller St001641Ascent tops of the flattened word with every smaller element earlier.
FlattenedMajorIndex St000748The major index of the permutation listing the blocks in order, each ascending.
HuangRhoadesMajorIndex St000747The major index of the flattened permutation, plus for each block ending before the next begins the elements so far beyond each block's first.
InterlacingNumber St001781Pairs of blocks whose i-th arcs interlace, each block closed by an arc to infinity.
InternalPoints St000562Elements with an earlier and a later element whose blocks open after theirs does.
IntertwiningNumber St000490Pairs a < b in different blocks with no element of either block between them.
LargestBlock St001062The size of the largest block.
LargestElementInFirstBlock St000505The largest element of the block containing 1.
LargestOpener St000839The largest of the blocks' smallest elements.
LastBlockSizeThe size of the final block.
LeftCloserBigger St000497Pairs of a closer j and a smaller element i of a later block.
LeftCloserSmaller St000498Pairs of a closer j and a larger element i of a later block.
LeftOpenerSmaller St000493Pairs of an opener j and a larger element i of a later block.
MahonianWordDescents St001840Descents of the Mahonian word.
MahonianWordExcedances St001839Excedances of the Mahonian word.
MahonianWordInversions St001841Inversions of the Mahonian word.
MahonianWordMajorIndex St001842The major index of the Mahonian word.
MahonianWordZIndex St001843The major indices of the Mahonian word restricted to each pair of letters, summed.
MaximalArcLength St000730The length of the longest arc; 0 when there is no arc.
MaximalBlockDifference St000503The largest difference between two elements of one block.
MinimalArcLength St000729The length of the shortest arc; the number of elements when there is no arc.
NestingNumber St000254The largest number of arcs that mutually nest.
Nestings St000233Pairs of arcs a < b < c < d with a~d and b~c.
OccurrencesOf12Over3 St000554Occurrences of the pattern 12/3: a < b in one block and c > b in another.
OccurrencesOf12Over34 St000560Occurrences of the pattern 12/34: a < b < c < d with a, b in one block and c, d in another.
OccurrencesOf13Over2 St000555Occurrences of the pattern 13/2: a < c in one block with b between them in another.
OccurrencesOf13Over24 St000559Occurrences of the pattern 13/24: a < b < c < d with a, c in one block and b, d in another.
OccurrencesOf13Over2Closers2 St000610Occurrences of the pattern 13/2 with 2 the greatest element of its block.
OccurrencesOf13Over2Closers2Arc13 St000585Occurrences of the pattern 13/2 with 2 the greatest element of its block and 1 and 3 consecutive in one block.
OccurrencesOf13Over2Openers1 St000602Occurrences of the pattern 13/2 with 1 the least element of its block.
OccurrencesOf13Over2Openers12Arc13 St000594Occurrences of the pattern 13/2 with 1 and 2 the least elements of their blocks and 1 and 3 consecutive in one block.
OccurrencesOf13Over2Openers1Arc13 St000600Occurrences of the pattern 13/2 with 1 the least element of its block and 1 and 3 consecutive in one block.
OccurrencesOf13Over2Openers1Closers2 St000581Occurrences of the pattern 13/2 with 1 the least element of its block and 2 the greatest element of its block.
OccurrencesOf13Over2Openers1Closers3Arc13 St000582Occurrences of the pattern 13/2 with 1 the least element of its block, 3 the greatest element of its block and 1 and 3 consecutive in one block.
OccurrencesOf13Over2Openers2Closers3Arc13 St000613Occurrences of the pattern 13/2 with 2 the least element of its block, 3 the greatest element of its block and 1 and 3 consecutive in one block.
OccurrencesOf1Over23 St000556Occurrences of the pattern 1/23: b < c in one block and a < b in another.
OccurrencesOf1Over23Arc23 St000599Occurrences of the pattern 1/23 with 2 and 3 consecutive in one block.
OccurrencesOf1Over23Closers1 St000611Occurrences of the pattern 1/23 with 1 the greatest element of its block.
OccurrencesOf1Over23Closers13Arc23 St000606Occurrences of the pattern 1/23 with 1 and 3 the greatest elements of their blocks and 2 and 3 consecutive in one block.
OccurrencesOf1Over23Closers1Arc23 St000589Occurrences of the pattern 1/23 with 1 the greatest element of its block and 2 and 3 consecutive in one block.
OccurrencesOf1Over23Closers3Arc23 St000605Occurrences of the pattern 1/23 with 3 the greatest element of its block and 2 and 3 consecutive in one block.
OccurrencesOf1Over23Openers1 St000595Occurrences of the pattern 1/23 with 1 the least element of its block.
OccurrencesOf1Over23Openers12 St000609Occurrences of the pattern 1/23 with 1 and 2 the least elements of their blocks.
OccurrencesOf1Over23Openers12Arc23 St000601Occurrences of the pattern 1/23 with 1 and 2 the least elements of their blocks and 2 and 3 consecutive in one block.
OccurrencesOf1Over23Openers12Closers3Arc23 St000598Occurrences of the pattern 1/23 with 1 and 2 the least elements of their blocks, 3 the greatest element of its block and 2 and 3 consecutive in one block.
