Your data matches 5 different statistics following compositions of up to 3 maps.
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Matching statistic: St000233
(load all 2 compositions to match this statistic)
Mapping
Mp00091: Set partitions —rotate increasing⟶ Set partitions
St000233: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => {{1}}
 => 0
{{1,2}}
 => {{1,2}}
 => 0
{{1},{2}}
 => {{1},{2}}
 => 0
{{1,2,3}}
 => {{1,2,3}}
 => 0
{{1,2},{3}}
 => {{1},{2,3}}
 => 0
{{1,3},{2}}
 => {{1,2},{3}}
 => 0
{{1},{2,3}}
 => {{1,3},{2}}
 => 0
{{1},{2},{3}}
 => {{1},{2},{3}}
 => 0
{{1,2,3,4}}
 => {{1,2,3,4}}
 => 0
{{1,2,3},{4}}
 => {{1},{2,3,4}}
 => 0
{{1,2,4},{3}}
 => {{1,2,3},{4}}
 => 0
{{1,2},{3,4}}
 => {{1,4},{2,3}}
 => 1
{{1,2},{3},{4}}
 => {{1},{2,3},{4}}
 => 0
{{1,3,4},{2}}
 => {{1,2,4},{3}}
 => 0
{{1,3},{2,4}}
 => {{1,3},{2,4}}
 => 0
{{1,3},{2},{4}}
 => {{1},{2,4},{3}}
 => 0
{{1,4},{2,3}}
 => {{1,2},{3,4}}
 => 0
{{1},{2,3,4}}
 => {{1,3,4},{2}}
 => 0
{{1},{2,3},{4}}
 => {{1},{2},{3,4}}
 => 0
{{1,4},{2},{3}}
 => {{1,2},{3},{4}}
 => 0
{{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => 0
{{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => 0
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => 0
{{1,2,3,4},{5}}
 => {{1},{2,3,4,5}}
 => 0
{{1,2,3,5},{4}}
 => {{1,2,3,4},{5}}
 => 0
{{1,2,3},{4,5}}
 => {{1,5},{2,3,4}}
 => 2
{{1,2,3},{4},{5}}
 => {{1},{2,3,4},{5}}
 => 0
{{1,2,4,5},{3}}
 => {{1,2,3,5},{4}}
 => 0
{{1,2,4},{3,5}}
 => {{1,4},{2,3,5}}
 => 1
{{1,2,4},{3},{5}}
 => {{1},{2,3,5},{4}}
 => 0
{{1,2,5},{3,4}}
 => {{1,2,3},{4,5}}
 => 0
{{1,2},{3,4,5}}
 => {{1,4,5},{2,3}}
 => 1
{{1,2},{3,4},{5}}
 => {{1},{2,3},{4,5}}
 => 0
{{1,2,5},{3},{4}}
 => {{1,2,3},{4},{5}}
 => 0
{{1,2},{3,5},{4}}
 => {{1,4},{2,3},{5}}
 => 1
{{1,2},{3},{4,5}}
 => {{1,5},{2,3},{4}}
 => 1
{{1,2},{3},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => 0
{{1,3,4,5},{2}}
 => {{1,2,4,5},{3}}
 => 0
{{1,3,4},{2,5}}
 => {{1,3},{2,4,5}}
 => 0
{{1,3,4},{2},{5}}
 => {{1},{2,4,5},{3}}
 => 0
{{1,3,5},{2,4}}
 => {{1,2,4},{3,5}}
 => 0
{{1,3},{2,4,5}}
 => {{1,3,5},{2,4}}
 => 0
{{1,3},{2,4},{5}}
 => {{1},{2,4},{3,5}}
 => 0
{{1,3,5},{2},{4}}
 => {{1,2,4},{3},{5}}
 => 0
{{1,3},{2,5},{4}}
 => {{1,3},{2,4},{5}}
 => 0
{{1,3},{2},{4,5}}
 => {{1,5},{2,4},{3}}
 => 1
{{1,3},{2},{4},{5}}
 => {{1},{2,4},{3},{5}}
 => 0
{{1,4,5},{2,3}}
 => {{1,2,5},{3,4}}
 => 1
{{1,4},{2,3,5}}
 => {{1,3,4},{2,5}}
 => 1
Description
The number of nestings of a set partition. This is given by the number of $i < i' < j' < j$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.
Matching statistic: St000232
(load all 3 compositions to match this statistic)
Mapping
Mp00091: Set partitions —rotate increasing⟶ Set partitions
Mp00115: Set partitions —Kasraoui-Zeng⟶ Set partitions
St000232: Set partitions ⟶ ℤResult quality: 44% ●values known / values provided: 87%●distinct values known / distinct values provided: 44%
Values
{{1}}
 => {{1}}
 => {{1}}
 => 0
{{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},{2,3}}
 => 0
{{1,3},{2}}
 => {{1,2},{3}}
 => {{1,2},{3}}
 => 0
{{1},{2,3}}
 => {{1,3},{2}}
 => {{1,3},{2}}
 => 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},{2,3,4}}
 => 0
{{1,2,4},{3}}
 => {{1,2,3},{4}}
 => {{1,2,3},{4}}
 => 0
{{1,2},{3,4}}
 => {{1,4},{2,3}}
 => {{1,3},{2,4}}
 => 1
{{1,2},{3},{4}}
 => {{1},{2,3},{4}}
 => {{1},{2,3},{4}}
 => 0
{{1,3,4},{2}}
 => {{1,2,4},{3}}
 => {{1,2,4},{3}}
 => 0
{{1,3},{2,4}}
 => {{1,3},{2,4}}
 => {{1,4},{2,3}}
 => 0
{{1,3},{2},{4}}
 => {{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => 0
{{1,4},{2,3}}
 => {{1,2},{3,4}}
 => {{1,2},{3,4}}
 => 0
{{1},{2,3,4}}
 => {{1,3,4},{2}}
 => {{1,3,4},{2}}
 => 0
{{1},{2,3},{4}}
 => {{1},{2},{3,4}}
 => {{1},{2},{3,4}}
 => 0
{{1,4},{2},{3}}
 => {{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => 0
