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1. Definition

A set partition of a set $\mathcal{S}$ is a collection of non-empty pairwise disjoint subsets (also known as the parts) of $\mathcal{S}$ whose union is $\mathcal{S}$. Mathematically we have $\mathcal{P} = \{P \subset S\}$ such that

2. Examples

Let $\mathcal{S} = \{1, 2, 3, 4\}$.

3. Properties

4. Remarks

5. Statistics

We have the following 94 statistics in the database:

St000105
Set partitions ⟶ ℤ
The number of blocks in the set partition.
St000163
Set partitions ⟶ ℤ
The size of the orbit of the set partition under rotation.
St000211
Set partitions ⟶ ℤ
The rank of the set partition.
St000229
Set partitions ⟶ ℤ
Sum of the difference between the maximal and the minimal elements of the blocks ....
St000230
Set partitions ⟶ ℤ
Sum of the minimal elements of the blocks of a set partition.
St000231
Set partitions ⟶ ℤ
Sum of the maximal elements of the blocks of a set partition.
St000232
Set partitions ⟶ ℤ
The number of crossings of a set partition.
St000233
Set partitions ⟶ ℤ
The number of nestings of a set partition.
St000247
Set partitions ⟶ ℤ
The number of singleton blocks of a set partition.
St000248
Set partitions ⟶ ℤ
The number of anti-singletons of a set partition.
St000249
Set partitions ⟶ ℤ
The number of singletons (St000247) plus the number of antisingletons (St000248) ....
St000250
Set partitions ⟶ ℤ
The number of blocks (St000105) plus the number of antisingletons (St000248) of a....
St000251
Set partitions ⟶ ℤ
The number of nonsingleton blocks of a set partition.
St000253
Set partitions ⟶ ℤ
The crossing number of a set partition.
St000254
Set partitions ⟶ ℤ
The nesting number of a set partition.
St000490
Set partitions ⟶ ℤ
The intertwining number of a set partition.
St000491
Set partitions ⟶ ℤ
The number of inversions of a set partition.
St000492
Set partitions ⟶ ℤ
The rob statistic of a set partition.
St000493
Set partitions ⟶ ℤ
The los statistic of a set partition.
St000496
Set partitions ⟶ ℤ
The rcs statistic of a set partition.
St000497
Set partitions ⟶ ℤ
The lcb statistic of a set partition.
St000498
Set partitions ⟶ ℤ
The lcs statistic of a set partition.
St000499
Set partitions ⟶ ℤ
The rcb statistic of a set partition.
St000502
Set partitions ⟶ ℤ
The number of successions of a set partitions.
St000503
Set partitions ⟶ ℤ
The maximal difference between two elements in a common block.
St000504
Set partitions ⟶ ℤ
The cardinality of the first block of a set partition.
St000505
Set partitions ⟶ ℤ
The biggest entry in the block containing the 1.
St000554
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1,2},{3}} in a set partition.
St000555
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1,3},{2}} in a set partition.
St000556
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2,3}} in a set partition.
St000557
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2},{3}} in a set partition.
St000558
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1,2}} in a set partition.
St000559
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1,3},{2,4}} in a set partition.
St000560
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1,2},{3,4}} in a set partition.
St000561
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1,2,3}} in a set partition.
St000562
Set partitions ⟶ ℤ
The number of internal points of a set partition.
St000563
Set partitions ⟶ ℤ
The number of overlapping pairs of blocks of a set partition.
St000564
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2}} in a set partition.
St000565
Set partitions ⟶ ℤ
The major index of a set partition.
St000572
Set partitions ⟶ ℤ
The dimension exponent of a set partition.
St000573
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 ....
St000574
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a ....
St000575
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element ....
St000576
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a ....
St000577
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element.....
St000578
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton.
St000579
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element.....
St000580
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is m....
St000581
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is ma....
St000582
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is ma....
St000583
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1,2 ar....
St000584
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is m....
St000585
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) a....
St000586
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal.
St000587
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal.
St000588
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 i....
St000589
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) a....
St000590
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is ma....
St000591
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal.
St000592
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal.
St000593
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal.
St000594
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3....
St000595
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal.
St000596
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is m....
St000597
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) a....
St000598
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is....
St000599
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive....
St000600
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) a....
St000601
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3....
St000602
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal.
St000603
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal.
St000604
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is m....
St000605
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) a....
St000606
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3....
St000607
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is ma....
St000608
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 i....
St000609
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal.
St000610
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal.
St000611
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal.
St000612
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) a....
St000613
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is ma....
St000614
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is ma....
St000615
Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal.
St000695
Set partitions ⟶ ℤ
The number of blocks in the first part of the atomic decomposition of a set parti....
St000728
Set partitions ⟶ ℤ
The dimension of a set partition.
St000729
Set partitions ⟶ ℤ
The minimal arc length of a set partition.
St000730
Set partitions ⟶ ℤ
The maximal arc length of a set partition.
St000747
Set partitions ⟶ ℤ
A variant of the major index of a set partition.
St000748
Set partitions ⟶ ℤ
The major index of the permutation obtained by flattening the set partition.
St000793
Set partitions ⟶ ℤ
The length of the longest partition in the vacillating tableau corresponding to a....
St000823
Set partitions ⟶ ℤ
The number of unsplittable factors of the set partition.
St000839
Set partitions ⟶ ℤ
The largest opener of a set partition.
St000925
Set partitions ⟶ ℤ
The number of topologically connected components of a set partition.
St000971
Set partitions ⟶ ℤ
The smallest closer of a set partition.

6. Maps

We have the following 8 maps in the database:

Mp00079
Set partitions ⟶ Integer partitions
shape
Mp00080
Set partitions ⟶ Permutations
to permutation
Mp00091
Set partitions ⟶ Set partitions
Cyclic rotation
Mp00092
Perfect matchings ⟶ Set partitions
to set partition
Mp00112
Set partitions ⟶ Set partitions
reverse
Mp00115
Set partitions ⟶ Set partitions
Kasraoui-Zeng
Mp00128
Set partitions ⟶ Integer compositions
to composition
Mp00138
Dyck paths ⟶ Set partitions
to noncrossing partition

7. References

8. Sage examples


CategoryCombinatorialCollection