OccurrencesOf1Over23Openers1Arc23 St000612Occurrences of the pattern 1/23 with 1 the least element of its block and 2 and 3 consecutive in one block.
OccurrencesOf1Over23Openers1Closers3Arc23 St000614Occurrences of the pattern 1/23 with 1 the least element of its block, 3 the greatest element of its block and 2 and 3 consecutive in one block.
OccurrencesOf1Over23Openers2 St000586Occurrences of the pattern 1/23 with 2 the least element of its block.
OccurrencesOf1Over23Openers2Arc23 St000597Occurrences of the pattern 1/23 with 2 the least element of its block and 2 and 3 consecutive in one block.
OccurrencesOf1Over23Openers2Closers1Arc23 St000590Occurrences of the pattern 1/23 with 2 the least element of its block, 1 the greatest element of its block and 2 and 3 consecutive in one block.
OccurrencesOf1Over23Openers2Closers3Arc23 St000607Occurrences of the pattern 1/23 with 2 the least element of its block, 3 the greatest element of its block and 2 and 3 consecutive in one block.
OccurrencesOf1Over2Over3 St000557Occurrences of the pattern 1/2/3: a < b < c in three different blocks.
OccurrencesOf1Over2Over3Closers1 St000592Occurrences of the pattern 1/2/3 with 1 the greatest element of its block.
OccurrencesOf1Over2Over3Closers13 St000615Occurrences of the pattern 1/2/3 with 1 and 3 the greatest elements of their blocks.
OccurrencesOf1Over2Over3Closers2 St000591Occurrences of the pattern 1/2/3 with 2 the greatest element of its block.
OccurrencesOf1Over2Over3Openers1 St000587Occurrences of the pattern 1/2/3 with 1 the least element of its block.
OccurrencesOf1Over2Over3Openers12 St000593Occurrences of the pattern 1/2/3 with 1 and 2 the least elements of their blocks.
OccurrencesOf1Over2Over3Openers12Closers3 St000608Occurrences of the pattern 1/2/3 with 1 and 2 the least elements of their blocks and 3 the greatest element of its block.
OccurrencesOf1Over2Over3Openers13Closers2 St000588Occurrences of the pattern 1/2/3 with 1 and 3 the least elements of their blocks and 2 the greatest element of its block.
OccurrencesOf1Over2Over3Openers1Closers3 St000584Occurrences of the pattern 1/2/3 with 1 the least element of its block and 3 the greatest element of its block.
OccurrencesOf1Over2Over3Openers23 St000603Occurrences of the pattern 1/2/3 with 2 and 3 the least elements of their blocks.
OccurrencesOf1Over2Over3Openers2Closers3 St000580Occurrences of the pattern 1/2/3 with 2 the least element of its block and 3 the greatest element of its block.
OccurrencesOf1Over2Over3Openers3Closers1 St000596Occurrences of the pattern 1/2/3 with 3 the least element of its block and 1 the greatest element of its block.
OccurrencesOf1Over2Over3Openers3Closers12 St000583Occurrences of the pattern 1/2/3 with 3 the least element of its block and 1 and 2 the greatest elements of their blocks.
OccurrencesOf1Over2Over3Openers3Closers2 St000604Occurrences of the pattern 1/2/3 with 3 the least element of its block and 2 the greatest element of its block.
OpenerBeforeCloserPairs St000574Pairs a < b in different blocks with a the least and b the greatest element of its block.
OverlappingBlockPairs St000563Pairs of blocks A, B with min A < min B < max A < max B.
RightCloserBigger St000499Pairs of a closer j and a smaller element i of an earlier block.
RightCloserSmaller St000496Pairs of a closer j and a larger element i of an earlier block.
RightOpenerBigger St000492Pairs of an opener j and a smaller element i of an earlier block.
RightOpenerSmaller St000491Pairs of an opener j and a larger element i of an earlier block.
RotationOrbitSize St000163The number of distinct partitions rotation i -> i + 1 (mod n) makes of this one.
SameBlockPairs St000558Pairs of elements in one block.
SameBlockTriples St000561Triples of elements in one block.
SetPartitionDimension St000728The sum of the lengths of the arcs, taking for each block its largest element minus its smallest.
SetPartitionRank St000211The number of elements minus the number of blocks.
SingletonBeforeCloserPairs St000573Pairs a < b in different blocks with a a singleton block and b the greatest element of its block.
SingletonBeforeLargerPairs St000578Pairs a < b in different blocks with a a singleton block.
SingletonBlocks St000247Blocks containing exactly one element.
SingletonsAndAntisingletons St000249Singleton blocks plus antisingletons.
SmallerBeforeCloserPairs St000579Pairs a < b in different blocks with b the greatest element of its block.
SmallestBlock St001075The size of the smallest block.
SmallestCloser St000971The smallest of the blocks' largest elements.
Successions St000502Elements i with i + 1 in the same block.
SumOfBlockMaxima St000231The sum of the largest elements of the blocks.
SumOfBlockMinima St000230The sum of the smallest elements of the blocks.
TerminalClosers St001050Closers e with every larger element also a closer.
TopologicallyConnectedComponents St000925The topologically connected components of the arc diagram: blocks joined when an arc of one crosses an arc of the other.
UnsplittableFactorCount St000823The number of unsplittable factors of the set partition under the split product.

