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Identifier

Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000008: Integer compositions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{1,2,3,4,5}} => {{1,2,3,4,5}} => [5] => [1,1,1,1,1] => 10

{{1,2,3,4},{5}} => {{1,2,3,4},{5}} => [4,1] => [2,1,1,1] => 9

{{1,2,3,5},{4}} => {{1,2,3},{4,5}} => [3,2] => [1,2,1,1] => 8

{{1,2,3},{4,5}} => {{1,2,3,5},{4}} => [4,1] => [2,1,1,1] => 9

{{1,2,3},{4},{5}} => {{1,2,3},{4},{5}} => [3,1,1] => [3,1,1] => 7

{{1,2,4,5},{3}} => {{1,2,5},{3,4}} => [3,2] => [1,2,1,1] => 8

{{1,2,4},{3,5}} => {{1,2},{3,4,5}} => [2,3] => [1,1,2,1] => 7

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

{{1,2,5},{3,4}} => {{1,2,4},{3,5}} => [3,2] => [1,2,1,1] => 8

{{1,2},{3,4,5}} => {{1,2,4,5},{3}} => [4,1] => [2,1,1,1] => 9

{{1,2},{3,4},{5}} => {{1,2,4},{3},{5}} => [3,1,1] => [3,1,1] => 7

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

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

{{1,2},{3},{4,5}} => {{1,2,5},{3},{4}} => [3,1,1] => [3,1,1] => 7

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

{{1,3,4,5},{2}} => {{1,4,5},{2,3}} => [3,2] => [1,2,1,1] => 8

{{1,3,4},{2,5}} => {{1,4},{2,3,5}} => [2,3] => [1,1,2,1] => 7

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

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

{{1,3},{2,4,5}} => {{1,5},{2,3,4}} => [2,3] => [1,1,2,1] => 7

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

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

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

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

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

{{1,4,5},{2,3}} => {{1,3,5},{2,4}} => [3,2] => [1,2,1,1] => 8

{{1,4},{2,3,5}} => {{1,3},{2,4,5}} => [2,3] => [1,1,2,1] => 7

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

{{1,5},{2,3,4}} => {{1,3,4},{2,5}} => [3,2] => [1,2,1,1] => 8

{{1},{2,3,4,5}} => {{1,3,4,5},{2}} => [4,1] => [2,1,1,1] => 9

{{1},{2,3,4},{5}} => {{1,3,4},{2},{5}} => [3,1,1] => [3,1,1] => 7

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

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

{{1},{2,3},{4,5}} => {{1,3,5},{2},{4}} => [3,1,1] => [3,1,1] => 7

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

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

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

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

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

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

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

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

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

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

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

{{1},{2},{3,4,5}} => {{1,4,5},{2},{3}} => [3,1,1] => [3,1,1] => 7

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

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

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

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

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

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

{{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => [6] => [1,1,1,1,1,1] => 15

{{1,2,3,4,5},{6}} => {{1,2,3,4,5},{6}} => [5,1] => [2,1,1,1,1] => 14

{{1,2,3,4,6},{5}} => {{1,2,3,4},{5,6}} => [4,2] => [1,2,1,1,1] => 13

{{1,2,3,4},{5,6}} => {{1,2,3,4,6},{5}} => [5,1] => [2,1,1,1,1] => 14

{{1,2,3,4},{5},{6}} => {{1,2,3,4},{5},{6}} => [4,1,1] => [3,1,1,1] => 12

{{1,2,3,5,6},{4}} => {{1,2,3,6},{4,5}} => [4,2] => [1,2,1,1,1] => 13

{{1,2,3,5},{4,6}} => {{1,2,3},{4,5,6}} => [3,3] => [1,1,2,1,1] => 12

{{1,2,3,5},{4},{6}} => {{1,2,3},{4,5},{6}} => [3,2,1] => [2,2,1,1] => 11

{{1,2,3,6},{4,5}} => {{1,2,3,5},{4,6}} => [4,2] => [1,2,1,1,1] => 13

{{1,2,3},{4,5,6}} => {{1,2,3,5,6},{4}} => [5,1] => [2,1,1,1,1] => 14

{{1,2,3},{4,5},{6}} => {{1,2,3,5},{4},{6}} => [4,1,1] => [3,1,1,1] => 12

{{1,2,3,6},{4},{5}} => {{1,2,3},{4},{5,6}} => [3,1,2] => [1,3,1,1] => 10

{{1,2,3},{4,6},{5}} => {{1,2,3},{4,6},{5}} => [3,2,1] => [2,2,1,1] => 11

{{1,2,3},{4},{5,6}} => {{1,2,3,6},{4},{5}} => [4,1,1] => [3,1,1,1] => 12

{{1,2,3},{4},{5},{6}} => {{1,2,3},{4},{5},{6}} => [3,1,1,1] => [4,1,1] => 9

{{1,2,4,5,6},{3}} => {{1,2,5,6},{3,4}} => [4,2] => [1,2,1,1,1] => 13

{{1,2,4,5},{3,6}} => {{1,2,5},{3,4,6}} => [3,3] => [1,1,2,1,1] => 12

{{1,2,4,5},{3},{6}} => {{1,2,5},{3,4},{6}} => [3,2,1] => [2,2,1,1] => 11

{{1,2,4,6},{3,5}} => {{1,2},{3,4,5,6}} => [2,4] => [1,1,1,2,1] => 11

{{1,2,4},{3,5,6}} => {{1,2,6},{3,4,5}} => [3,3] => [1,1,2,1,1] => 12

{{1,2,4},{3,5},{6}} => {{1,2},{3,4,5},{6}} => [2,3,1] => [2,1,2,1] => 10

{{1,2,4,6},{3},{5}} => {{1,2},{3,4,6},{5}} => [2,3,1] => [2,1,2,1] => 10

{{1,2,4},{3,6},{5}} => {{1,2},{3,4},{5,6}} => [2,2,2] => [1,2,2,1] => 9

{{1,2,4},{3},{5,6}} => {{1,2,6},{3,4},{5}} => [3,2,1] => [2,2,1,1] => 11

{{1,2,4},{3},{5},{6}} => {{1,2},{3,4},{5},{6}} => [2,2,1,1] => [3,2,1] => 8

{{1,2,5,6},{3,4}} => {{1,2,4,6},{3,5}} => [4,2] => [1,2,1,1,1] => 13

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Description

The major index of the composition.
The descents of a composition $[c_1,c_2,\dots,c_k]$ are the partial sums $c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$, excluding the sum of all parts. The major index of a composition is the sum of its descents.
For details about the major index see Permutations/Descents-Major.

Map

conjugate

Description

The conjugate of a composition.
The conjugate of a composition $C$ is defined as the complement (Mp00039complement) of the reversal (Mp00038reverse) of $C$.
Equivalently, the ribbon shape corresponding to the conjugate of $C$ is the conjugate of the ribbon shape of $C$.

Map

intertwining number to dual major index

Description

A bijection sending the intertwining number of a set partition to its dual major index.
More precisely, St000490The intertwining number of a set partition.$(P) = $St000493The los statistic of a set partition.$(\phi(P))$ for all set partitions $P$.

Map

to composition

Description

The integer composition of block sizes of a set partition.
For a set partition of $\{1,2,\dots,n\}$, this is the integer composition of $n$ obtained by sorting the blocks by their minimal element and then taking the block sizes.