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Your data matches 9 different statistics following compositions of up to 3 maps.
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Matching statistic: St000382
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00128: Set partitions —to composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> {{1}}
=> [1] => 1
{{1,2}}
=> {{1,2}}
=> [2] => 2
{{1},{2}}
=> {{1},{2}}
=> [1,1] => 1
{{1,2,3}}
=> {{1,2,3}}
=> [3] => 3
{{1,2},{3}}
=> {{1,2},{3}}
=> [2,1] => 2
{{1,3},{2}}
=> {{1},{2,3}}
=> [1,2] => 1
{{1},{2,3}}
=> {{1,3},{2}}
=> [2,1] => 2
{{1},{2},{3}}
=> {{1},{2},{3}}
=> [1,1,1] => 1
{{1,2,3,4}}
=> {{1,2,3,4}}
=> [4] => 4
{{1,2,3},{4}}
=> {{1,2,3},{4}}
=> [3,1] => 3
{{1,2,4},{3}}
=> {{1,2},{3,4}}
=> [2,2] => 2
{{1,2},{3,4}}
=> {{1,2,4},{3}}
=> [3,1] => 3
{{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> [2,1,1] => 2
{{1,3,4},{2}}
=> {{1,4},{2,3}}
=> [2,2] => 2
{{1,3},{2,4}}
=> {{1},{2,3,4}}
=> [1,3] => 1
{{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> [1,2,1] => 1
{{1,4},{2,3}}
=> {{1,3},{2,4}}
=> [2,2] => 2
{{1},{2,3,4}}
=> {{1,3,4},{2}}
=> [3,1] => 3
{{1},{2,3},{4}}
=> {{1,3},{2},{4}}
=> [2,1,1] => 2
{{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> [1,1,2] => 1
{{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> [1,2,1] => 1
{{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> [2,1,1] => 2
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [1,1,1,1] => 1
{{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> [5] => 5
{{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> [4,1] => 4
{{1,2,3,5},{4}}
=> {{1,2,3},{4,5}}
=> [3,2] => 3
{{1,2,3},{4,5}}
=> {{1,2,3,5},{4}}
=> [4,1] => 4
{{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> [3,1,1] => 3
{{1,2,4,5},{3}}
=> {{1,2,5},{3,4}}
=> [3,2] => 3
{{1,2,4},{3,5}}
=> {{1,2},{3,4,5}}
=> [2,3] => 2
{{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> [2,2,1] => 2
{{1,2,5},{3,4}}
=> {{1,2,4},{3,5}}
=> [3,2] => 3
{{1,2},{3,4,5}}
=> {{1,2,4,5},{3}}
=> [4,1] => 4
{{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> [3,1,1] => 3
{{1,2,5},{3},{4}}
=> {{1,2},{3},{4,5}}
=> [2,1,2] => 2
{{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> [2,2,1] => 2
{{1,2},{3},{4,5}}
=> {{1,2,5},{3},{4}}
=> [3,1,1] => 3
{{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> [2,1,1,1] => 2
{{1,3,4,5},{2}}
=> {{1,4,5},{2,3}}
=> [3,2] => 3
{{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> [2,3] => 2
{{1,3,4},{2},{5}}
=> {{1,4},{2,3},{5}}
=> [2,2,1] => 2
{{1,3,5},{2,4}}
=> {{1},{2,3,4,5}}
=> [1,4] => 1
{{1,3},{2,4,5}}
=> {{1,5},{2,3,4}}
=> [2,3] => 2
{{1,3},{2,4},{5}}
=> {{1},{2,3,4},{5}}
=> [1,3,1] => 1
{{1,3,5},{2},{4}}
=> {{1},{2,3,5},{4}}
=> [1,3,1] => 1
{{1,3},{2,5},{4}}
=> {{1},{2,3},{4,5}}
=> [1,2,2] => 1
{{1,3},{2},{4,5}}
=> {{1,5},{2,3},{4}}
=> [2,2,1] => 2
{{1,3},{2},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> [1,2,1,1] => 1
{{1,4,5},{2,3}}
=> {{1,3,5},{2,4}}
=> [3,2] => 3
{{1,4},{2,3,5}}
=> {{1,3},{2,4,5}}
=> [2,3] => 2
Description
The first part of an integer composition.
Matching statistic: St000383
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> {{1}}
=> [1] => [1] => 1
{{1,2}}
=> {{1,2}}
=> [2] => [2] => 2
{{1},{2}}
=> {{1},{2}}
=> [1,1] => [1,1] => 1
{{1,2,3}}
=> {{1,2,3}}
=> [3] => [3] => 3
{{1,2},{3}}
=> {{1,2},{3}}
=> [2,1] => [1,2] => 2
{{1,3},{2}}
=> {{1},{2,3}}
=> [1,2] => [2,1] => 1
{{1},{2,3}}
=> {{1,3},{2}}
=> [2,1] => [1,2] => 2
{{1},{2},{3}}
=> {{1},{2},{3}}
=> [1,1,1] => [1,1,1] => 1
{{1,2,3,4}}
=> {{1,2,3,4}}
=> [4] => [4] => 4
{{1,2,3},{4}}
=> {{1,2,3},{4}}
=> [3,1] => [1,3] => 3
{{1,2,4},{3}}
=> {{1,2},{3,4}}
=> [2,2] => [2,2] => 2
{{1,2},{3,4}}
=> {{1,2,4},{3}}
=> [3,1] => [1,3] => 3
{{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> [2,1,1] => [1,1,2] => 2
{{1,3,4},{2}}
=> {{1,4},{2,3}}
=> [2,2] => [2,2] => 2
{{1,3},{2,4}}
=> {{1},{2,3,4}}
=> [1,3] => [3,1] => 1
{{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> [1,2,1] => [1,2,1] => 1
{{1,4},{2,3}}
=> {{1,3},{2,4}}
=> [2,2] => [2,2] => 2
{{1},{2,3,4}}
=> {{1,3,4},{2}}
=> [3,1] => [1,3] => 3
{{1},{2,3},{4}}
=> {{1,3},{2},{4}}
=> [2,1,1] => [1,1,2] => 2
{{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> [1,1,2] => [2,1,1] => 1
{{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> [1,2,1] => [1,2,1] => 1
{{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> [2,1,1] => [1,1,2] => 2
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [1,1,1,1] => [1,1,1,1] => 1
{{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> [5] => [5] => 5
{{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> [4,1] => [1,4] => 4
{{1,2,3,5},{4}}
=> {{1,2,3},{4,5}}
=> [3,2] => [2,3] => 3
{{1,2,3},{4,5}}
=> {{1,2,3,5},{4}}
=> [4,1] => [1,4] => 4
{{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,3] => 3
{{1,2,4,5},{3}}
=> {{1,2,5},{3,4}}
=> [3,2] => [2,3] => 3
{{1,2,4},{3,5}}
=> {{1,2},{3,4,5}}
=> [2,3] => [3,2] => 2
{{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> [2,2,1] => [1,2,2] => 2
{{1,2,5},{3,4}}
=> {{1,2,4},{3,5}}
=> [3,2] => [2,3] => 3
{{1,2},{3,4,5}}
=> {{1,2,4,5},{3}}
=> [4,1] => [1,4] => 4
{{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,3] => 3
{{1,2,5},{3},{4}}
=> {{1,2},{3},{4,5}}
=> [2,1,2] => [2,1,2] => 2
{{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> [2,2,1] => [1,2,2] => 2
{{1,2},{3},{4,5}}
=> {{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,3] => 3
{{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,1,2] => 2
{{1,3,4,5},{2}}
=> {{1,4,5},{2,3}}
=> [3,2] => [2,3] => 3
{{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> [2,3] => [3,2] => 2
{{1,3,4},{2},{5}}
=> {{1,4},{2,3},{5}}