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

Enumeration

Examples

Basic #

The set partitions of an explicit list, in the family's own RGS order

live ↗

\mathrm{SetPartitions}(\bigl\lbrack a, b, c\bigr\rbrack)

Out

\bigl\lbrack\bigl\lbrack\bigl\lbrack a, b, c\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack a, b\bigr\rbrack, \bigl\lbrack c\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack a, c\bigr\rbrack, \bigl\lbrack b\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack a\bigr\rbrack, \bigl\lbrack b, c\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack a\bigr\rbrack, \bigl\lbrack b\bigr\rbrack, \bigl\lbrack c\bigr\rbrack\bigr\rbrack\bigr\rbrack

$B_4$

live ↗

\mathrm{Count}(\mathrm{SetPartitions}(4))

Out

15

Scope #

Exactly 2 blocks

live ↗

\mathrm{SetPartitions}(3, 2)

Out

\bigl\lbrack{\operatorname{SetPartition}}([[1, 2], [3]]), {\operatorname{SetPartition}}([[1, 3], [2]]), {\operatorname{SetPartition}}([[1], [2, 3]])\bigr\rbrack

The partitions of a 4-set into 2 blocks number $S(4, 2) = 7$

live ↗

\mathrm{Count}(\mathrm{SetPartitions}(4, 2))

Out

7

Properties #

The family has $B_4 = 15$ members. See BellNumber

live ↗

\mathrm{BellNumber}(4)

Out

15

Those with exactly 2 blocks number $S(4, 2) = 7$ . See Stirling

live ↗

\mathrm{Stirling}(4, 2)

Out

7

Statistics #

set partitions of 5 by blocks: the Stirling numbers of the second kind

live ↗

\mathrm{Tally}(\mathrm{CombinatorialStat}(\mathrm{SetPartitions}(5), \text{Blocks}))

Out

\operatorname{Tuple}(\bigl\lbrack1, 2, 3, 4, 5\bigr\rbrack,\bigl\lbrack1, 15, 25, 10, 1\bigr\rbrack)

by singleton blocks

live ↗

\mathrm{Tally}(\mathrm{CombinatorialStat}(\mathrm{SetPartitions}(5), \text{SingletonBlocks}))

Out

\operatorname{Tuple}(\bigl\lbrack0, 1, 2, 3, 5\bigr\rbrack,\bigl\lbrack11, 20, 10, 10, 1\bigr\rbrack)

by crossings: the 14 non-crossing ones, a Catalan number, and one more

live ↗

\mathrm{Tally}(\mathrm{CombinatorialStat}(\mathrm{SetPartitions}(4), \text{Crossings}))

Out

\operatorname{Tuple}(\bigl\lbrack0, 1\bigr\rbrack,\bigl\lbrack14, 1\bigr\rbrack)

About

Examples	9
Checked against	Combinatorica 2 agree2 differWolfram Language 2 agreeSageMath 2 agreeSymPy 2 agreemathlib4 2 agreeOscar 1 agreeJulia 2 agreempmath 1 agree
Crosswalk	Wikipedia MathWorld Wikidata Britannica OEIS ×3 Wolfram LanguageCombinatorica ×2SageMath SymPy mathlib4 ×2

Implementation

mappedcombinatoricapackages/oracle/src/mappings.ts

Combinatorica lists set partitions in a different order than ours, so a listing may differ where the count agrees.

mappedcombinatoricapackages/oracle/src/mappings.ts

Combinatorica lists set partitions in a different order than ours, so a listing may differ where the count agrees.

SetPartitions ​

Definition

Statistics

Maps

From SetPartition

To SetPartition

Enumeration

Examples

Implementation

SetPartitions