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Your data matches 3 different statistics following compositions of up to 3 maps.
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Matching statistic: St000683
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000683: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000683: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2] => [1,1,0,0]
=> [1,0,1,0]
=> 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 0
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 0
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 0
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 0
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 0
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 6
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
{{1,4},{2,3},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
Description
The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps.
Matching statistic: St000575
(load all 41 compositions to match this statistic)
(load all 41 compositions to match this statistic)
Mp00112: Set partitions —complement⟶ Set partitions
St000575: Set partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 100%
St000575: Set partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> {{1,2}}
=> 0
{{1},{2}}
=> {{1},{2}}
=> 1
{{1,2,3}}
=> {{1,2,3}}
=> 0
{{1,2},{3}}
=> {{1},{2,3}}
=> 0
{{1,3},{2}}
=> {{1,3},{2}}
=> 0
{{1},{2,3}}
=> {{1,2},{3}}
=> 1
{{1},{2},{3}}
=> {{1},{2},{3}}
=> 3
{{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> {{1},{2,3,4}}
=> 0
{{1,2,4},{3}}
=> {{1,3,4},{2}}
=> 0
{{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 0
{{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> 1
{{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}}
=> 1
{{1,4},{2,3}}
=> {{1,4},{2,3}}
=> 0
{{1},{2,3,4}}
=> {{1,2,3},{4}}
=> 1
{{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 2
{{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 1
{{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> 2
{{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> 3
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 6
{{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}}
=> {{1,3,4,5},{2}}
=> 0
{{1,2,3},{4,5}}
=> {{1,2},{3,4,5}}
=> 0
{{1,2,3},{4},{5}}
=> {{1},{2},{3,4,5}}
=> 1
{{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> 0
{{1,2,4},{3,5}}
=> {{1,3},{2,4,5}}
=> 0
{{1,2,4},{3},{5}}
=> {{1},{2,4,5},{3}}
=> 1
{{1,2,5},{3,4}}
=> {{1,4,5},{2,3}}
=> 0
{{1,2},{3,4,5}}
=> {{1,2,3},{4,5}}
=> 0
{{1,2},{3,4},{5}}
=> {{1},{2,3},{4,5}}
=> 0
{{1,2,5},{3},{4}}
=> {{1,4,5},{2},{3}}
=> 1
{{1,2},{3,5},{4}}
=> {{1,3},{2},{4,5}}
=> 0
{{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> 1
{{1,2},{3},{4},{5}}
=> {{1},{2},{3},{4,5}}
=> 3
{{1,3,4,5},{2}}
=> {{1,2,3,5},{4}}
=> 0
{{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> 0
{{1,3,4},{2},{5}}
=> {{1},{2,3,5},{4}}
=> 1
{{1,3,5},{2,4}}
=> {{1,3,5},{2,4}}
=> 0
{{1,3},{2,4,5}}
=> {{1,2,4},{3,5}}
=> 0
{{1,3},{2,4},{5}}
=> {{1},{2,4},{3,5}}
=> 0
{{1,3,5},{2},{4}}
=> {{1,3,5},{2},{4}}
=> 1
{{1,3},{2,5},{4}}
=> {{1,4},{2},{3,5}}
=> 0
{{1,3},{2},{4,5}}
=> {{1,2},{3,5},{4}}
=> 1
{{1,3},{2},{4},{5}}
=> {{1},{2},{3,5},{4}}
=> 3
{{1,4,5},{2,3}}
=> {{1,2,5},{3,4}}
=> 0
{{1,4},{2,3,5}}
=> {{1,3,4},{2,5}}
=> 0
{{1,4},{2,3},{5}}
=> {{1},{2,5},{3,4}}
=> 0
{{1},{2},{3},{4},{5},{6},{7,8}}
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 21