{{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => {{1,3},{2},{4}}
 => 0
{{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => {{1,4},{2},{3}}
 => 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},{2,3,4,5}}
 => 0
{{1,2,3,5},{4}}
 => {{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => 0
{{1,2,3},{4,5}}
 => {{1,5},{2,3,4}}
 => {{1,3,5},{2,4}}
 => 2
{{1,2,3},{4},{5}}
 => {{1},{2,3,4},{5}}
 => {{1},{2,3,4},{5}}
 => 0
{{1,2,4,5},{3}}
 => {{1,2,3,5},{4}}
 => {{1,2,3,5},{4}}
 => 0
{{1,2,4},{3,5}}
 => {{1,4},{2,3,5}}
 => {{1,3,4},{2,5}}
 => 1
{{1,2,4},{3},{5}}
 => {{1},{2,3,5},{4}}
 => {{1},{2,3,5},{4}}
 => 0
{{1,2,5},{3,4}}
 => {{1,2,3},{4,5}}
 => {{1,2,3},{4,5}}
 => 0
{{1,2},{3,4,5}}
 => {{1,4,5},{2,3}}
 => {{1,3},{2,4,5}}
 => 1
{{1,2},{3,4},{5}}
 => {{1},{2,3},{4,5}}
 => {{1},{2,3},{4,5}}
 => 0
{{1,2,5},{3},{4}}
 => {{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => 0
{{1,2},{3,5},{4}}
 => {{1,4},{2,3},{5}}
 => {{1,3},{2,4},{5}}
 => 1
{{1,2},{3},{4,5}}
 => {{1,5},{2,3},{4}}
 => {{1,3},{2,5},{4}}
 => 1
{{1,2},{3},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => 0
{{1,3,4,5},{2}}
 => {{1,2,4,5},{3}}
 => {{1,2,4,5},{3}}
 => 0
{{1,3,4},{2,5}}
 => {{1,3},{2,4,5}}
 => {{1,4,5},{2,3}}
 => 0
{{1,3,4},{2},{5}}
 => {{1},{2,4,5},{3}}
 => {{1},{2,4,5},{3}}
 => 0
{{1,3,5},{2,4}}
 => {{1,2,4},{3,5}}
 => {{1,2,5},{3,4}}
 => 0
{{1,3},{2,4,5}}
 => {{1,3,5},{2,4}}
 => {{1,5},{2,3,4}}
 => 0
{{1,3},{2,4},{5}}
 => {{1},{2,4},{3,5}}
 => {{1},{2,5},{3,4}}
 => 0
{{1,3,5},{2},{4}}
 => {{1,2,4},{3},{5}}
 => {{1,2,4},{3},{5}}
 => 0
{{1,3},{2,5},{4}}
 => {{1,3},{2,4},{5}}
 => {{1,4},{2,3},{5}}
 => 0
{{1,3},{2},{4,5}}
 => {{1,5},{2,4},{3}}
 => {{1,4},{2,5},{3}}
 => 1
{{1,3},{2},{4},{5}}
 => {{1},{2,4},{3},{5}}
 => {{1},{2,4},{3},{5}}
 => 0
{{1,4,5},{2,3}}
 => {{1,2,5},{3,4}}
 => {{1,2,4},{3,5}}
 => 1
{{1,4},{2,3,5}}
 => {{1,3,4},{2,5}}
 => {{1,4},{2,3,5}}
 => 1
{{1,2},{3,4},{5,6},{7,8},{9,10}}
 => {{1,10},{2,3},{4,5},{6,7},{8,9}}
 => {{1,3},{2,5},{4,7},{6,9},{8,10}}
 => ? = 4
{{1,4},{2,3},{5,6},{7,8},{9,10}}
 => {{1,10},{2,5},{3,4},{6,7},{8,9}}
 => {{1,4},{2,5},{3,7},{6,9},{8,10}}
 => ? = 5
{{1,6},{2,3},{4,5},{7,8},{9,10}}
 => {{1,10},{2,7},{3,4},{5,6},{8,9}}
 => {{1,4},{2,6},{3,7},{5,9},{8,10}}
 => ? = 6
{{1,8},{2,3},{4,5},{6,7},{9,10}}
 => {{1,10},{2,9},{3,4},{5,6},{7,8}}
 => {{1,4},{2,6},{3,8},{5,9},{7,10}}
 => ? = 7
{{1,2},{3,6},{4,5},{7,8},{9,10}}
 => {{1,10},{2,3},{4,7},{5,6},{8,9}}
 => {{1,3},{2,6},{4,7},{5,9},{8,10}}
 => ? = 5
{{1,6},{2,5},{3,4},{7,8},{9,10}}
 => {{1,10},{2,7},{3,6},{4,5},{8,9}}
 => {{1,5},{2,6},{3,7},{4,9},{8,10}}
 => ? = 7
{{1,8},{2,5},{3,4},{6,7},{9,10}}
 => {{1,10},{2,9},{3,6},{4,5},{7,8}}
 => {{1,5},{2,6},{3,8},{4,9},{7,10}}
 => ? = 8
{{1,10},{2,5},{3,4},{6,7},{8,9}}
 => {{1,2},{3,6},{4,5},{7,8},{9,10}}
 => {{1,2},{3,5},{4,6},{7,8},{9,10}}
 => ? = 1
{{1,2},{3,8},{4,5},{6,7},{9,10}}
 => {{1,10},{2,3},{4,9},{5,6},{7,8}}
 => {{1,3},{2,6},{4,8},{5,9},{7,10}}
 => ? = 6
{{1,8},{2,7},{3,4},{5,6},{9,10}}
 => {{1,10},{2,9},{3,8},{4,5},{6,7}}
 => {{1,5},{2,7},{3,8},{4,9},{6,10}}
 => ? = 9
{{1,10},{2,7},{3,4},{5,6},{8,9}}
 => {{1,2},{3,8},{4,5},{6,7},{9,10}}
 => {{1,2},{3,5},{4,7},{6,8},{9,10}}
 => ? = 2
{{1,2},{3,10},{4,5},{6,7},{8,9}}
 => {{1,4},{2,3},{5,6},{7,8},{9,10}}
 => {{1,3},{2,4},{5,6},{7,8},{9,10}}
 => ? = 1
{{1,10},{2,9},{3,4},{5,6},{7,8}}
 => {{1,2},{3,10},{4,5},{6,7},{8,9}}
 => {{1,2},{3,5},{4,7},{6,9},{8,10}}
 => ? = 3
{{1,2},{3,4},{5,8},{6,7},{9,10}}
 => {{1,10},{2,3},{4,5},{6,9},{7,8}}
 => {{1,3},{2,5},{4,8},{6,9},{7,10}}
 => ? = 5
{{1,4},{2,3},{5,8},{6,7},{9,10}}
 => {{1,10},{2,5},{3,4},{6,9},{7,8}}
 => {{1,4},{2,5},{3,8},{6,9},{7,10}}
 => ? = 6
{{1,8},{2,3},{4,7},{5,6},{9,10}}
 => {{1,10},{2,9},{3,4},{5,8},{6,7}}
 => {{1,4},{2,7},{3,8},{5,9},{6,10}}
 => ? = 8
{{1,10},{2,3},{4,7},{5,6},{8,9}}
