- FindStat

Identifier

Mp00112: Set partitions —complement⟶ Set partitions
St000556: Set partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => 0

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

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

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

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

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

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

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

{{1,2,3,6},{4,5}} => {{1,4,5,6},{2,3}} => 7

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

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

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

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

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

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

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

{{1,2,4,5},{3,6}} => {{1,4},{2,3,5,6}} => 7

{{1,2,4,5},{3},{6}} => {{1},{2,3,5,6},{4}} => 7

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

{{1,2,4},{3,5,6}} => {{1,2,4},{3,5,6}} => 7

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

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

{{1,2,4},{3,6},{5}} => {{1,4},{2},{3,5,6}} => 7

{{1,2,4},{3},{5,6}} => {{1,2},{3,5,6},{4}} => 7

{{1,2,4},{3},{5},{6}} => {{1},{2},{3,5,6},{4}} => 7

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

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

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$F_{2} = 2$

$F_{3} = 4 + q$

$F_{4} = 7 + 5\ q + 2\ q^{2} + q^{3}$

$F_{5} = 11 + 12\ q + 11\ q^{2} + 12\ q^{3} + 3\ q^{4} + 3\ q^{6}$

$F_{6} = 16 + 22\ q + 27\ q^{2} + 45\ q^{3} + 31\ q^{4} + 18\ q^{5} + 22\ q^{6} + 11\ q^{7} + q^{8} + 6\ q^{9} + 2\ q^{10} + 2\ q^{12}$

$F_{7} = 22 + 35\ q + 50\ q^{2} + 106\ q^{3} + 95\ q^{4} + 110\ q^{5} + 121\ q^{6} + 96\ q^{7} + 61\ q^{8} + 46\ q^{9} + 47\ q^{10} + 25\ q^{11} + 27\ q^{12} + 16\ q^{13} + 4\ q^{14} + 6\ q^{15} + 3\ q^{16} + 4\ q^{18} + q^{19} + 2\ q^{20}$

Description

The number of occurrences of the pattern {{1},{2,3}} in a set partition.

Map

complement

Description

The complement of a set partition obtained by replacing

$i$ with

$n+1-i$ .