{{1},{2},{3},{4},{5},{6,7,8}}
=> {{1,2,3},{4},{5},{6},{7},{8}}
=> ? = 15
{{1},{2},{3},{4},{5,7,8},{6}}
=> {{1,2,4},{3},{5},{6},{7},{8}}
=> ? = 14
{{1},{2},{3},{4},{5,6,7},{8}}
=> {{1},{2,3,4},{5},{6},{7},{8}}
=> ? = 14
{{1},{2},{3},{4},{5,6,8},{7}}
=> {{1,3,4},{2},{5},{6},{7},{8}}
=> ? = 14
{{1},{2},{3},{4},{5,6,7,8}}
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 10
{{1},{2},{3},{4,6,7,8},{5}}
=> {{1,2,3,5},{4},{6},{7},{8}}
=> ? = 9
{{1},{2},{3},{4,8},{5,6,7}}
=> {{1,5},{2,3,4},{6},{7},{8}}
=> ? = 9
{{1},{2},{3},{4,5,6,7,8}}
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 6
{{1},{2},{3,4},{5,6,7,8}}
=> {{1,2,3,4},{5,6},{7},{8}}
=> ? = 5
{{1},{2},{3,5},{4},{6,7,8}}
=> {{1,2,3},{4,6},{5},{7},{8}}
=> ? = 8
{{1},{2},{3,7,8},{4},{5},{6}}
=> {{1,2,6},{3},{4},{5},{7},{8}}
=> ? = 12
{{1},{2},{3,5,7,8},{4},{6}}
=> {{1,2,4,6},{3},{5},{7},{8}}
=> ? = 8
{{1},{2},{3,8},{4},{5,6,7}}
=> {{1,6},{2,3,4},{5},{7},{8}}
=> ? = 8
{{1},{2},{3,5,6,7,8},{4}}
=> {{1,2,3,4,6},{5},{7},{8}}
=> ? = 5
{{1},{2},{3,4,5},{6},{7},{8}}
=> {{1},{2},{3},{4,5,6},{7},{8}}
=> ? = 12
{{1},{2},{3,6,7,8},{4,5}}
=> {{1,2,3,6},{4,5},{7},{8}}
=> ? = 5
{{1},{2},{3,8},{4,6,7},{5}}
=> {{1,6},{2,3,5},{4},{7},{8}}
=> ? = 7
{{1},{2},{3,4,5,6},{7,8}}
=> {{1,2},{3,4,5,6},{7},{8}}
=> ? = 5
{{1},{2},{3,8},{4,5,6},{7}}
=> {{1,6},{2},{3,4,5},{7},{8}}
=> ? = 7
{{1},{2},{3,4,7,8},{5,6}}
=> {{1,2,5,6},{3,4},{7},{8}}
=> ? = 5
{{1},{2},{3,4,5,8},{6},{7}}
=> {{1,4,5,6},{2},{3},{7},{8}}
=> ? = 8
{{1},{2},{3,4,5,6,7},{8}}
=> {{1},{2,3,4,5,6},{7},{8}}
=> ? = 5
{{1},{2},{3,8},{4,5,6,7}}
=> {{1,6},{2,3,4,5},{7},{8}}
=> ? = 5
{{1},{2},{3,4,5,8},{6,7}}
=> {{1,4,5,6},{2,3},{7},{8}}
=> ? = 5
{{1},{2},{3,4,5,6,8},{7}}
=> {{1,3,4,5,6},{2},{7},{8}}
=> ? = 5
{{1},{2},{3,4,5,6,7,8}}
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 3
{{1},{2,3},{4},{5,6,7,8}}
=> {{1,2,3,4},{5},{6,7},{8}}
=> ? = 4
{{1},{2,3},{4,5},{6,7,8}}
=> {{1,2,3},{4,5},{6,7},{8}}
=> ? = 3
{{1},{2,3},{4,6,7},{5},{8}}
=> {{1},{2,3,5},{4},{6,7},{8}}
=> ? = 5
{{1},{2,3},{4,6,8},{5},{7}}
=> {{1,3,5},{2},{4},{6,7},{8}}
=> ? = 5
{{1},{2,3},{4,6,7,8},{5}}
=> {{1,2,3,5},{4},{6,7},{8}}
=> ? = 3
{{1},{2,3},{4,5,8},{6},{7}}
=> {{1,4,5},{2},{3},{6,7},{8}}
=> ? = 5
{{1},{2,3},{4,8},{5,6,7}}
=> {{1,5},{2,3,4},{6,7},{8}}
=> ? = 3
{{1},{2,3},{4,5,6,8},{7}}
=> {{1,3,4,5},{2},{6,7},{8}}
=> ? = 3
{{1},{2,3},{4,5,6,7,8}}
=> {{1,2,3,4,5},{6,7},{8}}
=> ? = 2
{{1},{2,4},{3},{5},{6,7,8}}
=> {{1,2,3},{4},{5,7},{6},{8}}
=> ? = 7
{{1},{2,4},{3},{5,7,8},{6}}
=> {{1,2,4},{3},{5,7},{6},{8}}
=> ? = 6
{{1},{2,4},{3},{5,6,7,8}}
=> {{1,2,3,4},{5,7},{6},{8}}
=> ? = 4
{{1},{2,7,8},{3},{4},{5},{6}}
=> {{1,2,7},{3},{4},{5},{6},{8}}
=> ? = 11
{{1},{2,5,8},{3},{4},{6},{7}}
=> {{1,4,7},{2},{3},{5},{6},{8}}
=> ? = 11
{{1},{2,8},{3},{4},{5,6,7}}
=> {{1,7},{2,3,4},{5},{6},{8}}
=> ? = 7
{{1},{2,5,6,8},{3},{4},{7}}
=> {{1,3,4,7},{2},{5},{6},{8}}
=> ? = 7
{{1},{2,5,6,7,8},{3},{4}}
=> {{1,2,3,4,7},{5},{6},{8}}
=> ? = 4
{{1},{2,6,7,8},{3},{4,5}}
=> {{1,2,3,7},{4,5},{6},{8}}
=> ? = 4
{{1},{2,4,6,7},{3},{5},{8}}
=> {{1},{2,3,5,7},{4},{6},{8}}
=> ? = 7
{{1},{2,8},{3},{4,6,7},{5}}
=> {{1,7},{2,3,5},{4},{6},{8}}
=> ? = 6
{{1},{2,4,6,8},{3},{5},{7}}
=> {{1,3,5,7},{2},{4},{6},{8}}
=> ? = 7
{{1},{2,4,6,7,8},{3},{5}}
=> {{1,2,3,5,7},{4},{6},{8}}
=> ? = 4
{{1},{2,4,7,8},{3},{5,6}}
=> {{1,2,5,7},{3,4},{6},{8}}
=> ? = 4
Description
The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton.
