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Identifier

Mp00249: Set partitions —Callan switch⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words

Images

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

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

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

{{1,2,3}} => {{1,3},{2}} => [3,2,1] => 00

{{1,2},{3}} => {{1,2},{3}} => [2,1,3] => 01

{{1,3},{2}} => {{1,2,3}} => [2,3,1] => 00

{{1},{2,3}} => {{1},{2,3}} => [1,3,2] => 10

{{1},{2},{3}} => {{1},{2},{3}} => [1,2,3] => 11

{{1,2,3,4}} => {{1,4},{2},{3}} => [4,2,3,1] => 000

{{1,2,3},{4}} => {{1,3},{2},{4}} => [3,2,1,4] => 001

{{1,2,4},{3}} => {{1,3,4},{2}} => [3,2,4,1] => 000

{{1,2},{3,4}} => {{1,2},{3,4}} => [2,1,4,3] => 010

{{1,2},{3},{4}} => {{1,2},{3},{4}} => [2,1,3,4] => 011

{{1,3,4},{2}} => {{1,2,4},{3}} => [2,4,3,1] => 000

{{1,3},{2,4}} => {{1,3},{2,4}} => [3,4,1,2] => 000

{{1,3},{2},{4}} => {{1,2,3},{4}} => [2,3,1,4] => 001

{{1,4},{2,3}} => {{1,4},{2,3}} => [4,3,2,1] => 000

{{1},{2,3,4}} => {{1},{2,3,4}} => [1,3,4,2] => 100

{{1},{2,3},{4}} => {{1},{2,3},{4}} => [1,3,2,4] => 101

{{1,4},{2},{3}} => {{1,2,3,4}} => [2,3,4,1] => 000

{{1},{2,4},{3}} => {{1},{2,4},{3}} => [1,4,3,2] => 100

{{1},{2},{3,4}} => {{1},{2},{3,4}} => [1,2,4,3] => 110

{{1},{2},{3},{4}} => {{1},{2},{3},{4}} => [1,2,3,4] => 111

{{1,2,3,4,5}} => {{1,5},{2},{3},{4}} => [5,2,3,4,1] => 0000

{{1,2,3,4},{5}} => {{1,4},{2},{3},{5}} => [4,2,3,1,5] => 0001

{{1,2,3,5},{4}} => {{1,4,5},{2},{3}} => [4,2,3,5,1] => 0000

{{1,2,3},{4,5}} => {{1,3},{2},{4,5}} => [3,2,1,5,4] => 0010

{{1,2,3},{4},{5}} => {{1,3},{2},{4},{5}} => [3,2,1,4,5] => 0011

{{1,2,4,5},{3}} => {{1,3,4,5},{2}} => [3,2,4,5,1] => 0000

{{1,2,4},{3,5}} => {{1,4},{2},{3,5}} => [4,2,5,1,3] => 0000

{{1,2,4},{3},{5}} => {{1,3,4},{2},{5}} => [3,2,4,1,5] => 0001

{{1,2,5},{3,4}} => {{1,5},{2},{3,4}} => [5,2,4,3,1] => 0000

{{1,2},{3,4,5}} => {{1,2},{3,4,5}} => [2,1,4,5,3] => 0100

{{1,2},{3,4},{5}} => {{1,2},{3,4},{5}} => [2,1,4,3,5] => 0101

{{1,2,5},{3},{4}} => {{1,3,5},{2},{4}} => [3,2,5,4,1] => 0000

{{1,2},{3,5},{4}} => {{1,2},{3,5},{4}} => [2,1,5,4,3] => 0100

{{1,2},{3},{4,5}} => {{1,2},{3},{4,5}} => [2,1,3,5,4] => 0110

{{1,2},{3},{4},{5}} => {{1,2},{3},{4},{5}} => [2,1,3,4,5] => 0111

{{1,3,4,5},{2}} => {{1,2,4,5},{3}} => [2,4,3,5,1] => 0000

{{1,3,4},{2,5}} => {{1,4},{2,5},{3}} => [4,5,3,1,2] => 0000

{{1,3,4},{2},{5}} => {{1,2,4},{3},{5}} => [2,4,3,1,5] => 0001

