- FindStat

Identifier

Mp00249: Set partitions —Callan switch⟶ Set partitions
Mp00174: Set partitions —dual major index to intertwining number⟶ Set partitions
St000729: Set partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$F_{4} = 10\ q + 3\ q^{2} + q^{3} + q^{4}$

$F_{5} = 37\ q + 10\ q^{2} + 3\ q^{3} + q^{4} + q^{5}$

$F_{6} = 151\ q + 37\ q^{2} + 10\ q^{3} + 3\ q^{4} + q^{5} + q^{6}$

$F_{7} = 674\ q + 151\ q^{2} + 37\ q^{3} + 10\ q^{4} + 3\ q^{5} + q^{6} + q^{7}$

Description

The minimal arc length of a set partition.
The arcs of a set partition are those

$i < j$ that are consecutive elements in the blocks. If there are no arcs, the minimal arc length is the size of the ground set (as the minimum of the empty set in the universe of arcs of length less than the size of the ground set).

Map

dual major index to intertwining number

Description

A bijection sending the dual major index of a set partition to its intertwining number.
More precisely, St000493The los statistic of a set partition.

$(P) =$ St000490The intertwining number of a set partition.

$(\phi(P))$ for all set partitions

$P$ .
This is the inverse of Mp00171intertwining number to dual major index.

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..