=> [2,2,1] => [1,2,2] => 2
{{1,3,5},{2,4}}
=> {{1},{2,3,4,5}}
=> [1,4] => [4,1] => 1
{{1,3},{2,4,5}}
=> {{1,5},{2,3,4}}
=> [2,3] => [3,2] => 2
{{1,3},{2,4},{5}}
=> {{1},{2,3,4},{5}}
=> [1,3,1] => [1,3,1] => 1
{{1,3,5},{2},{4}}
=> {{1},{2,3,5},{4}}
=> [1,3,1] => [1,3,1] => 1
{{1,3},{2,5},{4}}
=> {{1},{2,3},{4,5}}
=> [1,2,2] => [2,2,1] => 1
{{1,3},{2},{4,5}}
=> {{1,5},{2,3},{4}}
=> [2,2,1] => [1,2,2] => 2
{{1,3},{2},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> [1,2,1,1] => [1,1,2,1] => 1
{{1,4,5},{2,3}}
=> {{1,3,5},{2,4}}
=> [3,2] => [2,3] => 3
{{1,4},{2,3,5}}
=> {{1,3},{2,4,5}}
=> [2,3] => [3,2] => 2
{{1,2},{3,4},{5,6},{7,8}}
=> {{1,2,4,6,8},{3},{5},{7}}
=> [5,1,1,1] => [1,1,1,5] => ? = 5
{{1,4},{2,3},{5,6},{7,8}}
=> {{1,3,6,8},{2,4},{5},{7}}
=> [4,2,1,1] => [1,1,2,4] => ? = 4
{{1,8},{2,3},{4,5},{6,7}}
=> {{1,3,5,7},{2},{4},{6,8}}
=> [4,1,1,2] => [2,1,1,4] => ? = 4
{{1,2},{3,6},{4,5},{7,8}}
=> {{1,2,5,8},{3,6},{4},{7}}
=> [4,2,1,1] => [1,1,2,4] => ? = 4
{{1,4},{2,7},{3,6},{5,8}}
=> {{1},{2},{3,4,6,8},{5,7}}
=> [1,1,4,2] => [2,4,1,1] => ? = 1
{{1,5},{2,7},{3,6},{4,8}}
=> {{1},{2},{3,6},{4,5,7,8}}
=> [1,1,2,4] => [4,2,1,1] => ? = 1
{{1,8},{2,5},{3,6},{4,7}}
=> {{1},{2},{3,5,6,7},{4,8}}
=> [1,1,4,2] => [2,4,1,1] => ? = 1
{{1,6},{2,5},{3,7},{4,8}}
=> {{1},{2},{3,5},{4,6,7,8}}
=> [1,1,2,4] => [4,2,1,1] => ? = 1
{{1,4},{2,3},{5,7},{6,8}}
=> {{1,3},{2,4,7,8},{5},{6}}
=> [2,4,1,1] => [1,1,4,2] => ? = 2
{{1,2},{3,4},{5,8},{6,7}}
=> {{1,2,4,7},{3,8},{5},{6}}
=> [4,2,1,1] => [1,1,2,4] => ? = 4
{{1,3},{2,4},{5,8},{6,7}}
=> {{1,7},{2,3,4,8},{5},{6}}
=> [2,4,1,1] => [1,1,4,2] => ? = 2
{{1,5},{2,6},{3,8},{4,7}}
=> {{1},{2},{3,7},{4,5,6,8}}
=> [1,1,2,4] => [4,2,1,1] => ? = 1
{{1,6},{2,7},{3,8},{4,5}}
=> {{1,5},{2},{3},{4,6,7,8}}
=> [2,1,1,4] => [4,1,1,2] => ? = 2
{{1,2},{3,4},{5,6},{7,8},{9,10}}
=> {{1,2,4,6,8,10},{3},{5},{7},{9}}
=> [6,1,1,1,1] => [1,1,1,1,6] => ? = 6
{{1,4},{2,3},{5,6},{7,8},{9,10}}
=> {{1,3,6,8,10},{2,4},{5},{7},{9}}
=> [5,2,1,1,1] => [1,1,1,2,5] => ? = 5
{{1,2},{3,6},{4,5},{7,8},{9,10}}
=> {{1,2,5,8,10},{3,6},{4},{7},{9}}
=> [5,2,1,1,1] => [1,1,1,2,5] => ? = 5
{{1,6},{2,5},{3,4},{7,8},{9,10}}
=> {{1,4,8,10},{2,5},{3,6},{7},{9}}
=> [4,2,2,1,1] => [1,1,2,2,4] => ? = 4
{{1,2},{3,4},{5,8},{6,7},{9,10}}
=> {{1,2,4,7,10},{3,8},{5},{6},{9}}
=> [5,2,1,1,1] => [1,1,1,2,5] => ? = 5
{{1,4},{2,3},{5,8},{6,7},{9,10}}
=> {{1,3,7,10},{2,4,8},{5},{6},{9}}
=> [4,3,1,1,1] => [1,1,1,3,4] => ? = 4
{{1,2},{3,8},{4,7},{5,6},{9,10}}
=> {{1,2,6,10},{3,7},{4,8},{5},{9}}
=> [4,2,2,1,1] => [1,1,2,2,4] => ? = 4
{{1,8},{2,7},{3,6},{4,5},{9,10}}
=> {{1,5,10},{2,6},{3,7},{4,8},{9}}
=> [3,2,2,2,1] => [1,2,2,2,3] => ? = 3
{{1,4},{2,3},{5,10},{6,7},{8,9}}
=> {{1,3,7,9},{2,4},{5,10},{6},{8}}
=> [4,2,2,1,1] => [1,1,2,2,4] => ? = 4
{{1,2},{3,4},{5,6},{7,10},{8,9}}
=> {{1,2,4,6,9},{3,10},{5},{7},{8}}
=> [5,2,1,1,1] => [1,1,1,2,5] => ? = 5
{{1,4},{2,3},{5,6},{7,10},{8,9}}
=> {{1,3,6,9},{2,4,10},{5},{7},{8}}
=> [4,3,1,1,1] => [1,1,1,3,4] => ? = 4
{{1,2},{3,6},{4,5},{7,10},{8,9}}
=> {{1,2,5,9},{3,6,10},{4},{7},{8}}
=> [4,3,1,1,1] => [1,1,1,3,4] => ? = 4
{{1,6},{2,5},{3,4},{7,10},{8,9}}
=> {{1,4,9},{2,5,10},{3,6},{7},{8}}
=> [3,3,2,1,1] => [1,1,2,3,3] => ? = 3
{{1,2},{3,4},{5,10},{6,9},{7,8}}
=> {{1,2,4,8},{3,9},{5,10},{6},{7}}
=> [4,2,2,1,1] => [1,1,2,2,4] => ? = 4
{{1,4},{2,3},{5,10},{6,9},{7,8}}
=> {{1,3,8},{2,4,9},{5,10},{6},{7}}
=> [3,3,2,1,1] => [1,1,2,3,3] => ? = 3
{{1,2},{3,10},{4,9},{5,8},{6,7}}
=> {{1,2,7},{3,8},{4,9},{5,10},{6}}
=> [3,2,2,2,1] => [1,2,2,2,3] => ? = 3
{{1,10},{2,9},{3,8},{4,7},{5,6}}
=> {{1,6},{2,7},{3,8},{4,9},{5,10}}
=> [2,2,2,2,2] => [2,2,2,2,2] => ? = 2
{{1},{2},{3},{4},{5},{6},{7,8}}
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> [2,1,1,1,1,1,1] => [1,1,1,1,1,1,2] => ? = 2
{{1},{2},{3},{4},{5},{6,7,8}}
=> {{1,7,8},{2},{3},{4},{5},{6}}
=> [3,1,1,1,1,1] => [1,1,1,1,1,3] => ? = 3
{{1},{2},{3},{4},{5,6},{7},{8}}
=> {{1,6},{2},{3},{4},{5},{7},{8}}
=> [2,1,1,1,1,1,1] => [1,1,1,1,1,1,2] => ? = 2
{{1},{2},{3},{4},{5,6},{7,8}}
=> {{1,6,8},{2},{3},{4},{5},{7}}
=> [3,1,1,1,1,1] => [1,1,1,1,1,3] => ? = 3
{{1},{2},{3},{4},{5,6,7},{8}}
=> {{1,6,7},{2},{3},{4},{5},{8}}
=> [3,1,1,1,1,1] => [1,1,1,1,1,3] => ? = 3
{{1},{2},{3},{4},{5,6,7,8}}
=> {{1,6,7,8},{2},{3},{4},{5}}
=> [4,1,1,1,1] => [1,1,1,1,4] => ? = 4
{{1},{2},{3},{4,5},{6},{7,8}}
=> {{1,5,8},{2},{3},{4},{6},{7}}
=> [3,1,1,1,1,1] => [1,1,1,1,1,3] => ? = 3
{{1},{2},{3},{4,6,7,8},{5}}
=> {{1,7,8},{2,6},{3},{4},{5}}
=> [3,2,1,1,1] => [1,1,1,2,3] => ? = 3
{{1},{2},{3},{4,8},{5,6,7}}
=> {{1,6,7},{2,8},{3},{4},{5}}
=> [3,2,1,1,1] => [1,1,1,2,3] => ? = 3
{{1},{2},{3},{4,5,6,7,8}}
=> {{1,5,6,7,8},{2},{3},{4}}
=> [5,1,1,1] => [1,1,1,5] => ? = 5
{{1},{2},{3,4},{5,6},{7},{8}}
=> {{1,4,6},{2},{3},{5},{7},{8}}
=> [3,1,1,1,1,1] => [1,1,1,1,1,3] => ? = 3
{{1},{2},{3,5},{4},{6,7,8}}
=> {{1,7,8},{2,5},{3},{4},{6}}
=> [3,2,1,1,1] => [1,1,1,2,3] => ? = 3
{{1},{2},{3,5,6,7,8},{4}}
=> {{1,6,7,8},{2,5},{3},{4}}
=> [4,2,1,1] => [1,1,2,4] => ? = 4
{{1},{2},{3,4,5},{6},{7},{8}}
=> {{1,4,5},{2},{3},{6},{7},{8}}
=> [3,1,1,1,1,1] => [1,1,1,1,1,3] => ? = 3
{{1},{2},{3,6,7,8},{4,5}}
=> {{1,5,7,8},{2,6},{3},{4}}
=> [4,2,1,1] => [1,1,2,4] => ? = 4
{{1},{2},{3,4,5,6},{7,8}}
=> {{1,4,5,6,8},{2},{3},{7}}
=> [5,1,1,1] => [1,1,1,5] => ? = 5
{{1},{2},{3,8},{4,5,6},{7}}
=> {{1,5,6},{2},{3,8},{4},{7}}
=> [3,1,2,1,1] => [1,1,2,1,3] => ? = 3
{{1},{2},{3,7,8},{4,5,6}}
=> {{1,5,6,8},{2,7},{3},{4}}
=> [4,2,1,1] => [1,1,2,4] => ? = 4
{{1},{2},{3,4,7,8},{5,6}}
=> {{1,4,6,8},{2,7},{3},{5}}
=> [4,2,1,1] => [1,1,2,4] => ? = 4
{{1},{2},{3,4,5,6,7},{8}}
=> {{1,4,5,6,7},{2},{3},{8}}
=> [5,1,1,1] => [1,1,1,5] => ? = 5
Description
The last part of an integer composition.