This is the number of pairs $i\lt j$ in different blocks such that $i$ is the maximal element of a block and $j$ is a singleton block.
Matching statistic: St000573
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000573: Set partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000573: Set partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2] => [1,1,0,0]
=> {{1,2}}
=> 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> {{1},{2}}
=> 1
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 0
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 0
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 0
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 3
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 0
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 0
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 0
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 1
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 0
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 0
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 1
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 0
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 2
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 1
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 2
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 3
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 6
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 0
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 0
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 0
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 0
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 1
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 0
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 1
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 0
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 0
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 0
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 3
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 0
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 1
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 0
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 0
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 0
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 3
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 0
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 0
{{1,4},{2,3},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 0
{{1},{2},{3},{4},{5},{6},{7,8}}
=> [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 21
{{1},{2},{3},{4},{5},{6,7,8}}
=> [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3},{4},{5},{6,7,8}}
=> ? = 15
{{1},{2},{3},{4},{5,7,8},{6}}
=> [1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> {{1},{2},{3},{4},{5,6,7},{8}}
=> ? = 14
{{1},{2},{3},{4},{5,6,7},{8}}
=> [1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> {{1},{2},{3},{4},{5,6,7},{8}}
=> ? = 14
{{1},{2},{3},{4},{5,6,8},{7}}
=> [1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> {{1},{2},{3},{4},{5,6,7},{8}}
=> ? = 14
{{1},{2},{3},{4},{5,6,7,8}}
=> [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> {{1},{2},{3},{4},{5,6,7,8}}
=> ? = 10
{{1},{2},{3},{4,6,7,8},{5}}
=> [1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> {{1},{2},{3},{4,5,6,7},{8}}
=> ? = 9
{{1},{2},{3},{4,8},{5,6,7}}
=> [1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> {{1},{2},{3},{4,5},{6,7,8}}
=> ? = 9
{{1},{2},{3},{4,5,6,7,8}}
=> [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2},{3},{4,5,6,7,8}}
=> ? = 6
{{1},{2},{3,4},{5,6,7,8}}
=> [1,1,2,4] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> {{1},{2},{3,4},{5,6,7,8}}
=> ? = 5
{{1},{2},{3,5},{4},{6,7,8}}
=> [1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4},{5},{6,7,8}}
=> ? = 8
{{1},{2},{3,7,8},{4},{5},{6}}
=> [1,1,3,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> {{1},{2},{3,4,5},{6},{7},{8}}
=> ? = 12
{{1},{2},{3,5,7,8},{4},{6}}
=> [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> {{1},{2},{3,4,5,6},{7},{8}}
=> ? = 8
{{1},{2},{3,8},{4},{5,6,7}}
=> [1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4},{5},{6,7,8}}
=> ? = 8
{{1},{2},{3,5,6,7,8},{4}}
=> [1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1},{2},{3,4,5,6,7},{8}}
=> ? = 5
{{1},{2},{3,4,5},{6},{7},{8}}
=> [1,1,3,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> {{1},{2},{3,4,5},{6},{7},{8}}