 => {{1,2},{3,4},{5,8},{6,7},{9,10}}
 => {{1,2},{3,4},{5,7},{6,8},{9,10}}
 => ? = 1
{{1,2},{3,8},{4,7},{5,6},{9,10}}
 => {{1,10},{2,3},{4,9},{5,8},{6,7}}
 => {{1,3},{2,7},{4,8},{5,9},{6,10}}
 => ? = 7
{{1,8},{2,7},{3,6},{4,5},{9,10}}
 => {{1,10},{2,9},{3,8},{4,7},{5,6}}
 => {{1,6},{2,7},{3,8},{4,9},{5,10}}
 => ? = 10
{{1,10},{2,7},{3,6},{4,5},{8,9}}
 => {{1,2},{3,8},{4,7},{5,6},{9,10}}
 => {{1,2},{3,6},{4,7},{5,8},{9,10}}
 => ? = 3
{{1,2},{3,10},{4,7},{5,6},{8,9}}
 => {{1,4},{2,3},{5,8},{6,7},{9,10}}
 => {{1,3},{2,4},{5,7},{6,8},{9,10}}
 => ? = 2
{{1,10},{2,9},{3,6},{4,5},{7,8}}
 => {{1,2},{3,10},{4,7},{5,6},{8,9}}
 => {{1,2},{3,6},{4,7},{5,9},{8,10}}
 => ? = 4
{{1,2},{3,4},{5,10},{6,7},{8,9}}
 => {{1,6},{2,3},{4,5},{7,8},{9,10}}
 => {{1,3},{2,5},{4,6},{7,8},{9,10}}
 => ? = 2
{{1,4},{2,3},{5,10},{6,7},{8,9}}
 => {{1,6},{2,5},{3,4},{7,8},{9,10}}
 => {{1,4},{2,5},{3,6},{7,8},{9,10}}
 => ? = 3
{{1,10},{2,3},{4,9},{5,6},{7,8}}
 => {{1,2},{3,4},{5,10},{6,7},{8,9}}
 => {{1,2},{3,4},{5,7},{6,9},{8,10}}
 => ? = 2
{{1,2},{3,10},{4,9},{5,6},{7,8}}
 => {{1,4},{2,3},{5,10},{6,7},{8,9}}
 => {{1,3},{2,4},{5,7},{6,9},{8,10}}
 => ? = 3
{{1,10},{2,9},{3,8},{4,5},{6,7}}
 => {{1,2},{3,10},{4,9},{5,6},{7,8}}
 => {{1,2},{3,6},{4,8},{5,9},{7,10}}
 => ? = 5
{{1,2},{3,4},{5,6},{7,10},{8,9}}
 => {{1,8},{2,3},{4,5},{6,7},{9,10}}
 => {{1,3},{2,5},{4,7},{6,8},{9,10}}
 => ? = 3
{{1,4},{2,3},{5,6},{7,10},{8,9}}
 => {{1,8},{2,5},{3,4},{6,7},{9,10}}
 => {{1,4},{2,5},{3,7},{6,8},{9,10}}
 => ? = 4
{{1,6},{2,3},{4,5},{7,10},{8,9}}
 => {{1,8},{2,7},{3,4},{5,6},{9,10}}
 => {{1,4},{2,6},{3,7},{5,8},{9,10}}
 => ? = 5
{{1,10},{2,3},{4,5},{6,9},{7,8}}
 => {{1,2},{3,4},{5,6},{7,10},{8,9}}
 => {{1,2},{3,4},{5,6},{7,9},{8,10}}
 => ? = 1
{{1,2},{3,6},{4,5},{7,10},{8,9}}
 => {{1,8},{2,3},{4,7},{5,6},{9,10}}
 => {{1,3},{2,6},{4,7},{5,8},{9,10}}
 => ? = 4
{{1,6},{2,5},{3,4},{7,10},{8,9}}
 => {{1,8},{2,7},{3,6},{4,5},{9,10}}
 => {{1,5},{2,6},{3,7},{4,8},{9,10}}
 => ? = 6
{{1,10},{2,5},{3,4},{6,9},{7,8}}
 => {{1,2},{3,6},{4,5},{7,10},{8,9}}
 => {{1,2},{3,5},{4,6},{7,9},{8,10}}
 => ? = 2
{{1,2},{3,10},{4,5},{6,9},{7,8}}
 => {{1,4},{2,3},{5,6},{7,10},{8,9}}
 => {{1,3},{2,4},{5,6},{7,9},{8,10}}
 => ? = 2
{{1,10},{2,9},{3,4},{5,8},{6,7}}
 => {{1,2},{3,10},{4,5},{6,9},{7,8}}
 => {{1,2},{3,5},{4,8},{6,9},{7,10}}
 => ? = 4
{{1,2},{3,4},{5,10},{6,9},{7,8}}
 => {{1,6},{2,3},{4,5},{7,10},{8,9}}
 => {{1,3},{2,5},{4,6},{7,9},{8,10}}
 => ? = 3
{{1,4},{2,3},{5,10},{6,9},{7,8}}
 => {{1,6},{2,5},{3,4},{7,10},{8,9}}
 => {{1,4},{2,5},{3,6},{7,9},{8,10}}
 => ? = 4
{{1,10},{2,3},{4,9},{5,8},{6,7}}
 => {{1,2},{3,4},{5,10},{6,9},{7,8}}
 => {{1,2},{3,4},{5,8},{6,9},{7,10}}
 => ? = 3
{{1,2},{3,10},{4,9},{5,8},{6,7}}
 => {{1,4},{2,3},{5,10},{6,9},{7,8}}
 => {{1,3},{2,4},{5,8},{6,9},{7,10}}
 => ? = 4
{{1,10},{2,9},{3,8},{4,7},{5,6}}
 => {{1,2},{3,10},{4,9},{5,8},{6,7}}
 => {{1,2},{3,7},{4,8},{5,9},{6,10}}
 => ? = 6
{{1,4,5,6,7,8},{2,3}}
 => {{1,2,5,6,7,8},{3,4}}
 => {{1,2,4},{3,5,6,7,8}}
 => ? = 1
{{1,2,5,6,7,8},{3,4}}
 => {{1,2,3,6,7,8},{4,5}}
 => {{1,2,3,5},{4,6,7,8}}
 => ? = 1
{{1,2,3,6,8},{4,5},{7}}
 => {{1,2,3,4,7},{5,6},{8}}
 => {{1,2,3,4,6},{5,7},{8}}
 => ? = 1
{{1,2,3,6,7,8},{4,5}}
 => {{1,2,3,4,7,8},{5,6}}
 => {{1,2,3,4,6},{5,7,8}}
 => ? = 1
{{1,2,3,4,5,6},{7,8}}
 => {{1,8},{2,3,4,5,6,7}}
 => {{1,3,5,7},{2,4,6,8}}
 => ? = 5
{{1,2,3,7},{4,5,6},{8}}
 => {{1},{2,3,4,8},{5,6,7}}
 => {{1},{2,3,4,6,8},{5,7}}
 => ? = 2
{{1,2,3,4,7},{5,6},{8}}
 => {{1},{2,3,4,5,8},{6,7}}
 => {{1},{2,3,4,5,7},{6,8}}
 => ? = 1
{{1,3,5,6,7,8},{2,4}}
 => {{1,2,4,6,7,8},{3,5}}
 => {{1,2,6,7,8},{3,4,5}}
 => ? = 0
{{1,3,4,6,7,8},{2,5}}
 => {{1,2,4,5,7,8},{3,6}}
 => {{1,2,5,6},{3,4,7,8}}
 => ? = 1
Description
The number of crossings of a set partition. This is given by the number of $i < i' < j < j'$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.
