- FindStat

Identifier

Mp00112: Set partitions —complement⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000209: Permutations ⟶ ℤ

Values

{{1}} => {{1}} => [1] => 0

{{1,2}} => {{1,2}} => [2,1] => 1

{{1},{2}} => {{1},{2}} => [1,2] => 0

{{1,2,3}} => {{1,2,3}} => [2,3,1] => 2

{{1,2},{3}} => {{1},{2,3}} => [1,3,2] => 1

{{1,3},{2}} => {{1,3},{2}} => [3,2,1] => 2

{{1},{2,3}} => {{1,2},{3}} => [2,1,3] => 1

{{1},{2},{3}} => {{1},{2},{3}} => [1,2,3] => 0

{{1,2,3,4}} => {{1,2,3,4}} => [2,3,4,1] => 3

{{1,2,3},{4}} => {{1},{2,3,4}} => [1,3,4,2] => 2

{{1,2,4},{3}} => {{1,3,4},{2}} => [3,2,4,1] => 3

{{1,2},{3,4}} => {{1,2},{3,4}} => [2,1,4,3] => 1

{{1,2},{3},{4}} => {{1},{2},{3,4}} => [1,2,4,3] => 1

{{1,3,4},{2}} => {{1,2,4},{3}} => [2,4,3,1] => 3

{{1,3},{2,4}} => {{1,3},{2,4}} => [3,4,1,2] => 2

{{1,3},{2},{4}} => {{1},{2,4},{3}} => [1,4,3,2] => 2

{{1,4},{2,3}} => {{1,4},{2,3}} => [4,3,2,1] => 3

{{1},{2,3,4}} => {{1,2,3},{4}} => [2,3,1,4] => 2

{{1},{2,3},{4}} => {{1},{2,3},{4}} => [1,3,2,4] => 1

{{1,4},{2},{3}} => {{1,4},{2},{3}} => [4,2,3,1] => 3

{{1},{2,4},{3}} => {{1,3},{2},{4}} => [3,2,1,4] => 2

{{1},{2},{3,4}} => {{1,2},{3},{4}} => [2,1,3,4] => 1

{{1},{2},{3},{4}} => {{1},{2},{3},{4}} => [1,2,3,4] => 0

{{1,2,3,4,5}} => {{1,2,3,4,5}} => [2,3,4,5,1] => 4

{{1,2,3,4},{5}} => {{1},{2,3,4,5}} => [1,3,4,5,2] => 3

{{1,2,3,5},{4}} => {{1,3,4,5},{2}} => [3,2,4,5,1] => 4

{{1,2,3},{4,5}} => {{1,2},{3,4,5}} => [2,1,4,5,3] => 2

{{1,2,3},{4},{5}} => {{1},{2},{3,4,5}} => [1,2,4,5,3] => 2

{{1,2,4,5},{3}} => {{1,2,4,5},{3}} => [2,4,3,5,1] => 4

{{1,2,4},{3,5}} => {{1,3},{2,4,5}} => [3,4,1,5,2] => 3

{{1,2,4},{3},{5}} => {{1},{2,4,5},{3}} => [1,4,3,5,2] => 3

{{1,2,5},{3,4}} => {{1,4,5},{2,3}} => [4,3,2,5,1] => 4

{{1,2},{3,4,5}} => {{1,2,3},{4,5}} => [2,3,1,5,4] => 2

{{1,2},{3,4},{5}} => {{1},{2,3},{4,5}} => [1,3,2,5,4] => 1

{{1,2,5},{3},{4}} => {{1,4,5},{2},{3}} => [4,2,3,5,1] => 4

{{1,2},{3,5},{4}} => {{1,3},{2},{4,5}} => [3,2,1,5,4] => 2

{{1,2},{3},{4,5}} => {{1,2},{3},{4,5}} => [2,1,3,5,4] => 1

{{1,2},{3},{4},{5}} => {{1},{2},{3},{4,5}} => [1,2,3,5,4] => 1

{{1,3,4,5},{2}} => {{1,2,3,5},{4}} => [2,3,5,4,1] => 4

{{1,3,4},{2,5}} => {{1,4},{2,3,5}} => [4,3,5,1,2] => 3

{{1,3,4},{2},{5}} => {{1},{2,3,5},{4}} => [1,3,5,4,2] => 3