Matching statistic: St000439
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> {{1}}
=> [1] => [1,0]
=> 2 = 1 + 1
{{1,2}}
=> {{1,2}}
=> [2] => [1,1,0,0]
=> 3 = 2 + 1
{{1},{2}}
=> {{1},{2}}
=> [1,1] => [1,0,1,0]
=> 2 = 1 + 1
{{1,2,3}}
=> {{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> 4 = 3 + 1
{{1,2},{3}}
=> {{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> 3 = 2 + 1
{{1,3},{2}}
=> {{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
{{1},{2,3}}
=> {{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> 3 = 2 + 1
{{1},{2},{3}}
=> {{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> 2 = 1 + 1
{{1,2,3,4}}
=> {{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
{{1,2,3},{4}}
=> {{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
{{1,2,4},{3}}
=> {{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
{{1,2},{3,4}}
=> {{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
{{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
{{1,3,4},{2}}
=> {{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
{{1,3},{2,4}}
=> {{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
{{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
{{1,4},{2,3}}
=> {{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
{{1},{2,3,4}}
=> {{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
{{1},{2,3},{4}}
=> {{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
{{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
{{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
{{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
{{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
{{1,2,3,5},{4}}
=> {{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
{{1,2,3},{4,5}}
=> {{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
{{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
{{1,2,4,5},{3}}
=> {{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
{{1,2,4},{3,5}}
=> {{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
{{1,2,5},{3,4}}
=> {{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
{{1,2},{3,4,5}}
=> {{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
{{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
{{1,2,5},{3},{4}}
=> {{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
{{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
{{1,2},{3},{4,5}}
=> {{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
{{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
{{1,3,4,5},{2}}
=> {{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
{{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,3,4},{2},{5}}
=> {{1,4},{2,3},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
{{1,3,5},{2,4}}
=> {{1},{2,3,4,5}}
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
{{1,3},{2,4,5}}
=> {{1,5},{2,3,4}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,3},{2,4},{5}}
=> {{1},{2,3,4},{5}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
{{1,3,5},{2},{4}}
=> {{1},{2,3,5},{4}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
{{1,3},{2,5},{4}}
=> {{1},{2,3},{4,5}}
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
{{1,3},{2},{4,5}}
=> {{1,5},{2,3},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
{{1,3},{2},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
{{1,4,5},{2,3}}
=> {{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
{{1,4},{2,3,5}}
=> {{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2},{3,4},{5,6},{7,8}}
=> {{1,2,4,6,8},{3},{5},{7}}
=> [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 5 + 1
{{1,3},{2,4},{5,6},{7,8}}
=> {{1,6,8},{2,3,4},{5},{7}}
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3 + 1
{{1,4},{2,3},{5,6},{7,8}}
=> {{1,3,6,8},{2,4},{5},{7}}
=> [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 4 + 1
{{1,5},{2,3},{4,6},{7,8}}
=> {{1,3,8},{2,6},{4,5},{7}}
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
{{1,6},{2,3},{4,5},{7,8}}
=> {{1,3,5,8},{2},{4,6},{7}}
=> [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 + 1
{{1,7},{2,3},{4,5},{6,8}}
=> {{1,3,5},{2,8},{4},{6,7}}
=> [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
{{1,8},{2,3},{4,5},{6,7}}
=> {{1,3,5,7},{2},{4},{6,8}}
=> [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 4 + 1
{{1,2},{3,5},{4,6},{7,8}}
=> {{1,2,8},{3,5,6},{4},{7}}
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3 + 1
{{1,2},{3,6},{4,5},{7,8}}
=> {{1,2,5,8},{3,6},{4},{7}}
=> [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 4 + 1
{{1,3},{2,6},{4,5},{7,8}}
=> {{1,5,8},{2,3},{4,6},{7}}
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
{{1,5},{2,6},{3,4},{7,8}}
=> {{1,4,8},{2},{3,5,6},{7}}
=> [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 + 1
{{1,6},{2,5},{3,4},{7,8}}
=> {{1,4,8},{2,5},{3,6},{7}}
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
{{1,8},{2,5},{3,4},{6,7}}
=> {{1,4,7},{2,5},{3},{6,8}}
=> [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
{{1,2},{3,7},{4,5},{6,8}}
=> {{1,2,5},{3,8},{4,7},{6}}
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
{{1,2},{3,8},{4,5},{6,7}}
=> {{1,2,5,7},{3},{4,8},{6}}
=> [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 + 1
{{1,3},{2,8},{4,5},{6,7}}
=> {{1,5,7},{2,3},{4},{6,8}}
=> [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
{{1,5},{2,8},{3,4},{6,7}}
=> {{1,4,7},{2},{3,5},{6,8}}
=> [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 + 1
{{1,7},{2,8},{3,4},{5,6}}
=> {{1,4,6},{2},{3},{5,7,8}}
=> [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 + 1
{{1,8},{2,7},{3,4},{5,6}}
=> {{1,4,6},{2},{3,7},{5,8}}
=> [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 + 1
{{1,2},{3,4},{5,7},{6,8}}
=> {{1,2,4},{3,7,8},{5},{6}}
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3 + 1
{{1,2},{3,4},{5,8},{6,7}}
=> {{1,2,4,7},{3,8},{5},{6}}
=> [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 4 + 1
{{1,4},{2,3},{5,8},{6,7}}
=> {{1,3,7},{2,4,8},{5},{6}}
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3 + 1
{{1,5},{2,3},{4,8},{6,7}}
=> {{1,3,7},{2},{4,5,8},{6}}
=> [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 + 1
{{1,7},{2,3},{4,8},{5,6}}
=> {{1,3,6},{2},{4,8},{5,7}}
=> [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 + 1
{{1,8},{2,3},{4,7},{5,6}}
=> {{1,3,6},{2,7},{4},{5,8}}
=> [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
{{1,2},{3,5},{4,8},{6,7}}
=> {{1,2,7},{3,5},{4,8},{6}}
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
{{1,2},{3,7},{4,8},{5,6}}
=> {{1,2,6},{3},{4,7,8},{5}}
=> [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 + 1
{{1,2},{3,8},{4,7},{5,6}}
=> {{1,2,6},{3,7},{4,8},{5}}
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
{{1,2},{3,4},{5,6},{7,8},{9,10}}
=> {{1,2,4,6,8,10},{3},{5},{7},{9}}
=> [6,1,1,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 6 + 1
{{1,4},{2,3},{5,6},{7,8},{9,10}}
=> {{1,3,6,8,10},{2,4},{5},{7},{9}}
=> [5,2,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 5 + 1
{{1,2},{3,6},{4,5},{7,8},{9,10}}
=> {{1,2,5,8,10},{3,6},{4},{7},{9}}
=> [5,2,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 5 + 1
{{1,6},{2,5},{3,4},{7,8},{9,10}}
=> {{1,4,8,10},{2,5},{3,6},{7},{9}}
=> [4,2,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 4 + 1
{{1,2},{3,4},{5,8},{6,7},{9,10}}
=> {{1,2,4,7,10},{3,8},{5},{6},{9}}
=> [5,2,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 5 + 1
{{1,4},{2,3},{5,8},{6,7},{9,10}}
=> {{1,3,7,10},{2,4,8},{5},{6},{9}}
=> [4,3,1,1,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 4 + 1
{{1,2},{3,8},{4,7},{5,6},{9,10}}
=> {{1,2,6,10},{3,7},{4,8},{5},{9}}
=> [4,2,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 4 + 1
{{1,8},{2,7},{3,6},{4,5},{9,10}}
=> {{1,5,10},{2,6},{3,7},{4,8},{9}}
=> [3,2,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
{{1,4},{2,3},{5,10},{6,7},{8,9}}
