- FindStat

Identifier

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000030: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

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

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

$F_{5} = 16 + 12\ q^{2} + 14\ q^{3} + 8\ q^{4} + q^{5} + q^{6}$

$F_{6} = 32 + 32\ q^{2} + 48\ q^{3} + 42\ q^{4} + 21\ q^{5} + 9\ q^{6} + 15\ q^{7} + 4\ q^{8}$

Description

The sum of the descent differences of a permutations.
This statistic is given by

$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} (\pi_i-\pi_{i+1}).$
See St000111The sum of the descent tops (or Genocchi descents) of a permutation. and St000154The sum of the descent bottoms of a permutation. for the sum of the descent tops and the descent bottoms, respectively. This statistic was studied in [1] and [2] where is was called the drop of a permutation.

Map

to permutation

Description

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

Map

cycle-as-one-line notation

Description

Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.

Map

major-index to inversion-number bijection

Description

Return the permutation whose Lehmer code equals the major code of the preimage.
This map sends the major index to the number of inversions.