{{1,3,5},{2,4}} => {{1,3,5},{2,4}} => [3,4,5,2,1] => 4

{{1,3},{2,4,5}} => {{1,2,4},{3,5}} => [2,4,5,1,3] => 3

{{1,3},{2,4},{5}} => {{1},{2,4},{3,5}} => [1,4,5,2,3] => 2

{{1,3,5},{2},{4}} => {{1,3,5},{2},{4}} => [3,2,5,4,1] => 4

{{1,3},{2,5},{4}} => {{1,4},{2},{3,5}} => [4,2,5,1,3] => 3

{{1,3},{2},{4,5}} => {{1,2},{3,5},{4}} => [2,1,5,4,3] => 2

{{1,3},{2},{4},{5}} => {{1},{2},{3,5},{4}} => [1,2,5,4,3] => 2

{{1,4,5},{2,3}} => {{1,2,5},{3,4}} => [2,5,4,3,1] => 4

{{1,4},{2,3,5}} => {{1,3,4},{2,5}} => [3,5,4,1,2] => 3

{{1,4},{2,3},{5}} => {{1},{2,5},{3,4}} => [1,5,4,3,2] => 3

{{1,5},{2,3,4}} => {{1,5},{2,3,4}} => [5,3,4,2,1] => 4

{{1},{2,3,4,5}} => {{1,2,3,4},{5}} => [2,3,4,1,5] => 3

{{1},{2,3,4},{5}} => {{1},{2,3,4},{5}} => [1,3,4,2,5] => 2

{{1,5},{2,3},{4}} => {{1,5},{2},{3,4}} => [5,2,4,3,1] => 4

{{1},{2,3,5},{4}} => {{1,3,4},{2},{5}} => [3,2,4,1,5] => 3

{{1},{2,3},{4,5}} => {{1,2},{3,4},{5}} => [2,1,4,3,5] => 1

{{1},{2,3},{4},{5}} => {{1},{2},{3,4},{5}} => [1,2,4,3,5] => 1

{{1,4,5},{2},{3}} => {{1,2,5},{3},{4}} => [2,5,3,4,1] => 4

{{1,4},{2,5},{3}} => {{1,4},{2,5},{3}} => [4,5,3,1,2] => 3

{{1,4},{2},{3,5}} => {{1,3},{2,5},{4}} => [3,5,1,4,2] => 3

{{1,4},{2},{3},{5}} => {{1},{2,5},{3},{4}} => [1,5,3,4,2] => 3

{{1,5},{2,4},{3}} => {{1,5},{2,4},{3}} => [5,4,3,2,1] => 4

{{1},{2,4,5},{3}} => {{1,2,4},{3},{5}} => [2,4,3,1,5] => 3

{{1},{2,4},{3,5}} => {{1,3},{2,4},{5}} => [3,4,1,2,5] => 2

{{1},{2,4},{3},{5}} => {{1},{2,4},{3},{5}} => [1,4,3,2,5] => 2

{{1,5},{2},{3,4}} => {{1,5},{2,3},{4}} => [5,3,2,4,1] => 4

{{1},{2,5},{3,4}} => {{1,4},{2,3},{5}} => [4,3,2,1,5] => 3

{{1},{2},{3,4,5}} => {{1,2,3},{4},{5}} => [2,3,1,4,5] => 2

{{1},{2},{3,4},{5}} => {{1},{2,3},{4},{5}} => [1,3,2,4,5] => 1

{{1,5},{2},{3},{4}} => {{1,5},{2},{3},{4}} => [5,2,3,4,1] => 4

{{1},{2,5},{3},{4}} => {{1,4},{2},{3},{5}} => [4,2,3,1,5] => 3

{{1},{2},{3,5},{4}} => {{1,3},{2},{4},{5}} => [3,2,1,4,5] => 2

{{1},{2},{3},{4,5}} => {{1,2},{3},{4},{5}} => [2,1,3,4,5] => 1

{{1},{2},{3},{4},{5}} => {{1},{2},{3},{4},{5}} => [1,2,3,4,5] => 0

{{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 5

{{1,2,3,4,5},{6}} => {{1},{2,3,4,5,6}} => [1,3,4,5,6,2] => 4

{{1,2,3,4,6},{5}} => {{1,3,4,5,6},{2}} => [3,2,4,5,6,1] => 5