=> {{1,3,7,9},{2,4},{5,10},{6},{8}}
=> [4,2,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 4 + 1
{{1,2},{3,4},{5,6},{7,10},{8,9}}
=> {{1,2,4,6,9},{3,10},{5},{7},{8}}
=> [5,2,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 5 + 1
{{1,4},{2,3},{5,6},{7,10},{8,9}}
=> {{1,3,6,9},{2,4,10},{5},{7},{8}}
=> [4,3,1,1,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 4 + 1
{{1,2},{3,6},{4,5},{7,10},{8,9}}
=> {{1,2,5,9},{3,6,10},{4},{7},{8}}
=> [4,3,1,1,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 4 + 1
{{1,6},{2,5},{3,4},{7,10},{8,9}}
=> {{1,4,9},{2,5,10},{3,6},{7},{8}}
=> [3,3,2,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 3 + 1
{{1,2},{3,4},{5,10},{6,9},{7,8}}
=> {{1,2,4,8},{3,9},{5,10},{6},{7}}
=> [4,2,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 4 + 1
{{1,4},{2,3},{5,10},{6,9},{7,8}}
=> {{1,3,8},{2,4,9},{5,10},{6},{7}}
=> [3,3,2,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 3 + 1
{{1,2},{3,10},{4,9},{5,8},{6,7}}
=> {{1,2,7},{3,8},{4,9},{5,10},{6}}
=> [3,2,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
{{1,10},{2,9},{3,8},{4,7},{5,6}}
=> {{1,6},{2,7},{3,8},{4,9},{5,10}}
=> [2,2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2 + 1
{{1},{2},{3},{4},{5,6,7,8}}
=> {{1,6,7,8},{2},{3},{4},{5}}
=> [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
{{1},{2},{3},{4,6,7,8},{5}}
=> {{1,7,8},{2,6},{3},{4},{5}}
=> [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
{{1},{2},{3},{4,8},{5,6,7}}
=> {{1,6,7},{2,8},{3},{4},{5}}
=> [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
{{1},{2},{3},{4,5,6,7,8}}
=> {{1,5,6,7,8},{2},{3},{4}}
=> [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 5 + 1
{{1},{2},{3,5},{4},{6,7,8}}
=> {{1,7,8},{2,5},{3},{4},{6}}
=> [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000504
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000504: Set partitions ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 80%
St000504: Set partitions ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 80%
Values
{{1}}
=> {{1}}
=> ? = 1
{{1,2}}
=> {{1,2}}
=> 2
{{1},{2}}
=> {{1},{2}}
=> 1
{{1,2,3}}
=> {{1,2,3}}
=> 3
{{1,2},{3}}
=> {{1,2},{3}}
=> 2
{{1,3},{2}}
=> {{1},{2,3}}
=> 1
{{1},{2,3}}
=> {{1,3},{2}}
=> 2
{{1},{2},{3}}
=> {{1},{2},{3}}
=> 1
{{1,2,3,4}}
=> {{1,2,3,4}}
=> 4
{{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 3
{{1,2,4},{3}}
=> {{1,2},{3,4}}
=> 2
{{1,2},{3,4}}
=> {{1,2,4},{3}}
=> 3
{{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> 2
{{1,3,4},{2}}
=> {{1,4},{2,3}}
=> 2
{{1,3},{2,4}}
=> {{1},{2,3,4}}
=> 1
{{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> 1
{{1,4},{2,3}}
=> {{1,3},{2,4}}
=> 2
{{1},{2,3,4}}
=> {{1,3,4},{2}}
=> 3
{{1},{2,3},{4}}
=> {{1,3},{2},{4}}
=> 2
{{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> 1
{{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> 2
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 1
{{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 5
{{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> 4
{{1,2,3,5},{4}}
=> {{1,2,3},{4,5}}
=> 3
{{1,2,3},{4,5}}
=> {{1,2,3,5},{4}}
=> 4
{{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 3
{{1,2,4,5},{3}}
=> {{1,2,5},{3,4}}
=> 3
{{1,2,4},{3,5}}
=> {{1,2},{3,4,5}}
=> 2
{{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> 2
{{1,2,5},{3,4}}
=> {{1,2,4},{3,5}}
=> 3
{{1,2},{3,4,5}}
=> {{1,2,4,5},{3}}
=> 4
{{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> 3
{{1,2,5},{3},{4}}
=> {{1,2},{3},{4,5}}
=> 2
{{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> 2
{{1,2},{3},{4,5}}
=> {{1,2,5},{3},{4}}
=> 3
{{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> 2
{{1,3,4,5},{2}}
=> {{1,4,5},{2,3}}
=> 3
{{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> 2
{{1,3,4},{2},{5}}
=> {{1,4},{2,3},{5}}
=> 2
{{1,3,5},{2,4}}
=> {{1},{2,3,4,5}}
=> 1
{{1,3},{2,4,5}}
=> {{1,5},{2,3,4}}
=> 2
{{1,3},{2,4},{5}}
=> {{1},{2,3,4},{5}}
=> 1
{{1,3,5},{2},{4}}
=> {{1},{2,3,5},{4}}
=> 1
{{1,3},{2,5},{4}}
=> {{1},{2,3},{4,5}}
=> 1
{{1,3},{2},{4,5}}
=> {{1,5},{2,3},{4}}
=> 2
{{1,3},{2},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> 1
{{1,4,5},{2,3}}
=> {{1,3,5},{2,4}}
=> 3
{{1,4},{2,3,5}}
=> {{1,3},{2,4,5}}
=> 2
{{1,4},{2,3},{5}}
=> {{1,3},{2,4},{5}}
=> 2
{{1,2},{3,4},{5,6},{7,8}}
=> {{1,2,4,6,8},{3},{5},{7}}
=> ? = 5
{{1,3},{2,4},{5,6},{7,8}}
=> {{1,6,8},{2,3,4},{5},{7}}
=> ? = 3
{{1,4},{2,3},{5,6},{7,8}}
=> {{1,3,6,8},{2,4},{5},{7}}
=> ? = 4
{{1,5},{2,3},{4,6},{7,8}}
=> {{1,3,8},{2,6},{4,5},{7}}
=> ? = 3
{{1,6},{2,3},{4,5},{7,8}}
=> {{1,3,5,8},{2},{4,6},{7}}
=> ? = 4
{{1,7},{2,3},{4,5},{6,8}}
=> {{1,3,5},{2,8},{4},{6,7}}
=> ? = 3
{{1,8},{2,3},{4,5},{6,7}}
=> {{1,3,5,7},{2},{4},{6,8}}
=> ? = 4
{{1,8},{2,4},{3,5},{6,7}}
=> {{1,7},{2,4,5},{3},{6,8}}
=> ? = 2
{{1,7},{2,4},{3,5},{6,8}}
=> {{1},{2,4,5,8},{3},{6,7}}
=> ? = 1
{{1,6},{2,4},{3,5},{7,8}}
=> {{1,8},{2,4,5},{3,6},{7}}
=> ? = 2
{{1,5},{2,4},{3,6},{7,8}}
=> {{1,8},{2,4},{3,5,6},{7}}
=> ? = 2
{{1,4},{2,5},{3,6},{7,8}}
=> {{1,8},{2},{3,4,5,6},{7}}
=> ? = 2
{{1,3},{2,5},{4,6},{7,8}}
=> {{1,8},{2,3,6},{4,5},{7}}
=> ? = 2
{{1,2},{3,5},{4,6},{7,8}}
=> {{1,2,8},{3,5,6},{4},{7}}
=> ? = 3
{{1,2},{3,6},{4,5},{7,8}}
=> {{1,2,5,8},{3,6},{4},{7}}
=> ? = 4
{{1,3},{2,6},{4,5},{7,8}}
=> {{1,5,8},{2,3},{4,6},{7}}
=> ? = 3
{{1,4},{2,6},{3,5},{7,8}}
=> {{1,8},{2,5},{3,4,6},{7}}
=> ? = 2
{{1,5},{2,6},{3,4},{7,8}}
=> {{1,4,8},{2},{3,5,6},{7}}
=> ? = 3
{{1,6},{2,5},{3,4},{7,8}}
=> {{1,4,8},{2,5},{3,6},{7}}
=> ? = 3
{{1,7},{2,5},{3,4},{6,8}}
=> {{1,4},{2,5,8},{3},{6,7}}
=> ? = 2
{{1,8},{2,5},{3,4},{6,7}}
=> {{1,4,7},{2,5},{3},{6,8}}
=> ? = 3
{{1,8},{2,6},{3,4},{5,7}}
=> {{1,4},{2,7},{3,6},{5,8}}
=> ? = 2
{{1,7},{2,6},{3,4},{5,8}}
=> {{1,4},{2},{3,6,8},{5,7}}
=> ? = 2
{{1,6},{2,7},{3,4},{5,8}}
=> {{1,4},{2},{3,8},{5,6,7}}
=> ? = 2
{{1,5},{2,7},{3,4},{6,8}}
=> {{1,4},{2,8},{3,5},{6,7}}
=> ? = 2
{{1,4},{2,7},{3,5},{6,8}}
=> {{1},{2,5,8},{3,4},{6,7}}
=> ? = 1
{{1,3},{2,7},{4,5},{6,8}}
=> {{1,5},{2,3,8},{4},{6,7}}
=> ? = 2
{{1,2},{3,7},{4,5},{6,8}}
=> {{1,2,5},{3,8},{4,7},{6}}
=> ? = 3
{{1,2},{3,8},{4,5},{6,7}}
=> {{1,2,5,7},{3},{4,8},{6}}
=> ? = 4
{{1,3},{2,8},{4,5},{6,7}}
=> {{1,5,7},{2,3},{4},{6,8}}
=> ? = 3
{{1,4},{2,8},{3,5},{6,7}}
=> {{1,7},{2,5},{3,4},{6,8}}
=> ? = 2
{{1,5},{2,8},{3,4},{6,7}}
=> {{1,4,7},{2},{3,5},{6,8}}
=> ? = 3
{{1,6},{2,8},{3,4},{5,7}}
=> {{1,4},{2,7},{3},{5,6,8}}
=> ? = 2
{{1,7},{2,8},{3,4},{5,6}}
=> {{1,4,6},{2},{3},{5,7,8}}
=> ? = 3
{{1,8},{2,7},{3,4},{5,6}}
=> {{1,4,6},{2},{3,7},{5,8}}
=> ? = 3
{{1,8},{2,7},{3,5},{4,6}}
=> {{1},{2,5,6},{3,7},{4,8}}
=> ? = 1
{{1,7},{2,8},{3,5},{4,6}}
=> {{1},{2,5,6},{3},{4,7,8}}
=> ? = 1
{{1,6},{2,8},{3,5},{4,7}}
=> {{1},{2,5},{3,7},{4,6,8}}
=> ? = 1
{{1,5},{2,8},{3,6},{4,7}}
=> {{1},{2},{3,6,7},{4,5,8}}
=> ? = 1
{{1,4},{2,8},{3,6},{5,7}}
=> {{1},{2,7},{3,4,6},{5,8}}
=> ? = 1
{{1,3},{2,8},{4,6},{5,7}}
=> {{1},{2,3,6,7},{4},{5,8}}
=> ? = 1
{{1,2},{3,8},{4,6},{5,7}}
=> {{1,2},{3,6,7},{4,8},{5}}
=> ? = 2
{{1,2},{3,7},{4,6},{5,8}}
=> {{1,2},{3,6},{4,7,8},{5}}
=> ? = 2
{{1,3},{2,7},{4,6},{5,8}}
=> {{1},{2,3,6},{4,8},{5,7}}
=> ? = 1
{{1,4},{2,7},{3,6},{5,8}}
=> {{1},{2},{3,4,6,8},{5,7}}
=> ? = 1
{{1,5},{2,7},{3,6},{4,8}}
=> {{1},{2},{3,6},{4,5,7,8}}
=> ? = 1
{{1,6},{2,7},{3,5},{4,8}}
=> {{1},{2,5},{3},{4,6,7,8}}
=> ? = 1
{{1,7},{2,6},{3,5},{4,8}}
=> {{1},{2,5},{3,6},{4,7,8}}
=> ? = 1
{{1,8},{2,6},{3,5},{4,7}}
=> {{1},{2,5},{3,6,7},{4,8}}
=> ? = 1
Description
The cardinality of the first block of a set partition.
The number of partitions of $\{1,\ldots,n\}$ into $k$ blocks in which the first block has cardinality $j+1$ is given by $\binom{n-1}{j}S(n-j-1,k-1)$, see [1, Theorem 1.1] and the references therein. Here, $S(n,k)$ are the ''Stirling numbers of the second kind'' counting all set partitions of $\{1,\ldots,n\}$ into $k$ blocks [2].