Matching statistic: St001513
(load all 2 compositions to match this statistic)
Mapping
Mp00091: Set partitions —rotate increasing⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St001513: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 27%●distinct values known / distinct values provided: 25%
Values
{{1}}
 => {{1}}
 => [1] => 0
{{1,2}}
 => {{1,2}}
 => [2,1] => 0
{{1},{2}}
 => {{1},{2}}
 => [1,2] => 0
{{1,2,3}}
 => {{1,2,3}}
 => [2,3,1] => 0
{{1,2},{3}}
 => {{1},{2,3}}
 => [1,3,2] => 0
{{1,3},{2}}
 => {{1,2},{3}}
 => [2,1,3] => 0
{{1},{2,3}}
 => {{1,3},{2}}
 => [3,2,1] => 0
{{1},{2},{3}}
 => {{1},{2},{3}}
 => [1,2,3] => 0
{{1,2,3,4}}
 => {{1,2,3,4}}
 => [2,3,4,1] => 0
{{1,2,3},{4}}
 => {{1},{2,3,4}}
 => [1,3,4,2] => 0
{{1,2,4},{3}}
 => {{1,2,3},{4}}
 => [2,3,1,4] => 0
{{1,2},{3,4}}
 => {{1,4},{2,3}}
 => [4,3,2,1] => 1
{{1,2},{3},{4}}
 => {{1},{2,3},{4}}
 => [1,3,2,4] => 0
{{1,3,4},{2}}
 => {{1,2,4},{3}}
 => [2,4,3,1] => 0
{{1,3},{2,4}}
 => {{1,3},{2,4}}
 => [3,4,1,2] => 0
{{1,3},{2},{4}}
 => {{1},{2,4},{3}}
 => [1,4,3,2] => 0
{{1,4},{2,3}}
 => {{1,2},{3,4}}
 => [2,1,4,3] => 0
{{1},{2,3,4}}
 => {{1,3,4},{2}}
 => [3,2,4,1] => 0
{{1},{2,3},{4}}
 => {{1},{2},{3,4}}
 => [1,2,4,3] => 0
{{1,4},{2},{3}}
 => {{1,2},{3},{4}}
 => [2,1,3,4] => 0
{{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => [3,2,1,4] => 0
{{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => [4,2,3,1] => 0
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => [1,2,3,4] => 0
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => [2,3,4,5,1] => 0
{{1,2,3,4},{5}}
 => {{1},{2,3,4,5}}
 => [1,3,4,5,2] => 0
{{1,2,3,5},{4}}
 => {{1,2,3,4},{5}}
 => [2,3,4,1,5] => 0
{{1,2,3},{4,5}}
 => {{1,5},{2,3,4}}
 => [5,3,4,2,1] => 2
{{1,2,3},{4},{5}}
 => {{1},{2,3,4},{5}}
 => [1,3,4,2,5] => 0
{{1,2,4,5},{3}}
 => {{1,2,3,5},{4}}
 => [2,3,5,4,1] => 0
{{1,2,4},{3,5}}
 => {{1,4},{2,3,5}}
 => [4,3,5,1,2] => 1
{{1,2,4},{3},{5}}
 => {{1},{2,3,5},{4}}
 => [1,3,5,4,2] => 0
{{1,2,5},{3,4}}
 => {{1,2,3},{4,5}}
 => [2,3,1,5,4] => 0
{{1,2},{3,4,5}}
 => {{1,4,5},{2,3}}
 => [4,3,2,5,1] => 1
{{1,2},{3,4},{5}}
 => {{1},{2,3},{4,5}}
 => [1,3,2,5,4] => 0
{{1,2,5},{3},{4}}
 => {{1,2,3},{4},{5}}
 => [2,3,1,4,5] => 0
{{1,2},{3,5},{4}}
 => {{1,4},{2,3},{5}}
 => [4,3,2,1,5] => 1
{{1,2},{3},{4,5}}
 => {{1,5},{2,3},{4}}
 => [5,3,2,4,1] => 1
{{1,2},{3},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => 0
{{1,3,4,5},{2}}
 => {{1,2,4,5},{3}}
 => [2,4,3,5,1] => 0
{{1,3,4},{2,5}}
 => {{1,3},{2,4,5}}
 => [3,4,1,5,2] => 0
{{1,3,4},{2},{5}}
 => {{1},{2,4,5},{3}}
 => [1,4,3,5,2] => 0
{{1,3,5},{2,4}}
 => {{1,2,4},{3,5}}
 => [2,4,5,1,3] => 0
{{1,3},{2,4,5}}
 => {{1,3,5},{2,4}}
 => [3,4,5,2,1] => 0
{{1,3},{2,4},{5}}
 => {{1},{2,4},{3,5}}
 => [1,4,5,2,3] => 0
{{1,3,5},{2},{4}}
 => {{1,2,4},{3},{5}}
 => [2,4,3,1,5] => 0
{{1,3},{2,5},{4}}
 => {{1,3},{2,4},{5}}
 => [3,4,1,2,5] => 0
{{1,3},{2},{4,5}}
 => {{1,5},{2,4},{3}}
 => [5,4,3,2,1] => 1
{{1,3},{2},{4},{5}}
 => {{1},{2,4},{3},{5}}
 => [1,4,3,2,5] => 0
{{1,4,5},{2,3}}
 => {{1,2,5},{3,4}}
 => [2,5,4,3,1] => 1
{{1,4},{2,3,5}}
 => {{1,3,4},{2,5}}
 => [3,5,4,1,2] => 1
{{1,2,3,4,5,6,7}}
 => {{1,2,3,4,5,6,7}}
 => [2,3,4,5,6,7,1] => ? = 0
{{1,2,3,4,5,7},{6}}
 => {{1,2,3,4,5,6},{7}}
 => [2,3,4,5,6,1,7] => ? = 0
{{1,2,3,4,5},{6,7}}
 => {{1,7},{2,3,4,5,6}}
 => [7,3,4,5,6,2,1] => ? = 4
{{1,2,3,4,6,7},{5}}
 => {{1,2,3,4,5,7},{6}}
 => [2,3,4,5,7,6,1] => ? = 0
{{1,2,3,4,6},{5,7}}
 => {{1,6},{2,3,4,5,7}}
 => [6,3,4,5,7,1,2] => ? = 3
{{1,2,3,4,7},{5,6}}
 => {{1,2,3,4,5},{6,7}}
 => [2,3,4,5,1,7,6] => ? = 0
