- FindStat

Identifier

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000060: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{2} = 2\ q$

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

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

$F_{5} = q + 8\ q^{2} + 7\ q^{3} + 36\ q^{4}$

$F_{6} = 22\ q^{2} + 21\ q^{3} + 19\ q^{4} + 141\ q^{5}$

Description

The greater neighbor of the maximum.
Han [2] showed that this statistic is (up to a shift) equidistributed on zigzag permutations (permutations

$\pi$ such that

$\pi(1) < \pi(2) > \pi(3) \cdots$ ) with the smallest path leaf label of the binary tree associated to a permutation (St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation.), see also [3].

Map

to permutation

Description

Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.

Map

Simion-Schmidt map

Description

The Simion-Schmidt map sends any permutation to a

$123$ -avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between

$132$ -avoiding permutations and

$123$ -avoiding permutations, see [1, Proposition 19].

Map

Corteel

Description

Corteel's map interchanging the number of crossings and the number of nestings of a permutation.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.