- FindStat

Identifier

Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000383: Integer compositions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

$F_{2} = q + q^{2}$

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

$F_{4} = 5\ q + 6\ q^{2} + 3\ q^{3} + q^{4}$

$F_{5} = 15\ q + 20\ q^{2} + 12\ q^{3} + 4\ q^{4} + q^{5}$

$F_{6} = 52\ q + 75\ q^{2} + 50\ q^{3} + 20\ q^{4} + 5\ q^{5} + q^{6}$

$F_{7} = 203\ q + 312\ q^{2} + 225\ q^{3} + 100\ q^{4} + 30\ q^{5} + 6\ q^{6} + q^{7}$

Description

The last part of an integer composition.

Map

reverse

Description

Return the reversal of a composition.
That is, the composition

$(i_1, i_2, \ldots, i_k)$ is sent to

$(i_k, i_{k-1}, \ldots, i_1)$ .

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.