{{1,2,3,4},{5,6,7}}
 => {{1,6,7},{2,3,4,5}}
 => [6,3,4,5,2,7,1] => ? = 3
{{1,2,3,4,7},{5},{6}}
 => {{1,2,3,4,5},{6},{7}}
 => [2,3,4,5,1,6,7] => ? = 0
{{1,2,3,4},{5,7},{6}}
 => {{1,6},{2,3,4,5},{7}}
 => [6,3,4,5,2,1,7] => ? = 3
{{1,2,3,4},{5},{6,7}}
 => {{1,7},{2,3,4,5},{6}}
 => [7,3,4,5,2,6,1] => ? = 3
{{1,2,3,5,6,7},{4}}
 => {{1,2,3,4,6,7},{5}}
 => [2,3,4,6,5,7,1] => ? = 0
{{1,2,3,5,6},{4,7}}
 => {{1,5},{2,3,4,6,7}}
 => [5,3,4,6,1,7,2] => ? = 2
{{1,2,3,5,7},{4,6}}
 => {{1,2,3,4,6},{5,7}}
 => [2,3,4,6,7,1,5] => ? = 0
{{1,2,3,5},{4,6,7}}
 => {{1,5,7},{2,3,4,6}}
 => [5,3,4,6,7,2,1] => ? = 2
{{1,2,3,5,7},{4},{6}}
 => {{1,2,3,4,6},{5},{7}}
 => [2,3,4,6,5,1,7] => ? = 0
{{1,2,3,5},{4,7},{6}}
 => {{1,5},{2,3,4,6},{7}}
 => [5,3,4,6,1,2,7] => ? = 2
{{1,2,3,5},{4},{6,7}}
 => {{1,7},{2,3,4,6},{5}}
 => [7,3,4,6,5,2,1] => ? = 3
{{1,2,3,6,7},{4,5}}
 => {{1,2,3,4,7},{5,6}}
 => [2,3,4,7,6,5,1] => ? = 1
{{1,2,3,6},{4,5,7}}
 => {{1,5,6},{2,3,4,7}}
 => [5,3,4,7,6,1,2] => ? = 3
{{1,2,3,7},{4,5,6}}
 => {{1,2,3,4},{5,6,7}}
 => [2,3,4,1,6,7,5] => ? = 0
{{1,2,3},{4,5,6,7}}
 => {{1,5,6,7},{2,3,4}}
 => [5,3,4,2,6,7,1] => ? = 2
{{1,2,3,7},{4,5},{6}}
 => {{1,2,3,4},{5,6},{7}}
 => [2,3,4,1,6,5,7] => ? = 0
{{1,2,3},{4,5,7},{6}}
 => {{1,5,6},{2,3,4},{7}}
 => [5,3,4,2,6,1,7] => ? = 2
{{1,2,3},{4,5},{6,7}}
 => {{1,7},{2,3,4},{5,6}}
 => [7,3,4,2,6,5,1] => ? = 3
{{1,2,3,6,7},{4},{5}}
 => {{1,2,3,4,7},{5},{6}}
 => [2,3,4,7,5,6,1] => ? = 0
{{1,2,3,6},{4,7},{5}}
 => {{1,5},{2,3,4,7},{6}}
 => [5,3,4,7,1,6,2] => ? = 2
{{1,2,3,6},{4},{5,7}}
 => {{1,6},{2,3,4,7},{5}}
 => [6,3,4,7,5,1,2] => ? = 2
{{1,2,3,7},{4,6},{5}}
 => {{1,2,3,4},{5,7},{6}}
 => [2,3,4,1,7,6,5] => ? = 0
{{1,2,3},{4,6,7},{5}}
 => {{1,5,7},{2,3,4},{6}}
 => [5,3,4,2,7,6,1] => ? = 2
{{1,2,3},{4,6},{5,7}}
 => {{1,6},{2,3,4},{5,7}}
 => [6,3,4,2,7,1,5] => ? = 2
{{1,2,3,7},{4},{5,6}}
 => {{1,2,3,4},{5},{6,7}}
 => [2,3,4,1,5,7,6] => ? = 0
{{1,2,3},{4,7},{5,6}}
 => {{1,5},{2,3,4},{6,7}}
 => [5,3,4,2,1,7,6] => ? = 2
{{1,2,3},{4},{5,6,7}}
 => {{1,6,7},{2,3,4},{5}}
 => [6,3,4,2,5,7,1] => ? = 2
{{1,2,3,7},{4},{5},{6}}
 => {{1,2,3,4},{5},{6},{7}}
 => [2,3,4,1,5,6,7] => ? = 0
{{1,2,3},{4,7},{5},{6}}
 => {{1,5},{2,3,4},{6},{7}}
 => [5,3,4,2,1,6,7] => ? = 2
{{1,2,3},{4},{5,7},{6}}
 => {{1,6},{2,3,4},{5},{7}}
 => [6,3,4,2,5,1,7] => ? = 2
{{1,2,3},{4},{5},{6,7}}
 => {{1,7},{2,3,4},{5},{6}}
 => [7,3,4,2,5,6,1] => ? = 2
{{1,2,4,5,6,7},{3}}
 => {{1,2,3,5,6,7},{4}}
 => [2,3,5,4,6,7,1] => ? = 0
{{1,2,4,5,6},{3,7}}
 => {{1,4},{2,3,5,6,7}}
 => [4,3,5,1,6,7,2] => ? = 1
{{1,2,4,5,7},{3,6}}
 => {{1,2,3,5,6},{4,7}}
 => [2,3,5,7,6,1,4] => ? = 1
{{1,2,4,5},{3,6,7}}
 => {{1,4,7},{2,3,5,6}}
 => [4,3,5,7,6,2,1] => ? = 2
{{1,2,4,5,7},{3},{6}}
 => {{1,2,3,5,6},{4},{7}}
 => [2,3,5,4,6,1,7] => ? = 0
{{1,2,4,5},{3,7},{6}}
 => {{1,4},{2,3,5,6},{7}}
 => [4,3,5,1,6,2,7] => ? = 1
{{1,2,4,5},{3},{6,7}}
 => {{1,7},{2,3,5,6},{4}}
 => [7,3,5,4,6,2,1] => ? = 3
{{1,2,4,6,7},{3,5}}
 => {{1,2,3,5,7},{4,6}}
 => [2,3,5,6,7,4,1] => ? = 0
{{1,2,4,6},{3,5,7}}
 => {{1,4,6},{2,3,5,7}}
 => [4,3,5,6,7,1,2] => ? = 1
{{1,2,4,7},{3,5,6}}
 => {{1,2,3,5},{4,6,7}}
 => [2,3,5,6,1,7,4] => ? = 0
{{1,2,4},{3,5,6,7}}
 => {{1,4,6,7},{2,3,5}}
 => [4,3,5,6,2,7,1] => ? = 1
{{1,2,4,7},{3,5},{6}}
 => {{1,2,3,5},{4,6},{7}}
 => [2,3,5,6,1,4,7] => ? = 0
{{1,2,4},{3,5,7},{6}}
 => {{1,4,6},{2,3,5},{7}}
 => [4,3,5,6,2,1,7] => ? = 1
Description
The number of nested exceedences of a permutation. For a permutation $\pi$, this is the number of pairs $i,j$ such that $i < j < \pi(j) < \pi(i)$. For exceedences, see [[St000155]].