=> ? = 12
{{1},{2},{3,6,7,8},{4,5}}
=> [1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> {{1},{2},{3,4,5,6},{7,8}}
=> ? = 5
{{1},{2},{3,8},{4,6,7},{5}}
=> [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> {{1},{2},{3,4},{5,6,7},{8}}
=> ? = 7
{{1},{2},{3,4,5,6},{7,8}}
=> [1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> {{1},{2},{3,4,5,6},{7,8}}
=> ? = 5
{{1},{2},{3,8},{4,5,6},{7}}
=> [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> {{1},{2},{3,4},{5,6,7},{8}}
=> ? = 7
{{1},{2},{3,4,7,8},{5,6}}
=> [1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> {{1},{2},{3,4,5,6},{7,8}}
=> ? = 5
{{1},{2},{3,4,5,8},{6},{7}}
=> [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> {{1},{2},{3,4,5,6},{7},{8}}
=> ? = 8
{{1},{2},{3,4,5,6,7},{8}}
=> [1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1},{2},{3,4,5,6,7},{8}}
=> ? = 5
{{1},{2},{3,8},{4,5,6,7}}
=> [1,1,2,4] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> {{1},{2},{3,4},{5,6,7,8}}
=> ? = 5
{{1},{2},{3,4,5,8},{6,7}}
=> [1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> {{1},{2},{3,4,5,6},{7,8}}
=> ? = 5
{{1},{2},{3,4,5,6,8},{7}}
=> [1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1},{2},{3,4,5,6,7},{8}}
=> ? = 5
{{1},{2,3},{4},{5,6,7,8}}
=> [1,2,1,4] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3},{4},{5,6,7,8}}
=> ? = 4
{{1},{2,3},{4,5},{6,7,8}}
=> [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> {{1},{2,3},{4,5},{6,7,8}}
=> ? = 3
{{1},{2,3},{4,6,7},{5},{8}}
=> [1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> {{1},{2,3},{4,5,6},{7},{8}}
=> ? = 5
{{1},{2,3},{4,6,8},{5},{7}}
=> [1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> {{1},{2,3},{4,5,6},{7},{8}}
=> ? = 5
{{1},{2,3},{4,6,7,8},{5}}
=> [1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> {{1},{2,3},{4,5,6,7},{8}}
=> ? = 3
{{1},{2,3},{4,5,8},{6},{7}}
=> [1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> {{1},{2,3},{4,5,6},{7},{8}}
=> ? = 5
{{1},{2,3},{4,8},{5,6,7}}
=> [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> {{1},{2,3},{4,5},{6,7,8}}
=> ? = 3
{{1},{2,3},{4,5,6,8},{7}}
=> [1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> {{1},{2,3},{4,5,6,7},{8}}
=> ? = 3
{{1},{2,3},{4,5,6,7,8}}
=> [1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2,3},{4,5,6,7,8}}
=> ? = 2
{{1},{2,4},{3},{5},{6,7,8}}
=> [1,2,1,1,3] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> {{1},{2,3},{4},{5},{6,7,8}}
=> ? = 7
{{1},{2,4},{3},{5,7,8},{6}}
=> [1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3},{4},{5,6,7},{8}}
=> ? = 6
{{1},{2,4},{3},{5,6,7,8}}
=> [1,2,1,4] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3},{4},{5,6,7,8}}
=> ? = 4
{{1},{2,7,8},{3},{4},{5},{6}}
=> [1,3,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> {{1},{2,3,4},{5},{6},{7},{8}}
=> ? = 11
{{1},{2,5,8},{3},{4},{6},{7}}
=> [1,3,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> {{1},{2,3,4},{5},{6},{7},{8}}
=> ? = 11
{{1},{2,8},{3},{4},{5,6,7}}
=> [1,2,1,1,3] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> {{1},{2,3},{4},{5},{6,7,8}}
=> ? = 7
{{1},{2,5,6,8},{3},{4},{7}}
=> [1,4,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> {{1},{2,3,4,5},{6},{7},{8}}
=> ? = 7
{{1},{2,5,6,7,8},{3},{4}}
=> [1,5,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> {{1},{2,3,4,5,6},{7},{8}}
=> ? = 4
{{1},{2,6,7,8},{3},{4,5}}
=> [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> {{1},{2,3,4,5},{6},{7,8}}
=> ? = 4
{{1},{2,4,6,7},{3},{5},{8}}
=> [1,4,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> {{1},{2,3,4,5},{6},{7},{8}}
=> ? = 7
{{1},{2,8},{3},{4,6,7},{5}}
=> [1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3},{4},{5,6,7},{8}}
=> ? = 6
{{1},{2,4,6,8},{3},{5},{7}}
=> [1,4,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> {{1},{2,3,4,5},{6},{7},{8}}
=> ? = 7
{{1},{2,4,6,7,8},{3},{5}}
=> [1,5,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> {{1},{2,3,4,5,6},{7},{8}}
=> ? = 4
{{1},{2,4,7,8},{3},{5,6}}
=> [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> {{1},{2,3,4,5},{6},{7,8}}
=> ? = 4
{{1},{2,4,5,7,8},{3},{6}}
=> [1,5,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> {{1},{2,3,4,5,6},{7},{8}}
=> ? = 4
Description
The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element.
This is the number of pairs $i\lt j$ in different blocks such that $i$ is a singleton block and $j$ is the maximal element of a block.
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