Matching statistic: St000502
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
St000502: Set partitions ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 80%
Values
{{1}}
=> ? = 1 - 1
{{1,2}}
=> 1 = 2 - 1
{{1},{2}}
=> 0 = 1 - 1
{{1,2,3}}
=> 2 = 3 - 1
{{1,2},{3}}
=> 1 = 2 - 1
{{1,3},{2}}
=> 0 = 1 - 1
{{1},{2,3}}
=> 1 = 2 - 1
{{1},{2},{3}}
=> 0 = 1 - 1
{{1,2,3,4}}
=> 3 = 4 - 1
{{1,2,3},{4}}
=> 2 = 3 - 1
{{1,2,4},{3}}
=> 1 = 2 - 1
{{1,2},{3,4}}
=> 2 = 3 - 1
{{1,2},{3},{4}}
=> 1 = 2 - 1
{{1,3,4},{2}}
=> 1 = 2 - 1
{{1,3},{2,4}}
=> 0 = 1 - 1
{{1,3},{2},{4}}
=> 0 = 1 - 1
{{1,4},{2,3}}
=> 1 = 2 - 1
{{1},{2,3,4}}
=> 2 = 3 - 1
{{1},{2,3},{4}}
=> 1 = 2 - 1
{{1,4},{2},{3}}
=> 0 = 1 - 1
{{1},{2,4},{3}}
=> 0 = 1 - 1
{{1},{2},{3,4}}
=> 1 = 2 - 1
{{1},{2},{3},{4}}
=> 0 = 1 - 1
{{1,2,3,4,5}}
=> 4 = 5 - 1
{{1,2,3,4},{5}}
=> 3 = 4 - 1
{{1,2,3,5},{4}}
=> 2 = 3 - 1
{{1,2,3},{4,5}}
=> 3 = 4 - 1
{{1,2,3},{4},{5}}
=> 2 = 3 - 1
{{1,2,4,5},{3}}
=> 2 = 3 - 1
{{1,2,4},{3,5}}
=> 1 = 2 - 1
{{1,2,4},{3},{5}}
=> 1 = 2 - 1
{{1,2,5},{3,4}}
=> 2 = 3 - 1
{{1,2},{3,4,5}}
=> 3 = 4 - 1
{{1,2},{3,4},{5}}
=> 2 = 3 - 1
{{1,2,5},{3},{4}}
=> 1 = 2 - 1
{{1,2},{3,5},{4}}
=> 1 = 2 - 1
{{1,2},{3},{4,5}}
=> 2 = 3 - 1
{{1,2},{3},{4},{5}}
=> 1 = 2 - 1
{{1,3,4,5},{2}}
=> 2 = 3 - 1
{{1,3,4},{2,5}}
=> 1 = 2 - 1
{{1,3,4},{2},{5}}
=> 1 = 2 - 1
{{1,3,5},{2,4}}
=> 0 = 1 - 1
{{1,3},{2,4,5}}
=> 1 = 2 - 1
{{1,3},{2,4},{5}}
=> 0 = 1 - 1
{{1,3,5},{2},{4}}
=> 0 = 1 - 1
{{1,3},{2,5},{4}}
=> 0 = 1 - 1
{{1,3},{2},{4,5}}
=> 1 = 2 - 1
{{1,3},{2},{4},{5}}
=> 0 = 1 - 1
{{1,4,5},{2,3}}
=> 2 = 3 - 1
{{1,4},{2,3,5}}
=> 1 = 2 - 1
{{1,4},{2,3},{5}}
=> 1 = 2 - 1
{{1,2},{3,4},{5,6},{7,8}}
=> ? = 5 - 1
{{1,3},{2,4},{5,6},{7,8}}
=> ? = 3 - 1
{{1,4},{2,3},{5,6},{7,8}}
=> ? = 4 - 1
{{1,5},{2,3},{4,6},{7,8}}
=> ? = 3 - 1
{{1,6},{2,3},{4,5},{7,8}}
=> ? = 4 - 1
{{1,7},{2,3},{4,5},{6,8}}
=> ? = 3 - 1
{{1,8},{2,3},{4,5},{6,7}}
=> ? = 4 - 1
{{1,8},{2,4},{3,5},{6,7}}
=> ? = 2 - 1
{{1,7},{2,4},{3,5},{6,8}}
=> ? = 1 - 1
{{1,6},{2,4},{3,5},{7,8}}
=> ? = 2 - 1
{{1,5},{2,4},{3,6},{7,8}}
=> ? = 2 - 1
{{1,4},{2,5},{3,6},{7,8}}
=> ? = 2 - 1
{{1,3},{2,5},{4,6},{7,8}}
=> ? = 2 - 1
{{1,2},{3,5},{4,6},{7,8}}
=> ? = 3 - 1
{{1,2},{3,6},{4,5},{7,8}}
=> ? = 4 - 1
{{1,3},{2,6},{4,5},{7,8}}
=> ? = 3 - 1
{{1,4},{2,6},{3,5},{7,8}}
=> ? = 2 - 1
{{1,5},{2,6},{3,4},{7,8}}
=> ? = 3 - 1
{{1,6},{2,5},{3,4},{7,8}}
=> ? = 3 - 1
{{1,7},{2,5},{3,4},{6,8}}
=> ? = 2 - 1
{{1,8},{2,5},{3,4},{6,7}}
=> ? = 3 - 1
{{1,8},{2,6},{3,4},{5,7}}
=> ? = 2 - 1
{{1,7},{2,6},{3,4},{5,8}}
=> ? = 2 - 1
{{1,6},{2,7},{3,4},{5,8}}
=> ? = 2 - 1
{{1,5},{2,7},{3,4},{6,8}}
=> ? = 2 - 1
{{1,4},{2,7},{3,5},{6,8}}
=> ? = 1 - 1
{{1,3},{2,7},{4,5},{6,8}}
=> ? = 2 - 1
{{1,2},{3,7},{4,5},{6,8}}
=> ? = 3 - 1
{{1,2},{3,8},{4,5},{6,7}}
=> ? = 4 - 1
{{1,3},{2,8},{4,5},{6,7}}
=> ? = 3 - 1
{{1,4},{2,8},{3,5},{6,7}}
=> ? = 2 - 1
{{1,5},{2,8},{3,4},{6,7}}
=> ? = 3 - 1
{{1,6},{2,8},{3,4},{5,7}}
=> ? = 2 - 1
{{1,7},{2,8},{3,4},{5,6}}
=> ? = 3 - 1
{{1,8},{2,7},{3,4},{5,6}}
=> ? = 3 - 1
{{1,8},{2,7},{3,5},{4,6}}
=> ? = 1 - 1
{{1,7},{2,8},{3,5},{4,6}}
=> ? = 1 - 1
{{1,6},{2,8},{3,5},{4,7}}
=> ? = 1 - 1
{{1,5},{2,8},{3,6},{4,7}}
=> ? = 1 - 1
{{1,4},{2,8},{3,6},{5,7}}
=> ? = 1 - 1
{{1,3},{2,8},{4,6},{5,7}}
=> ? = 1 - 1
{{1,2},{3,8},{4,6},{5,7}}
=> ? = 2 - 1
{{1,2},{3,7},{4,6},{5,8}}
=> ? = 2 - 1
{{1,3},{2,7},{4,6},{5,8}}
=> ? = 1 - 1
{{1,4},{2,7},{3,6},{5,8}}
=> ? = 1 - 1
{{1,5},{2,7},{3,6},{4,8}}
=> ? = 1 - 1
{{1,6},{2,7},{3,5},{4,8}}
=> ? = 1 - 1
{{1,7},{2,6},{3,5},{4,8}}
=> ? = 1 - 1
{{1,8},{2,6},{3,5},{4,7}}
=> ? = 1 - 1
Description
The number of successions of a set partitions.
This is the number of indices $i$ such that $i$ and $i+1$ belonging to the same block.
Matching statistic: St000025
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 70%
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 70%
Values
{{1}}
=> {{1}}
=> [1] => [1,0]
=> 1
{{1,2}}
=> {{1,2}}
=> [2] => [1,1,0,0]
=> 2
{{1},{2}}
=> {{1},{2}}
=> [1,1] => [1,0,1,0]
=> 1
{{1,2,3}}
=> {{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> 3
{{1,2},{3}}
=> {{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> 2
{{1,3},{2}}
=> {{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> 1
{{1},{2,3}}
=> {{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> 2
{{1},{2},{3}}
=> {{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> 1
{{1,2,3,4}}
=> {{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> 4
{{1,2,3},{4}}
=> {{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
{{1,2,4},{3}}
=> {{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
{{1,2},{3,4}}
=> {{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
{{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
{{1,3,4},{2}}
=> {{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
{{1,3},{2,4}}
=> {{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
{{1,4},{2,3}}
=> {{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
{{1},{2,3,4}}
=> {{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
{{1},{2,3},{4}}
=> {{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
{{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
{{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
{{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
{{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
{{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
{{1,2,3,5},{4}}
=> {{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
{{1,2,3},{4,5}}
=> {{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
{{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
{{1,2,4,5},{3}}
=> {{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
{{1,2,4},{3,5}}
=> {{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
{{1,2,5},{3,4}}
=> {{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
{{1,2},{3,4,5}}
=> {{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
{{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
{{1,2,5},{3},{4}}
=> {{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
{{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
{{1,2},{3},{4,5}}
=> {{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
{{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
{{1,3,4,5},{2}}
=> {{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
{{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,3,4},{2},{5}}
=> {{1,4},{2,3},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
{{1,3,5},{2,4}}
=> {{1},{2,3,4,5}}
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,3},{2,4,5}}
=> {{1,5},{2,3,4}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,3},{2,4},{5}}
=> {{1},{2,3,4},{5}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
{{1,3,5},{2},{4}}
=> {{1},{2,3,5},{4}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
{{1,3},{2,5},{4}}
=> {{1},{2,3},{4,5}}
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,3},{2},{4,5}}
=> {{1,5},{2,3},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
{{1,3},{2},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
{{1,4,5},{2,3}}
=> {{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
{{1,4},{2,3,5}}
=> {{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,2},{3,4},{5,6},{7,8}}
=> {{1,2,4,6,8},{3},{5},{7}}
=> [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 5
{{1,3},{2,4},{5,6},{7,8}}
=> {{1,6,8},{2,3,4},{5},{7}}
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
{{1,4},{2,3},{5,6},{7,8}}
=> {{1,3,6,8},{2,4},{5},{7}}
=> [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 4
{{1,5},{2,3},{4,6},{7,8}}
=> {{1,3,8},{2,6},{4,5},{7}}
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
{{1,6},{2,3},{4,5},{7,8}}
=> {{1,3,5,8},{2},{4,6},{7}}
=> [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4
{{1,7},{2,3},{4,5},{6,8}}
=> {{1,3,5},{2,8},{4},{6,7}}
=> [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3
{{1,8},{2,3},{4,5},{6,7}}
=> {{1,3,5,7},{2},{4},{6,8}}
=> [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 4
{{1,8},{2,4},{3,5},{6,7}}
=> {{1,7},{2,4,5},{3},{6,8}}
=> [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 2
{{1,7},{2,4},{3,5},{6,8}}
=> {{1},{2,4,5,8},{3},{6,7}}
=> [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 1
{{1,6},{2,4},{3,5},{7,8}}
=> {{1,8},{2,4,5},{3,6},{7}}
=> [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 2
{{1,5},{2,4},{3,6},{7,8}}
=> {{1,8},{2,4},{3,5,6},{7}}
=> [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 2
{{1,4},{2,5},{3,6},{7,8}}
=> {{1,8},{2},{3,4,5,6},{7}}
=> [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
{{1,3},{2,5},{4,6},{7,8}}
=> {{1,8},{2,3,6},{4,5},{7}}
=> [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 2
{{1,2},{3,5},{4,6},{7,8}}
=> {{1,2,8},{3,5,6},{4},{7}}
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
{{1,2},{3,6},{4,5},{7,8}}
=> {{1,2,5,8},{3,6},{4},{7}}
=> [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 4
{{1,3},{2,6},{4,5},{7,8}}
=> {{1,5,8},{2,3},{4,6},{7}}
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
{{1,4},{2,6},{3,5},{7,8}}
=> {{1,8},{2,5},{3,4,6},{7}}
=> [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 2
{{1,5},{2,6},{3,4},{7,8}}
=> {{1,4,8},{2},{3,5,6},{7}}
=> [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3
{{1,6},{2,5},{3,4},{7,8}}
=> {{1,4,8},{2,5},{3,6},{7}}
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
{{1,7},{2,5},{3,4},{6,8}}
=> {{1,4},{2,5,8},{3},{6,7}}
=> [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 2
{{1,8},{2,5},{3,4},{6,7}}
=> {{1,4,7},{2,5},{3},{6,8}}
=> [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3
{{1,8},{2,6},{3,4},{5,7}}
=> {{1,4},{2,7},{3,6},{5,8}}
=> [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
{{1,7},{2,6},{3,4},{5,8}}
=> {{1,4},{2},{3,6,8},{5,7}}
=> [2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 2
{{1,6},{2,7},{3,4},{5,8}}
=> {{1,4},{2},{3,8},{5,6,7}}
=> [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2
{{1,5},{2,7},{3,4},{6,8}}
=> {{1,4},{2,8},{3,5},{6,7}}
=> [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
{{1,4},{2,7},{3,5},{6,8}}
=> {{1},{2,5,8},{3,4},{6,7}}
=> [1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
{{1,3},{2,7},{4,5},{6,8}}
=> {{1,5},{2,3,8},{4},{6,7}}
=> [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 2
{{1,2},{3,7},{4,5},{6,8}}
=> {{1,2,5},{3,8},{4,7},{6}}
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
{{1,2},{3,8},{4,5},{6,7}}
=> {{1,2,5,7},{3},{4,8},{6}}
=> [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4
{{1,3},{2,8},{4,5},{6,7}}
=> {{1,5,7},{2,3},{4},{6,8}}
=> [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3
{{1,4},{2,8},{3,5},{6,7}}
=> {{1,7},{2,5},{3,4},{6,8}}
=> [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
{{1,5},{2,8},{3,4},{6,7}}
=> {{1,4,7},{2},{3,5},{6,8}}
=> [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3
{{1,6},{2,8},{3,4},{5,7}}
=> {{1,4},{2,7},{3},{5,6,8}}
=> [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 2
{{1,7},{2,8},{3,4},{5,6}}
=> {{1,4,6},{2},{3},{5,7,8}}
=> [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3
{{1,8},{2,7},{3,4},{5,6}}
=> {{1,4,6},{2},{3,7},{5,8}}