Matching statistic: St001549
Mapping
Mp00091: Set partitions —rotate increasing⟶ Set partitions
Mp00115: Set partitions —Kasraoui-Zeng⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St001549: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 27%●distinct values known / distinct values provided: 25%
Values
{{1}}
 => {{1}}
 => {{1}}
 => [1] => 0
{{1,2}}
 => {{1,2}}
 => {{1,2}}
 => [2,1] => 0
{{1},{2}}
 => {{1},{2}}
 => {{1},{2}}
 => [1,2] => 0
{{1,2,3}}
 => {{1,2,3}}
 => {{1,2,3}}
 => [2,3,1] => 0
{{1,2},{3}}
 => {{1},{2,3}}
 => {{1},{2,3}}
 => [1,3,2] => 0
{{1,3},{2}}
 => {{1,2},{3}}
 => {{1,2},{3}}
 => [2,1,3] => 0
{{1},{2,3}}
 => {{1,3},{2}}
 => {{1,3},{2}}
 => [3,2,1] => 0
{{1},{2},{3}}
 => {{1},{2},{3}}
 => {{1},{2},{3}}
 => [1,2,3] => 0
{{1,2,3,4}}
 => {{1,2,3,4}}
 => {{1,2,3,4}}
 => [2,3,4,1] => 0
{{1,2,3},{4}}
 => {{1},{2,3,4}}
 => {{1},{2,3,4}}
 => [1,3,4,2] => 0
{{1,2,4},{3}}
 => {{1,2,3},{4}}
 => {{1,2,3},{4}}
 => [2,3,1,4] => 0
{{1,2},{3,4}}
 => {{1,4},{2,3}}
 => {{1,3},{2,4}}
 => [3,4,1,2] => 1
{{1,2},{3},{4}}
 => {{1},{2,3},{4}}
 => {{1},{2,3},{4}}
 => [1,3,2,4] => 0
{{1,3,4},{2}}
 => {{1,2,4},{3}}
 => {{1,2,4},{3}}
 => [2,4,3,1] => 0
{{1,3},{2,4}}
 => {{1,3},{2,4}}
 => {{1,4},{2,3}}
 => [4,3,2,1] => 0
{{1,3},{2},{4}}
 => {{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => [1,4,3,2] => 0
{{1,4},{2,3}}
 => {{1,2},{3,4}}
 => {{1,2},{3,4}}
 => [2,1,4,3] => 0
{{1},{2,3,4}}
 => {{1,3,4},{2}}
 => {{1,3,4},{2}}
 => [3,2,4,1] => 0
{{1},{2,3},{4}}
 => {{1},{2},{3,4}}
 => {{1},{2},{3,4}}
 => [1,2,4,3] => 0
{{1,4},{2},{3}}
 => {{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => [2,1,3,4] => 0
{{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => {{1,3},{2},{4}}
 => [3,2,1,4] => 0
{{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => {{1,4},{2},{3}}
 => [4,2,3,1] => 0
{{1},{2},{3},{4}}
 => {{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}}
 => [2,3,4,5,1] => 0
{{1,2,3,4},{5}}
 => {{1},{2,3,4,5}}
 => {{1},{2,3,4,5}}
 => [1,3,4,5,2] => 0
{{1,2,3,5},{4}}
 => {{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => [2,3,4,1,5] => 0
{{1,2,3},{4,5}}
 => {{1,5},{2,3,4}}
 => {{1,3,5},{2,4}}
 => [3,4,5,2,1] => 2
{{1,2,3},{4},{5}}
 => {{1},{2,3,4},{5}}
 => {{1},{2,3,4},{5}}
 => [1,3,4,2,5] => 0
{{1,2,4,5},{3}}
 => {{1,2,3,5},{4}}
 => {{1,2,3,5},{4}}
 => [2,3,5,4,1] => 0
{{1,2,4},{3,5}}
 => {{1,4},{2,3,5}}
 => {{1,3,4},{2,5}}
 => [3,5,4,1,2] => 1
{{1,2,4},{3},{5}}
 => {{1},{2,3,5},{4}}
 => {{1},{2,3,5},{4}}
 => [1,3,5,4,2] => 0
{{1,2,5},{3,4}}
 => {{1,2,3},{4,5}}
 => {{1,2,3},{4,5}}
 => [2,3,1,5,4] => 0
{{1,2},{3,4,5}}
 => {{1,4,5},{2,3}}
 => {{1,3},{2,4,5}}
 => [3,4,1,5,2] => 1
{{1,2},{3,4},{5}}
 => {{1},{2,3},{4,5}}
 => {{1},{2,3},{4,5}}
 => [1,3,2,5,4] => 0
{{1,2,5},{3},{4}}
 => {{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => [2,3,1,4,5] => 0
{{1,2},{3,5},{4}}
 => {{1,4},{2,3},{5}}
 => {{1,3},{2,4},{5}}
 => [3,4,1,2,5] => 1
{{1,2},{3},{4,5}}
 => {{1,5},{2,3},{4}}
 => {{1,3},{2,5},{4}}
 => [3,5,1,4,2] => 1
{{1,2},{3},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => 0
{{1,3,4,5},{2}}
 => {{1,2,4,5},{3}}
 => {{1,2,4,5},{3}}
 => [2,4,3,5,1] => 0
{{1,3,4},{2,5}}
 => {{1,3},{2,4,5}}
 => {{1,4,5},{2,3}}
 => [4,3,2,5,1] => 0
{{1,3,4},{2},{5}}
 => {{1},{2,4,5},{3}}
 => {{1},{2,4,5},{3}}
 => [1,4,3,5,2] => 0
{{1,3,5},{2,4}}
 => {{1,2,4},{3,5}}
 => {{1,2,5},{3,4}}
 => [2,5,4,3,1] => 0
{{1,3},{2,4,5}}
 => {{1,3,5},{2,4}}
 => {{1,5},{2,3,4}}
 => [5,3,4,2,1] => 0
{{1,3},{2,4},{5}}
 => {{1},{2,4},{3,5}}
 => {{1},{2,5},{3,4}}
 => [1,5,4,3,2] => 0
{{1,3,5},{2},{4}}
 => {{1,2,4},{3},{5}}
 => {{1,2,4},{3},{5}}
 => [2,4,3,1,5] => 0
{{1,3},{2,5},{4}}
 => {{1,3},{2,4},{5}}
 => {{1,4},{2,3},{5}}
 => [4,3,2,1,5] => 0
{{1,3},{2},{4,5}}
 => {{1,5},{2,4},{3}}
 => {{1,4},{2,5},{3}}
 => [4,5,3,1,2] => 1
{{1,3},{2},{4},{5}}
 => {{1},{2,4},{3},{5}}
 => {{1},{2,4},{3},{5}}
 => [1,4,3,2,5] => 0
{{1,4,5},{2,3}}
 => {{1,2,5},{3,4}}
 => {{1,2,4},{3,5}}
 => [2,4,5,1,3] => 1
{{1,4},{2,3,5}}
 => {{1,3,4},{2,5}}
 => {{1,4},{2,3,5}}
 => [4,3,5,1,2] => 1
{{1,2,3,4,5,6,7}}
 => {{1,2,3,4,5,6,7}}
 => {{1,2,3,4,5,6,7}}
 => [2,3,4,5,6,7,1] => ? = 0
{{1,2,3,4,5,7},{6}}
 => {{1,2,3,4,5,6},{7}}
 => {{1,2,3,4,5,6},{7}}
 => [2,3,4,5,6,1,7] => ? = 0
{{1,2,3,4,5},{6,7}}
 => {{1,7},{2,3,4,5,6}}
 => {{1,3,5,7},{2,4,6}}
 => [3,4,5,6,7,2,1] => ? = 4
{{1,2,3,4,6,7},{5}}
 => {{1,2,3,4,5,7},{6}}
 => {{1,2,3,4,5,7},{6}}
 => [2,3,4,5,7,6,1] => ? = 0
{{1,2,3,4,6},{5,7}}
 => {{1,6},{2,3,4,5,7}}
 => {{1,3,5,6},{2,4,7}}
 => [3,4,5,7,6,1,2] => ? = 3
{{1,2,3,4,7},{5,6}}
 => {{1,2,3,4,5},{6,7}}
 => {{1,2,3,4,5},{6,7}}
 => [2,3,4,5,1,7,6] => ? = 0
{{1,2,3,4},{5,6,7}}
 => {{1,6,7},{2,3,4,5}}
 => {{1,3,5},{2,4,6,7}}
