- FindStat

Identifier

Mp00112: Set partitions —complement⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000607: Set partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{2} = 2$

$F_{3} = 4 + q$

$F_{4} = 9 + 4\ q + 2\ q^{2}$

$F_{5} = 25 + 12\ q + 8\ q^{2} + 6\ q^{3} + q^{4}$

$F_{6} = 83 + 36\ q + 27\ q^{2} + 30\ q^{3} + 17\ q^{4} + 8\ q^{5} + 2\ q^{6}$

$F_{7} = 307 + 125\ q + 100\ q^{2} + 123\ q^{3} + 86\ q^{4} + 73\ q^{5} + 34\ q^{6} + 20\ q^{7} + 8\ q^{8} + q^{9}$

Description

The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block.

Map

complement

Description

The complement of a set partition obtained by replacing

$i$ with

$n+1-i$ .

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

$P$ .