{{1,2,3,4},{5,6}} => {{1,2},{3,4,5,6}} => [2,1,4,5,6,3] => 3

{{1,2,3,4},{5},{6}} => {{1},{2},{3,4,5,6}} => [1,2,4,5,6,3] => 3

{{1,2,3,5,6},{4}} => {{1,2,4,5,6},{3}} => [2,4,3,5,6,1] => 5

{{1,2,3,5},{4,6}} => {{1,3},{2,4,5,6}} => [3,4,1,5,6,2] => 4

{{1,2,3,5},{4},{6}} => {{1},{2,4,5,6},{3}} => [1,4,3,5,6,2] => 4

{{1,2,3,6},{4,5}} => {{1,4,5,6},{2,3}} => [4,3,2,5,6,1] => 5

{{1,2,3},{4,5,6}} => {{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => 2

{{1,2,3},{4,5},{6}} => {{1},{2,3},{4,5,6}} => [1,3,2,5,6,4] => 2

{{1,2,3,6},{4},{5}} => {{1,4,5,6},{2},{3}} => [4,2,3,5,6,1] => 5

{{1,2,3},{4,6},{5}} => {{1,3},{2},{4,5,6}} => [3,2,1,5,6,4] => 2

{{1,2,3},{4},{5,6}} => {{1,2},{3},{4,5,6}} => [2,1,3,5,6,4] => 2

{{1,2,3},{4},{5},{6}} => {{1},{2},{3},{4,5,6}} => [1,2,3,5,6,4] => 2

{{1,2,4,5,6},{3}} => {{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => 5

{{1,2,4,5},{3,6}} => {{1,4},{2,3,5,6}} => [4,3,5,1,6,2] => 4

{{1,2,4,5},{3},{6}} => {{1},{2,3,5,6},{4}} => [1,3,5,4,6,2] => 4

{{1,2,4,6},{3,5}} => {{1,3,5,6},{2,4}} => [3,4,5,2,6,1] => 5

{{1,2,4},{3,5,6}} => {{1,2,4},{3,5,6}} => [2,4,5,1,6,3] => 3

{{1,2,4},{3,5},{6}} => {{1},{2,4},{3,5,6}} => [1,4,5,2,6,3] => 3

{{1,2,4,6},{3},{5}} => {{1,3,5,6},{2},{4}} => [3,2,5,4,6,1] => 5

{{1,2,4},{3,6},{5}} => {{1,4},{2},{3,5,6}} => [4,2,5,1,6,3] => 3

{{1,2,4},{3},{5,6}} => {{1,2},{3,5,6},{4}} => [2,1,5,4,6,3] => 3

{{1,2,4},{3},{5},{6}} => {{1},{2},{3,5,6},{4}} => [1,2,5,4,6,3] => 3

{{1,2,5,6},{3,4}} => {{1,2,5,6},{3,4}} => [2,5,4,3,6,1] => 5

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + 2\ q + 2\ q^{2}$

$F_{4} = 1 + 4\ q + 5\ q^{2} + 5\ q^{3}$

$F_{5} = 1 + 7\ q + 12\ q^{2} + 17\ q^{3} + 15\ q^{4}$

$F_{6} = 1 + 12\ q + 29\ q^{2} + 45\ q^{3} + 64\ q^{4} + 52\ q^{5}$

Description

Maximum difference of elements in cycles.
Given a cycle

$C$ in a permutation, we can compute the maximum distance between elements in the cycle, that is

$\max \{ a_i-a_j | a_i, a_j \in C \}$ .
The statistic is then the maximum of this value over all cycles in the permutation.

Map

to permutation

Description

Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.

Map

complement

Description

The complement of a set partition obtained by replacing

$i$ with

$n+1-i$ .