 => [3,4,5,6,1,7,2] => ? = 3
{{1,2,3,4,7},{5},{6}}
 => {{1,2,3,4,5},{6},{7}}
 => {{1,2,3,4,5},{6},{7}}
 => [2,3,4,5,1,6,7] => ? = 0
{{1,2,3,4},{5,7},{6}}
 => {{1,6},{2,3,4,5},{7}}
 => {{1,3,5},{2,4,6},{7}}
 => [3,4,5,6,1,2,7] => ? = 3
{{1,2,3,4},{5},{6,7}}
 => {{1,7},{2,3,4,5},{6}}
 => {{1,3,5},{2,4,7},{6}}
 => [3,4,5,7,1,6,2] => ? = 3
{{1,2,3,5,6,7},{4}}
 => {{1,2,3,4,6,7},{5}}
 => {{1,2,3,4,6,7},{5}}
 => [2,3,4,6,5,7,1] => ? = 0
{{1,2,3,5,6},{4,7}}
 => {{1,5},{2,3,4,6,7}}
 => {{1,3,6,7},{2,4,5}}
 => [3,4,6,5,2,7,1] => ? = 2
{{1,2,3,5,7},{4,6}}
 => {{1,2,3,4,6},{5,7}}
 => {{1,2,3,4,7},{5,6}}
 => [2,3,4,7,6,5,1] => ? = 0
{{1,2,3,5},{4,6,7}}
 => {{1,5,7},{2,3,4,6}}
 => {{1,3,7},{2,4,5,6}}
 => [3,4,7,5,6,2,1] => ? = 2
{{1,2,3,5,7},{4},{6}}
 => {{1,2,3,4,6},{5},{7}}
 => {{1,2,3,4,6},{5},{7}}
 => [2,3,4,6,5,1,7] => ? = 0
{{1,2,3,5},{4,7},{6}}
 => {{1,5},{2,3,4,6},{7}}
 => {{1,3,6},{2,4,5},{7}}
 => [3,4,6,5,2,1,7] => ? = 2
{{1,2,3,5},{4},{6,7}}
 => {{1,7},{2,3,4,6},{5}}
 => {{1,3,6},{2,4,7},{5}}
 => [3,4,6,7,5,1,2] => ? = 3
{{1,2,3,6,7},{4,5}}
 => {{1,2,3,4,7},{5,6}}
 => {{1,2,3,4,6},{5,7}}
 => [2,3,4,6,7,1,5] => ? = 1
{{1,2,3,6},{4,5,7}}
 => {{1,5,6},{2,3,4,7}}
 => {{1,3,6},{2,4,5,7}}
 => [3,4,6,5,7,1,2] => ? = 3
{{1,2,3,7},{4,5,6}}
 => {{1,2,3,4},{5,6,7}}
 => {{1,2,3,4},{5,6,7}}
 => [2,3,4,1,6,7,5] => ? = 0
{{1,2,3},{4,5,6,7}}
 => {{1,5,6,7},{2,3,4}}
 => {{1,3,5,6,7},{2,4}}
 => [3,4,5,2,6,7,1] => ? = 2
{{1,2,3,7},{4,5},{6}}
 => {{1,2,3,4},{5,6},{7}}
 => {{1,2,3,4},{5,6},{7}}
 => [2,3,4,1,6,5,7] => ? = 0
{{1,2,3},{4,5,7},{6}}
 => {{1,5,6},{2,3,4},{7}}
 => {{1,3,5,6},{2,4},{7}}
 => [3,4,5,2,6,1,7] => ? = 2
{{1,2,3},{4,5},{6,7}}
 => {{1,7},{2,3,4},{5,6}}
 => {{1,3,6},{2,4},{5,7}}
 => [3,4,6,2,7,1,5] => ? = 3
{{1,2,3,6,7},{4},{5}}
 => {{1,2,3,4,7},{5},{6}}
 => {{1,2,3,4,7},{5},{6}}
 => [2,3,4,7,5,6,1] => ? = 0
{{1,2,3,6},{4,7},{5}}
 => {{1,5},{2,3,4,7},{6}}
 => {{1,3,7},{2,4,5},{6}}
 => [3,4,7,5,2,6,1] => ? = 2
{{1,2,3,6},{4},{5,7}}
 => {{1,6},{2,3,4,7},{5}}
 => {{1,3,7},{2,4,6},{5}}
 => [3,4,7,6,5,2,1] => ? = 2
{{1,2,3,7},{4,6},{5}}
 => {{1,2,3,4},{5,7},{6}}
 => {{1,2,3,4},{5,7},{6}}
 => [2,3,4,1,7,6,5] => ? = 0
{{1,2,3},{4,6,7},{5}}
 => {{1,5,7},{2,3,4},{6}}
 => {{1,3,5,7},{2,4},{6}}
 => [3,4,5,2,7,6,1] => ? = 2
{{1,2,3},{4,6},{5,7}}
 => {{1,6},{2,3,4},{5,7}}
 => {{1,3,7},{2,4},{5,6}}
 => [3,4,7,2,6,5,1] => ? = 2
{{1,2,3,7},{4},{5,6}}
 => {{1,2,3,4},{5},{6,7}}
 => {{1,2,3,4},{5},{6,7}}
 => [2,3,4,1,5,7,6] => ? = 0
{{1,2,3},{4,7},{5,6}}
 => {{1,5},{2,3,4},{6,7}}
 => {{1,3,5},{2,4},{6,7}}
 => [3,4,5,2,1,7,6] => ? = 2
{{1,2,3},{4},{5,6,7}}
 => {{1,6,7},{2,3,4},{5}}
 => {{1,3,6,7},{2,4},{5}}
 => [3,4,6,2,5,7,1] => ? = 2
{{1,2,3,7},{4},{5},{6}}
 => {{1,2,3,4},{5},{6},{7}}
 => {{1,2,3,4},{5},{6},{7}}
 => [2,3,4,1,5,6,7] => ? = 0
{{1,2,3},{4,7},{5},{6}}
 => {{1,5},{2,3,4},{6},{7}}
 => {{1,3,5},{2,4},{6},{7}}
 => [3,4,5,2,1,6,7] => ? = 2
{{1,2,3},{4},{5,7},{6}}
 => {{1,6},{2,3,4},{5},{7}}
 => {{1,3,6},{2,4},{5},{7}}
 => [3,4,6,2,5,1,7] => ? = 2
{{1,2,3},{4},{5},{6,7}}
 => {{1,7},{2,3,4},{5},{6}}
 => {{1,3,7},{2,4},{5},{6}}
 => [3,4,7,2,5,6,1] => ? = 2
{{1,2,4,5,6,7},{3}}
 => {{1,2,3,5,6,7},{4}}
 => {{1,2,3,5,6,7},{4}}
 => [2,3,5,4,6,7,1] => ? = 0
{{1,2,4,5,6},{3,7}}
 => {{1,4},{2,3,5,6,7}}
 => {{1,3,4},{2,5,6,7}}
 => [3,5,4,1,6,7,2] => ? = 1
{{1,2,4,5,7},{3,6}}
 => {{1,2,3,5,6},{4,7}}
 => {{1,2,3,6},{4,5,7}}
 => [2,3,6,5,7,1,4] => ? = 1
{{1,2,4,5},{3,6,7}}
 => {{1,4,7},{2,3,5,6}}
 => {{1,3,4,5,7},{2,6}}
 => [3,6,4,5,7,2,1] => ? = 2
{{1,2,4,5,7},{3},{6}}
 => {{1,2,3,5,6},{4},{7}}
 => {{1,2,3,5,6},{4},{7}}
 => [2,3,5,4,6,1,7] => ? = 0
{{1,2,4,5},{3,7},{6}}
 => {{1,4},{2,3,5,6},{7}}
 => {{1,3,4},{2,5,6},{7}}
 => [3,5,4,1,6,2,7] => ? = 1
{{1,2,4,5},{3},{6,7}}
 => {{1,7},{2,3,5,6},{4}}
 => {{1,3,6},{2,5,7},{4}}
 => [3,5,6,4,7,1,2] => ? = 3
{{1,2,4,6,7},{3,5}}
 => {{1,2,3,5,7},{4,6}}
 => {{1,2,3,7},{4,5,6}}
 => [2,3,7,5,6,4,1] => ? = 0
{{1,2,4,6},{3,5,7}}
 => {{1,4,6},{2,3,5,7}}
 => {{1,3,4,5,6},{2,7}}
 => [3,7,4,5,6,1,2] => ? = 1
{{1,2,4,7},{3,5,6}}
 => {{1,2,3,5},{4,6,7}}
 => {{1,2,3,6,7},{4,5}}
 => [2,3,6,5,4,7,1] => ? = 0
{{1,2,4},{3,5,6,7}}
 => {{1,4,6,7},{2,3,5}}
 => {{1,3,4,5},{2,6,7}}
 => [3,6,4,5,1,7,2] => ? = 1
{{1,2,4,7},{3,5},{6}}
 => {{1,2,3,5},{4,6},{7}}
 => {{1,2,3,6},{4,5},{7}}
 => [2,3,6,5,4,1,7] => ? = 0
{{1,2,4},{3,5,7},{6}}
 => {{1,4,6},{2,3,5},{7}}
 => {{1,3,4,5},{2,6},{7}}
 => [3,6,4,5,1,2,7] => ? = 1
Description
The number of restricted non-inversions between exceedances. This is for a permutation $\sigma$ of length $n$ given by $$\operatorname{nie}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(i) < \sigma(j) \}.$$
Matching statistic: St001867