=> [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3
{{1,8},{2,7},{3,5},{4,6}}
=> {{1},{2,5,6},{3,7},{4,8}}
=> [1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
{{1,7},{2,8},{3,5},{4,6}}
=> {{1},{2,5,6},{3},{4,7,8}}
=> [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 1
{{1,6},{2,8},{3,5},{4,7}}
=> {{1},{2,5},{3,7},{4,6,8}}
=> [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
{{1,5},{2,8},{3,6},{4,7}}
=> {{1},{2},{3,6,7},{4,5,8}}
=> [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 1
{{1,4},{2,8},{3,6},{5,7}}
=> {{1},{2,7},{3,4,6},{5,8}}
=> [1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
{{1,3},{2,8},{4,6},{5,7}}
=> {{1},{2,3,6,7},{4},{5,8}}
=> [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 1
{{1,2},{3,8},{4,6},{5,7}}
=> {{1,2},{3,6,7},{4,8},{5}}
=> [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 2
{{1,2},{3,7},{4,6},{5,8}}
=> {{1,2},{3,6},{4,7,8},{5}}
=> [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 2
{{1,3},{2,7},{4,6},{5,8}}
=> {{1},{2,3,6},{4,8},{5,7}}
=> [1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
{{1,4},{2,7},{3,6},{5,8}}
=> {{1},{2},{3,4,6,8},{5,7}}
=> [1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 1
{{1,5},{2,7},{3,6},{4,8}}
=> {{1},{2},{3,6},{4,5,7,8}}
=> [1,1,2,4] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
{{1,6},{2,7},{3,5},{4,8}}
=> {{1},{2,5},{3},{4,6,7,8}}
=> [1,2,1,4] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
{{1,7},{2,6},{3,5},{4,8}}
=> {{1},{2,5},{3,6},{4,7,8}}
=> [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
{{1,8},{2,6},{3,5},{4,7}}
=> {{1},{2,5},{3,6,7},{4,8}}
=> [1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
{{1,8},{2,5},{3,6},{4,7}}
=> {{1},{2},{3,5,6,7},{4,8}}
=> [1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 1
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
Matching statistic: St000026
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 70%
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 70%
Values
{{1}}
=> {{1}}
=> [1] => [1,0]
=> 1
{{1,2}}
=> {{1,2}}
=> [2] => [1,1,0,0]
=> 2
{{1},{2}}
=> {{1},{2}}
=> [1,1] => [1,0,1,0]
=> 1
{{1,2,3}}
=> {{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> 3
{{1,2},{3}}
=> {{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> 2
{{1,3},{2}}
=> {{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> 1
{{1},{2,3}}
=> {{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> 2
{{1},{2},{3}}
=> {{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> 1
{{1,2,3,4}}
=> {{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> 4
{{1,2,3},{4}}
=> {{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
{{1,2,4},{3}}
=> {{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
{{1,2},{3,4}}
=> {{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
{{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
{{1,3,4},{2}}
=> {{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
{{1,3},{2,4}}
=> {{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
{{1,4},{2,3}}
=> {{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
{{1},{2,3,4}}
=> {{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
{{1},{2,3},{4}}
=> {{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
{{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
{{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
{{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
{{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
{{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
{{1,2,3,5},{4}}
=> {{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
{{1,2,3},{4,5}}
=> {{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
{{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
{{1,2,4,5},{3}}
=> {{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
{{1,2,4},{3,5}}
=> {{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
{{1,2,5},{3,4}}
=> {{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
{{1,2},{3,4,5}}
=> {{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
{{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
{{1,2,5},{3},{4}}
=> {{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
{{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
{{1,2},{3},{4,5}}
=> {{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
{{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
{{1,3,4,5},{2}}
=> {{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
{{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,3,4},{2},{5}}
=> {{1,4},{2,3},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
{{1,3,5},{2,4}}
=> {{1},{2,3,4,5}}
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,3},{2,4,5}}
=> {{1,5},{2,3,4}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,3},{2,4},{5}}
=> {{1},{2,3,4},{5}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
{{1,3,5},{2},{4}}
=> {{1},{2,3,5},{4}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
{{1,3},{2,5},{4}}
=> {{1},{2,3},{4,5}}
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,3},{2},{4,5}}
=> {{1,5},{2,3},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
{{1,3},{2},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
{{1,4,5},{2,3}}
=> {{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
{{1,4},{2,3,5}}
=> {{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,2},{3,4},{5,6},{7,8}}
=> {{1,2,4,6,8},{3},{5},{7}}
=> [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 5
{{1,3},{2,4},{5,6},{7,8}}
=> {{1,6,8},{2,3,4},{5},{7}}
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
{{1,4},{2,3},{5,6},{7,8}}
=> {{1,3,6,8},{2,4},{5},{7}}
=> [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 4
{{1,5},{2,3},{4,6},{7,8}}
=> {{1,3,8},{2,6},{4,5},{7}}
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
{{1,6},{2,3},{4,5},{7,8}}
=> {{1,3,5,8},{2},{4,6},{7}}
=> [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4
{{1,7},{2,3},{4,5},{6,8}}
=> {{1,3,5},{2,8},{4},{6,7}}
=> [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3
{{1,8},{2,3},{4,5},{6,7}}
=> {{1,3,5,7},{2},{4},{6,8}}
=> [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 4
{{1,8},{2,4},{3,5},{6,7}}
=> {{1,7},{2,4,5},{3},{6,8}}
=> [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 2
{{1,7},{2,4},{3,5},{6,8}}
=> {{1},{2,4,5,8},{3},{6,7}}
=> [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 1
{{1,6},{2,4},{3,5},{7,8}}
=> {{1,8},{2,4,5},{3,6},{7}}
=> [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 2
{{1,5},{2,4},{3,6},{7,8}}
=> {{1,8},{2,4},{3,5,6},{7}}
=> [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 2
{{1,4},{2,5},{3,6},{7,8}}
=> {{1,8},{2},{3,4,5,6},{7}}
=> [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
{{1,3},{2,5},{4,6},{7,8}}
=> {{1,8},{2,3,6},{4,5},{7}}
=> [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 2
{{1,2},{3,5},{4,6},{7,8}}
=> {{1,2,8},{3,5,6},{4},{7}}
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
{{1,2},{3,6},{4,5},{7,8}}
=> {{1,2,5,8},{3,6},{4},{7}}
=> [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 4
{{1,3},{2,6},{4,5},{7,8}}
=> {{1,5,8},{2,3},{4,6},{7}}
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
{{1,4},{2,6},{3,5},{7,8}}
=> {{1,8},{2,5},{3,4,6},{7}}
=> [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 2
{{1,5},{2,6},{3,4},{7,8}}
=> {{1,4,8},{2},{3,5,6},{7}}
=> [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3
{{1,6},{2,5},{3,4},{7,8}}
=> {{1,4,8},{2,5},{3,6},{7}}
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
{{1,7},{2,5},{3,4},{6,8}}
=> {{1,4},{2,5,8},{3},{6,7}}
=> [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 2
{{1,8},{2,5},{3,4},{6,7}}
=> {{1,4,7},{2,5},{3},{6,8}}
=> [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3
{{1,8},{2,6},{3,4},{5,7}}
=> {{1,4},{2,7},{3,6},{5,8}}
=> [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
{{1,7},{2,6},{3,4},{5,8}}
=> {{1,4},{2},{3,6,8},{5,7}}
=> [2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 2
{{1,6},{2,7},{3,4},{5,8}}
=> {{1,4},{2},{3,8},{5,6,7}}
=> [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2
{{1,5},{2,7},{3,4},{6,8}}
=> {{1,4},{2,8},{3,5},{6,7}}
=> [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
{{1,4},{2,7},{3,5},{6,8}}
=> {{1},{2,5,8},{3,4},{6,7}}
=> [1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
{{1,3},{2,7},{4,5},{6,8}}
=> {{1,5},{2,3,8},{4},{6,7}}
=> [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 2
{{1,2},{3,7},{4,5},{6,8}}
=> {{1,2,5},{3,8},{4,7},{6}}
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
{{1,2},{3,8},{4,5},{6,7}}
=> {{1,2,5,7},{3},{4,8},{6}}
=> [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4
{{1,3},{2,8},{4,5},{6,7}}
=> {{1,5,7},{2,3},{4},{6,8}}
=> [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3
{{1,4},{2,8},{3,5},{6,7}}
=> {{1,7},{2,5},{3,4},{6,8}}
=> [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
{{1,5},{2,8},{3,4},{6,7}}
=> {{1,4,7},{2},{3,5},{6,8}}
=> [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3
{{1,6},{2,8},{3,4},{5,7}}
=> {{1,4},{2,7},{3},{5,6,8}}
=> [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 2
{{1,7},{2,8},{3,4},{5,6}}
=> {{1,4,6},{2},{3},{5,7,8}}
=> [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3
{{1,8},{2,7},{3,4},{5,6}}
=> {{1,4,6},{2},{3,7},{5,8}}
=> [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3
{{1,8},{2,7},{3,5},{4,6}}
=> {{1},{2,5,6},{3,7},{4,8}}
=> [1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
{{1,7},{2,8},{3,5},{4,6}}
=> {{1},{2,5,6},{3},{4,7,8}}
=> [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 1
{{1,6},{2,8},{3,5},{4,7}}
=> {{1},{2,5},{3,7},{4,6,8}}
=> [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
{{1,5},{2,8},{3,6},{4,7}}
=> {{1},{2},{3,6,7},{4,5,8}}
=> [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 1
{{1,4},{2,8},{3,6},{5,7}}
=> {{1},{2,7},{3,4,6},{5,8}}
=> [1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
{{1,3},{2,8},{4,6},{5,7}}
=> {{1},{2,3,6,7},{4},{5,8}}
=> [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 1
{{1,2},{3,8},{4,6},{5,7}}
=> {{1,2},{3,6,7},{4,8},{5}}
=> [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 2
{{1,2},{3,7},{4,6},{5,8}}
=> {{1,2},{3,6},{4,7,8},{5}}
=> [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 2
{{1,3},{2,7},{4,6},{5,8}}
=> {{1},{2,3,6},{4,8},{5,7}}
=> [1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
{{1,4},{2,7},{3,6},{5,8}}
=> {{1},{2},{3,4,6,8},{5,7}}
=> [1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 1
{{1,5},{2,7},{3,6},{4,8}}
=> {{1},{2},{3,6},{4,5,7,8}}
=> [1,1,2,4] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
{{1,6},{2,7},{3,5},{4,8}}
=> {{1},{2,5},{3},{4,6,7,8}}
=> [1,2,1,4] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
{{1,7},{2,6},{3,5},{4,8}}
=> {{1},{2,5},{3,6},{4,7,8}}
=> [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
{{1,8},{2,6},{3,5},{4,7}}
=> {{1},{2,5},{3,6,7},{4,8}}
=> [1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
{{1,8},{2,5},{3,6},{4,7}}
=> {{1},{2},{3,5,6,7},{4,8}}
=> [1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 1
Description
The position of the first return of a Dyck path.