{{1,3,5},{2,4}} => {{1,5},{2,4},{3}} => [5,4,3,2,1] => 0000

{{1,3},{2,4,5}} => {{1,3},{2,4,5}} => [3,4,1,5,2] => 0000

{{1,3},{2,4},{5}} => {{1,3},{2,4},{5}} => [3,4,1,2,5] => 0001

{{1,3,5},{2},{4}} => {{1,2,5},{3},{4}} => [2,5,3,4,1] => 0000

{{1,3},{2,5},{4}} => {{1,3},{2,5},{4}} => [3,5,1,4,2] => 0000

{{1,3},{2},{4,5}} => {{1,2,3},{4,5}} => [2,3,1,5,4] => 0010

{{1,3},{2},{4},{5}} => {{1,2,3},{4},{5}} => [2,3,1,4,5] => 0011

{{1,4,5},{2,3}} => {{1,4,5},{2,3}} => [4,3,2,5,1] => 0000

{{1,4},{2,3,5}} => {{1,4},{2,3,5}} => [4,3,5,1,2] => 0000

{{1,4},{2,3},{5}} => {{1,4},{2,3},{5}} => [4,3,2,1,5] => 0001

{{1,5},{2,3,4}} => {{1,5},{2,3,4}} => [5,3,4,2,1] => 0000

{{1},{2,3,4,5}} => {{1},{2,3,4,5}} => [1,3,4,5,2] => 1000

{{1},{2,3,4},{5}} => {{1},{2,3,4},{5}} => [1,3,4,2,5] => 1001

{{1,5},{2,3},{4}} => {{1,5},{2,3},{4}} => [5,3,2,4,1] => 0000

{{1},{2,3,5},{4}} => {{1},{2,3,5},{4}} => [1,3,5,4,2] => 1000

{{1},{2,3},{4,5}} => {{1},{2,3},{4,5}} => [1,3,2,5,4] => 1010

{{1},{2,3},{4},{5}} => {{1},{2,3},{4},{5}} => [1,3,2,4,5] => 1011

{{1,4,5},{2},{3}} => {{1,2,3,5},{4}} => [2,3,5,4,1] => 0000

{{1,4},{2,5},{3}} => {{1,3,4},{2,5}} => [3,5,4,1,2] => 0000

{{1,4},{2},{3,5}} => {{1,2,4},{3,5}} => [2,4,5,1,3] => 0000

{{1,4},{2},{3},{5}} => {{1,2,3,4},{5}} => [2,3,4,1,5] => 0001

{{1,5},{2,4},{3}} => {{1,3,5},{2,4}} => [3,4,5,2,1] => 0000

{{1},{2,4,5},{3}} => {{1},{2,4,5},{3}} => [1,4,3,5,2] => 1000

{{1},{2,4},{3,5}} => {{1},{2,4},{3,5}} => [1,4,5,2,3] => 1000

{{1},{2,4},{3},{5}} => {{1},{2,4},{3},{5}} => [1,4,3,2,5] => 1001

{{1,5},{2},{3,4}} => {{1,2,5},{3,4}} => [2,5,4,3,1] => 0000

{{1},{2,5},{3,4}} => {{1},{2,5},{3,4}} => [1,5,4,3,2] => 1000

{{1},{2},{3,4,5}} => {{1},{2},{3,4,5}} => [1,2,4,5,3] => 1100

{{1},{2},{3,4},{5}} => {{1},{2},{3,4},{5}} => [1,2,4,3,5] => 1101

{{1,5},{2},{3},{4}} => {{1,2,3,4,5}} => [2,3,4,5,1] => 0000

{{1},{2,5},{3},{4}} => {{1},{2,5},{3},{4}} => [1,5,3,4,2] => 1000

{{1},{2},{3,5},{4}} => {{1},{2},{3,5},{4}} => [1,2,5,4,3] => 1100

{{1},{2},{3},{4,5}} => {{1},{2},{3},{4,5}} => [1,2,3,5,4] => 1110

{{1},{2},{3},{4},{5}} => {{1},{2},{3},{4},{5}} => [1,2,3,4,5] => 1111

{{1,2,3,4,5,6}} => {{1,6},{2},{3},{4},{5}} => [6,2,3,4,5,1] => 00000

{{1,2,3,4,5},{6}} => {{1,5},{2},{3},{4},{6}} => [5,2,3,4,1,6] => 00001

{{1,2,3,4,6},{5}} => {{1,5,6},{2},{3},{4}} => [5,2,3,4,6,1] => 00000

{{1,2,3,4},{5,6}} => {{1,4},{2},{3},{5,6}} => [4,2,3,1,6,5] => 00010

{{1,2,3,4},{5},{6}} => {{1,4},{2},{3},{5},{6}} => [4,2,3,1,5,6] => 00011

