- FindStat

Identifier

Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000728: Set partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The dimension of a set partition.
This is the sum of the lengths of the arcs of a set partition. Equivalently, one obtains that this is the sum of the maximal entries of the blocks minus the sum of the minimal entries of the blocks.
A slightly shifted definition of the dimension is St000572The dimension exponent of a set partition..

Map

weak exceedance partition

Description

The set partition induced by the weak exceedances of a permutation.
This is the coarsest set partition that contains all arcs $(i, \pi(i))$ with $i\leq\pi(i)$.