Matching statistic: St000237
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00112: Set partitions —complement⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 100%
Mp00080: Set partitions —to permutation⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> {{1}}
=> [1] => 0 = 1 - 1
{{1,2}}
=> {{1,2}}
=> [2,1] => 1 = 2 - 1
{{1},{2}}
=> {{1},{2}}
=> [1,2] => 0 = 1 - 1
{{1,2,3}}
=> {{1,2,3}}
=> [2,3,1] => 2 = 3 - 1
{{1,2},{3}}
=> {{1},{2,3}}
=> [1,3,2] => 1 = 2 - 1
{{1,3},{2}}
=> {{1,3},{2}}
=> [3,2,1] => 0 = 1 - 1
{{1},{2,3}}
=> {{1,2},{3}}
=> [2,1,3] => 1 = 2 - 1
{{1},{2},{3}}
=> {{1},{2},{3}}
=> [1,2,3] => 0 = 1 - 1
{{1,2,3,4}}
=> {{1,2,3,4}}
=> [2,3,4,1] => 3 = 4 - 1
{{1,2,3},{4}}
=> {{1},{2,3,4}}
=> [1,3,4,2] => 2 = 3 - 1
{{1,2,4},{3}}
=> {{1,3,4},{2}}
=> [3,2,4,1] => 1 = 2 - 1
{{1,2},{3,4}}
=> {{1,2},{3,4}}
=> [2,1,4,3] => 2 = 3 - 1
{{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> [1,2,4,3] => 1 = 2 - 1
{{1,3,4},{2}}
=> {{1,2,4},{3}}
=> [2,4,3,1] => 1 = 2 - 1
{{1,3},{2,4}}
=> {{1,3},{2,4}}
=> [3,4,1,2] => 0 = 1 - 1
{{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> [1,4,3,2] => 0 = 1 - 1
{{1,4},{2,3}}
=> {{1,4},{2,3}}
=> [4,3,2,1] => 1 = 2 - 1
{{1},{2,3,4}}
=> {{1,2,3},{4}}
=> [2,3,1,4] => 2 = 3 - 1
{{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> [1,3,2,4] => 1 = 2 - 1
{{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> [4,2,3,1] => 0 = 1 - 1
{{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> [3,2,1,4] => 0 = 1 - 1
{{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> [2,1,3,4] => 1 = 2 - 1
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [1,2,3,4] => 0 = 1 - 1
{{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> [2,3,4,5,1] => 4 = 5 - 1
{{1,2,3,4},{5}}
=> {{1},{2,3,4,5}}
=> [1,3,4,5,2] => 3 = 4 - 1
{{1,2,3,5},{4}}
=> {{1,3,4,5},{2}}
=> [3,2,4,5,1] => 2 = 3 - 1
{{1,2,3},{4,5}}
=> {{1,2},{3,4,5}}
=> [2,1,4,5,3] => 3 = 4 - 1
{{1,2,3},{4},{5}}
=> {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => 2 = 3 - 1
{{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> [2,4,3,5,1] => 2 = 3 - 1
{{1,2,4},{3,5}}
=> {{1,3},{2,4,5}}
=> [3,4,1,5,2] => 1 = 2 - 1
{{1,2,4},{3},{5}}
=> {{1},{2,4,5},{3}}
=> [1,4,3,5,2] => 1 = 2 - 1
{{1,2,5},{3,4}}
=> {{1,4,5},{2,3}}
=> [4,3,2,5,1] => 2 = 3 - 1
{{1,2},{3,4,5}}
=> {{1,2,3},{4,5}}
=> [2,3,1,5,4] => 3 = 4 - 1
{{1,2},{3,4},{5}}
=> {{1},{2,3},{4,5}}
=> [1,3,2,5,4] => 2 = 3 - 1
{{1,2,5},{3},{4}}
=> {{1,4,5},{2},{3}}
=> [4,2,3,5,1] => 1 = 2 - 1
{{1,2},{3,5},{4}}
=> {{1,3},{2},{4,5}}
=> [3,2,1,5,4] => 1 = 2 - 1
{{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> [2,1,3,5,4] => 2 = 3 - 1
{{1,2},{3},{4},{5}}
=> {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => 1 = 2 - 1
{{1,3,4,5},{2}}
=> {{1,2,3,5},{4}}
=> [2,3,5,4,1] => 2 = 3 - 1
{{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> [4,3,5,1,2] => 1 = 2 - 1
{{1,3,4},{2},{5}}
=> {{1},{2,3,5},{4}}
=> [1,3,5,4,2] => 1 = 2 - 1
{{1,3,5},{2,4}}
=> {{1,3,5},{2,4}}
=> [3,4,5,2,1] => 0 = 1 - 1
{{1,3},{2,4,5}}
=> {{1,2,4},{3,5}}
=> [2,4,5,1,3] => 1 = 2 - 1
{{1,3},{2,4},{5}}
=> {{1},{2,4},{3,5}}
=> [1,4,5,2,3] => 0 = 1 - 1
{{1,3,5},{2},{4}}
=> {{1,3,5},{2},{4}}
=> [3,2,5,4,1] => 0 = 1 - 1
{{1,3},{2,5},{4}}
=> {{1,4},{2},{3,5}}
=> [4,2,5,1,3] => 0 = 1 - 1
{{1,3},{2},{4,5}}
=> {{1,2},{3,5},{4}}
=> [2,1,5,4,3] => 1 = 2 - 1
{{1,3},{2},{4},{5}}
=> {{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => 0 = 1 - 1
{{1,4,5},{2,3}}
=> {{1,2,5},{3,4}}
=> [2,5,4,3,1] => 2 = 3 - 1
{{1,4},{2,3,5}}
=> {{1,3,4},{2,5}}
=> [3,5,4,1,2] => 1 = 2 - 1
{{1,2,3,4,5,6},{7}}
=> {{1},{2,3,4,5,6,7}}
=> [1,3,4,5,6,7,2] => ? = 6 - 1
{{1,2,3,4,5},{6,7}}
=> {{1,2},{3,4,5,6,7}}
=> [2,1,4,5,6,7,3] => ? = 6 - 1
{{1,2,3,4,5},{6},{7}}
=> {{1},{2},{3,4,5,6,7}}
=> [1,2,4,5,6,7,3] => ? = 5 - 1
{{1,2,3,4,6,7},{5}}
=> {{1,2,4,5,6,7},{3}}
=> [2,4,3,5,6,7,1] => ? = 5 - 1
{{1,2,3,4,6},{5,7}}
=> {{1,3},{2,4,5,6,7}}
=> [3,4,1,5,6,7,2] => ? = 4 - 1
{{1,2,3,4,6},{5},{7}}
=> {{1},{2,4,5,6,7},{3}}
=> [1,4,3,5,6,7,2] => ? = 4 - 1
{{1,2,3,4},{5,6,7}}
=> {{1,2,3},{4,5,6,7}}
=> [2,3,1,5,6,7,4] => ? = 6 - 1
{{1,2,3,4},{5,6},{7}}
=> {{1},{2,3},{4,5,6,7}}
=> [1,3,2,5,6,7,4] => ? = 5 - 1
{{1,2,3,4},{5,7},{6}}
=> {{1,3},{2},{4,5,6,7}}
=> [3,2,1,5,6,7,4] => ? = 4 - 1
{{1,2,3,4},{5},{6,7}}
=> {{1,2},{3},{4,5,6,7}}
=> [2,1,3,5,6,7,4] => ? = 5 - 1
{{1,2,3,4},{5},{6},{7}}
=> {{1},{2},{3},{4,5,6,7}}
=> [1,2,3,5,6,7,4] => ? = 4 - 1
{{1,2,3,5,6,7},{4}}
=> {{1,2,3,5,6,7},{4}}
=> [2,3,5,4,6,7,1] => ? = 5 - 1
{{1,2,3,5,6},{4,7}}
=> {{1,4},{2,3,5,6,7}}
=> [4,3,5,1,6,7,2] => ? = 4 - 1
{{1,2,3,5,6},{4},{7}}
=> {{1},{2,3,5,6,7},{4}}
=> [1,3,5,4,6,7,2] => ? = 4 - 1
{{1,2,3,5},{4,6,7}}
=> {{1,2,4},{3,5,6,7}}
=> [2,4,5,1,6,7,3] => ? = 4 - 1
{{1,2,3,5},{4,6},{7}}
=> {{1},{2,4},{3,5,6,7}}
=> [1,4,5,2,6,7,3] => ? = 3 - 1
{{1,2,3,5,7},{4},{6}}
=> {{1,3,5,6,7},{2},{4}}
=> [3,2,5,4,6,7,1] => ? = 3 - 1
{{1,2,3,5},{4,7},{6}}
=> {{1,4},{2},{3,5,6,7}}
=> [4,2,5,1,6,7,3] => ? = 3 - 1
{{1,2,3,5},{4},{6,7}}
=> {{1,2},{3,5,6,7},{4}}
=> [2,1,5,4,6,7,3] => ? = 4 - 1
{{1,2,3,5},{4},{6},{7}}
=> {{1},{2},{3,5,6,7},{4}}
=> [1,2,5,4,6,7,3] => ? = 3 - 1
{{1,2,3,6,7},{4,5}}
=> {{1,2,5,6,7},{3,4}}
=> [2,5,4,3,6,7,1] => ? = 5 - 1
{{1,2,3,6},{4,5,7}}
=> {{1,3,4},{2,5,6,7}}
=> [3,5,4,1,6,7,2] => ? = 4 - 1
{{1,2,3,6},{4,5},{7}}
=> {{1},{2,5,6,7},{3,4}}
=> [1,5,4,3,6,7,2] => ? = 4 - 1
{{1,2,3,7},{4,5,6}}
=> {{1,5,6,7},{2,3,4}}
=> [5,3,4,2,6,7,1] => ? = 5 - 1
{{1,2,3},{4,5,6,7}}
=> {{1,2,3,4},{5,6,7}}
=> [2,3,4,1,6,7,5] => ? = 6 - 1
{{1,2,3},{4,5,6},{7}}
=> {{1},{2,3,4},{5,6,7}}
=> [1,3,4,2,6,7,5] => ? = 5 - 1
{{1,2,3,7},{4,5},{6}}
=> {{1,5,6,7},{2},{3,4}}
=> [5,2,4,3,6,7,1] => ? = 4 - 1
{{1,2,3},{4,5,7},{6}}
=> {{1,3,4},{2},{5,6,7}}
=> [3,2,4,1,6,7,5] => ? = 4 - 1
{{1,2,3},{4,5},{6,7}}
=> {{1,2},{3,4},{5,6,7}}
=> [2,1,4,3,6,7,5] => ? = 5 - 1
{{1,2,3},{4,5},{6},{7}}
=> {{1},{2},{3,4},{5,6,7}}
=> [1,2,4,3,6,7,5] => ? = 4 - 1
{{1,2,3,6,7},{4},{5}}
=> {{1,2,5,6,7},{3},{4}}
=> [2,5,3,4,6,7,1] => ? = 4 - 1