(load all 2 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001867: Signed permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 12%
Values
{{1}}
 => [1] => [1] => [1] => 0
{{1,2}}
 => [2,1] => [2,1] => [2,1] => 0
{{1},{2}}
 => [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => [2,3,1] => [2,3,1] => 0
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [2,1,3] => 0
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => [3,2,1] => 0
{{1},{2,3}}
 => [1,3,2] => [3,1,2] => [3,1,2] => 0
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [2,3,4,1] => [2,3,4,1] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [4,2,3,1] => [4,2,3,1] => 0
{{1,2},{3,4}}
 => [2,1,4,3] => [2,4,1,3] => [2,4,1,3] => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [3,2,4,1] => [3,2,4,1] => 0
{{1,3},{2,4}}
 => [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [3,4,1,2] => [3,4,1,2] => 0
{{1},{2,3},{4}}
 => [1,3,2,4] => [3,1,2,4] => [3,1,2,4] => 0
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,4,3,1] => [2,4,3,1] => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => [4,3,1,2] => [4,3,1,2] => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 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] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 0
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 2
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 0
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [4,2,3,5,1] => [4,2,3,5,1] => ? = 0
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,5,2,1,3] => [4,5,2,1,3] => ? = 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [4,2,3,1,5] => [4,2,3,1,5] => ? = 0
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 0
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 0
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [2,5,3,4,1] => [2,5,3,4,1] => ? = 0
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [5,2,4,1,3] => [5,2,4,1,3] => ? = 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 0
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 0
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [5,3,1,4,2] => [5,3,1,4,2] => ? = 0
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [3,2,4,1,5] => [3,2,4,1,5] => ? = 0
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 0
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,1,4,5,2] => [3,1,4,5,2] => ? = 0
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 0
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [3,5,2,4,1] => [3,5,2,4,1] => ? = 0
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [3,5,1,4,2] => [3,5,1,4,2] => ? = 0
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [4,3,2,5,1] => [4,3,2,5,1] => ? = 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,3,1,5,2] => [4,3,1,5,2] => ? = 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
{{1,5},{2,3,4}}
 => [5,3,4,2,1] => [3,5,4,2,1] => [3,5,4,2,1] => ? = 0
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 0
{{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 0
{{1,5},{2,3},{4}}
 => [5,3,2,4,1] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 0
{{1},{2,3,5},{4}}
 => [1,3,5,4,2] => [5,3,4,1,2] => [5,3,4,1,2] => ? = 0
{{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 1
{{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 0
{{1,4,5},{2},{3}}
 => [4,2,3,5,1] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 0
{{1,4},{2,5},{3}}
 => [4,5,3,1,2] => [4,1,5,3,2] => [4,1,5,3,2] => ? = 0
{{1,4},{2},{3,5}}
 => [4,2,5,1,3] => [2,4,1,5,3] => [2,4,1,5,3] => ? = 0
{{1,4},{2},{3},{5}}
 => [4,2,3,1,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 0
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
{{1},{2,4,5},{3}}
 => [1,4,3,5,2] => [4,3,5,1,2] => [4,3,5,1,2] => ? = 0
{{1},{2,4},{3,5}}
 => [1,4,5,2,3] => [4,1,5,2,3] => [4,1,5,2,3] => ? = 0
{{1},{2,4},{3},{5}}
 => [1,4,3,2,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 0
{{1,5},{2},{3,4}}
 => [5,2,4,3,1] => [5,2,4,3,1] => [5,2,4,3,1] => ? = 0
{{1},{2,5},{3,4}}
 => [1,5,4,3,2] => [5,4,3,1,2] => [5,4,3,1,2] => ? = 0
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 0
{{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 0
{{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 0
{{1},{2,5},{3},{4}}
 => [1,5,3,4,2] => [3,5,4,1,2] => [3,5,4,1,2] => ? = 0
{{1},{2},{3,5},{4}}
 => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 0
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
Description
The number of alignments of type EN of a signed permutation. An alignment of type EN of a signed permutation π∈Hn is a pair −n≤i≤j≤n, i,j≠0, such that one of the following conditions hold: * $-i < 0 < -\pi(i) < \pi(j) < j$ * $i \leq\pi(i) < \pi(j) < j$.