{{1,2,3,5,6},{4}} => {{1,4,5,6},{2},{3}} => [4,2,3,5,6,1] => 00000

{{1,2,3,5},{4,6}} => {{1,5},{2},{3},{4,6}} => [5,2,3,6,1,4] => 00000

{{1,2,3,5},{4},{6}} => {{1,4,5},{2},{3},{6}} => [4,2,3,5,1,6] => 00001

{{1,2,3,6},{4,5}} => {{1,6},{2},{3},{4,5}} => [6,2,3,5,4,1] => 00000

{{1,2,3},{4,5,6}} => {{1,3},{2},{4,5,6}} => [3,2,1,5,6,4] => 00100

{{1,2,3},{4,5},{6}} => {{1,3},{2},{4,5},{6}} => [3,2,1,5,4,6] => 00101

{{1,2,3,6},{4},{5}} => {{1,4,6},{2},{3},{5}} => [4,2,3,6,5,1] => 00000

{{1,2,3},{4,6},{5}} => {{1,3},{2},{4,6},{5}} => [3,2,1,6,5,4] => 00100

{{1,2,3},{4},{5,6}} => {{1,3},{2},{4},{5,6}} => [3,2,1,4,6,5] => 00110

{{1,2,3},{4},{5},{6}} => {{1,3},{2},{4},{5},{6}} => [3,2,1,4,5,6] => 00111

{{1,2,4,5,6},{3}} => {{1,3,4,5,6},{2}} => [3,2,4,5,6,1] => 00000

{{1,2,4,5},{3,6}} => {{1,5},{2},{3,6},{4}} => [5,2,6,4,1,3] => 00000

{{1,2,4,5},{3},{6}} => {{1,3,4,5},{2},{6}} => [3,2,4,5,1,6] => 00001

{{1,2,4,6},{3,5}} => {{1,6},{2},{3,5},{4}} => [6,2,5,4,3,1] => 00000

{{1,2,4},{3,5,6}} => {{1,4},{2},{3,5,6}} => [4,2,5,1,6,3] => 00000

{{1,2,4},{3,5},{6}} => {{1,4},{2},{3,5},{6}} => [4,2,5,1,3,6] => 00001

{{1,2,4,6},{3},{5}} => {{1,3,4,6},{2},{5}} => [3,2,4,6,5,1] => 00000

{{1,2,4},{3,6},{5}} => {{1,4},{2},{3,6},{5}} => [4,2,6,1,5,3] => 00000

{{1,2,4},{3},{5,6}} => {{1,3,4},{2},{5,6}} => [3,2,4,1,6,5] => 00010

{{1,2,4},{3},{5},{6}} => {{1,3,4},{2},{5},{6}} => [3,2,4,1,5,6] => 00011

{{1,2,5,6},{3,4}} => {{1,5,6},{2},{3,4}} => [5,2,4,3,6,1] => 00000

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Map

Callan switch

Description

Switch the first closer and the minimum of the smallest closer and the second element of the block containing 1 in a set partition.
More precisely, this involution implements a joint symmetry between St000971The smallest closer of a set partition. and St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition..

Map

to permutation

Description

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

Map

connectivity set

Description

The connectivity set of a permutation as a binary word.
According to [2], also known as the global ascent set.
The connectivity set is
$$C(\pi)=\{i\in [n-1] | \forall 1 \leq j \leq i < k \leq n : \pi(j) < \pi(k)\}.$$
For $n > 1$ it can also be described as the set of occurrences of the mesh pattern
$$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$$
or equivalently
$$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$$
see [3].
The permutation is connected, when the connectivity set is empty.