{{1,2,3,6},{4,7},{5}}
=> {{1,4},{2,5,6,7},{3}}
=> [4,5,3,1,6,7,2] => ? = 3 - 1
{{1,2,3,6},{4},{5,7}}
=> {{1,3},{2,5,6,7},{4}}
=> [3,5,1,4,6,7,2] => ? = 3 - 1
{{1,2,3,6},{4},{5},{7}}
=> {{1},{2,5,6,7},{3},{4}}
=> [1,5,3,4,6,7,2] => ? = 3 - 1
{{1,2,3,7},{4,6},{5}}
=> {{1,5,6,7},{2,4},{3}}
=> [5,4,3,2,6,7,1] => ? = 3 - 1
{{1,2,3},{4,6,7},{5}}
=> {{1,2,4},{3},{5,6,7}}
=> [2,4,3,1,6,7,5] => ? = 4 - 1
{{1,2,3},{4,6},{5,7}}
=> {{1,3},{2,4},{5,6,7}}
=> [3,4,1,2,6,7,5] => ? = 3 - 1
{{1,2,3},{4,6},{5},{7}}
=> {{1},{2,4},{3},{5,6,7}}
=> [1,4,3,2,6,7,5] => ? = 3 - 1
{{1,2,3},{4,7},{5,6}}
=> {{1,4},{2,3},{5,6,7}}
=> [4,3,2,1,6,7,5] => ? = 4 - 1
{{1,2,3},{4},{5,6,7}}
=> {{1,2,3},{4},{5,6,7}}
=> [2,3,1,4,6,7,5] => ? = 5 - 1
{{1,2,3},{4},{5,6},{7}}
=> {{1},{2,3},{4},{5,6,7}}
=> [1,3,2,4,6,7,5] => ? = 4 - 1
{{1,2,3},{4,7},{5},{6}}
=> {{1,4},{2},{3},{5,6,7}}
=> [4,2,3,1,6,7,5] => ? = 3 - 1
{{1,2,3},{4},{5,7},{6}}
=> {{1,3},{2},{4},{5,6,7}}
=> [3,2,1,4,6,7,5] => ? = 3 - 1
{{1,2,3},{4},{5},{6,7}}
=> {{1,2},{3},{4},{5,6,7}}
=> [2,1,3,4,6,7,5] => ? = 4 - 1
{{1,2,3},{4},{5},{6},{7}}
=> {{1},{2},{3},{4},{5,6,7}}
=> [1,2,3,4,6,7,5] => ? = 3 - 1
{{1,2,4,5,6,7},{3}}
=> {{1,2,3,4,6,7},{5}}
=> [2,3,4,6,5,7,1] => ? = 5 - 1
{{1,2,4,5,6},{3,7}}
=> {{1,5},{2,3,4,6,7}}
=> [5,3,4,6,1,7,2] => ? = 4 - 1
{{1,2,4,5,6},{3},{7}}
=> {{1},{2,3,4,6,7},{5}}
=> [1,3,4,6,5,7,2] => ? = 4 - 1
{{1,2,4,5,7},{3,6}}
=> {{1,3,4,6,7},{2,5}}
=> [3,5,4,6,2,7,1] => ? = 3 - 1
{{1,2,4,5},{3,6,7}}
=> {{1,2,5},{3,4,6,7}}
=> [2,5,4,6,1,7,3] => ? = 4 - 1
Description
The number of small exceedances.
This is the number of indices $i$ such that $\pi_i=i+1$.
Matching statistic: St000214
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1] => 0 = 1 - 1
{{1,2}}
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
{{1},{2}}
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
{{1,2,3}}
=> [2,3,1] => [3,1,2] => [3,2,1] => 2 = 3 - 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [2,3,1] => 0 = 1 - 1
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => [4,3,2,1] => 3 = 4 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => [3,2,1,4] => 2 = 3 - 1
{{1,2,4},{3}}
=> [2,4,3,1] => [4,1,3,2] => [3,4,2,1] => 1 = 2 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2 = 3 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1 = 2 - 1
{{1,3,4},{2}}
=> [3,2,4,1] => [4,2,1,3] => [2,4,3,1] => 1 = 2 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => [3,1,4,2] => 0 = 1 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 0 = 1 - 1
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 1 = 2 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => [1,4,3,2] => 2 = 3 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => [2,3,4,1] => 0 = 1 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 0 = 1 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => [5,4,3,2,1] => 4 = 5 - 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => [4,3,2,1,5] => 3 = 4 - 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,1,2,4,3] => [4,5,3,2,1] => 2 = 3 - 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => [3,2,1,5,4] => 3 = 4 - 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => [3,2,1,4,5] => 2 = 3 - 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [5,1,3,2,4] => [3,5,4,2,1] => 2 = 3 - 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,5,2,3] => [4,2,1,5,3] => 1 = 2 - 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,1,3,2,5] => [3,4,2,1,5] => 1 = 2 - 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,1,4,3,2] => [4,3,5,2,1] => 2 = 3 - 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,4,3] => 3 = 4 - 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2 = 3 - 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [5,1,3,4,2] => [3,4,5,2,1] => 1 = 2 - 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => [2,1,4,5,3] => 1 = 2 - 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2 = 3 - 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1 = 2 - 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [5,2,1,3,4] => [2,5,4,3,1] => 2 = 3 - 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,5,1,3,2] => [4,3,1,5,2] => 1 = 2 - 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [4,2,1,3,5] => [2,4,3,1,5] => 1 = 2 - 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,4,1,2,3] => [4,2,5,3,1] => 0 = 1 - 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,5,1,2,4] => [3,1,5,4,2] => 1 = 2 - 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2,5] => [3,1,4,2,5] => 0 = 1 - 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [5,2,1,4,3] => [2,4,5,3,1] => 0 = 1 - 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => [3,1,4,5,2] => 0 = 1 - 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 1 = 2 - 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => [2,3,1,4,5] => 0 = 1 - 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [5,3,2,1,4] => [3,2,5,4,1] => 2 = 3 - 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,5,2,1,3] => [4,1,5,3,2] => 1 = 2 - 1
{{1,2,3,4,5,7},{6}}
=> [2,3,4,5,7,6,1] => [7,1,2,3,4,6,5] => [6,7,5,4,3,2,1] => ? = 5 - 1
{{1,2,3,4,6,7},{5}}
=> [2,3,4,6,5,7,1] => [7,1,2,3,5,4,6] => [5,7,6,4,3,2,1] => ? = 5 - 1
{{1,2,3,4,6},{5,7}}
=> [2,3,4,6,7,1,5] => [6,1,2,3,7,4,5] => [6,4,3,2,1,7,5] => ? = 4 - 1
{{1,2,3,4,7},{5,6}}
=> [2,3,4,7,6,5,1] => [7,1,2,3,6,5,4] => [6,5,7,4,3,2,1] => ? = 5 - 1
{{1,2,3,4},{5,6,7}}
=> [2,3,4,1,6,7,5] => [4,1,2,3,7,5,6] => [4,3,2,1,7,6,5] => ? = 6 - 1
{{1,2,3,4},{5,6},{7}}
=> [2,3,4,1,6,5,7] => [4,1,2,3,6,5,7] => [4,3,2,1,6,5,7] => ? = 5 - 1
{{1,2,3,4,7},{5},{6}}
=> [2,3,4,7,5,6,1] => [7,1,2,3,5,6,4] => [5,6,7,4,3,2,1] => ? = 4 - 1
{{1,2,3,4},{5,7},{6}}
=> [2,3,4,1,7,6,5] => [4,1,2,3,7,6,5] => [4,3,2,1,6,7,5] => ? = 4 - 1
{{1,2,3,4},{5},{6,7}}
=> [2,3,4,1,5,7,6] => [4,1,2,3,5,7,6] => [4,3,2,1,5,7,6] => ? = 5 - 1
{{1,2,3,5,6,7},{4}}
=> [2,3,5,4,6,7,1] => [7,1,2,4,3,5,6] => [4,7,6,5,3,2,1] => ? = 5 - 1
{{1,2,3,5,6},{4,7}}
=> [2,3,5,7,6,1,4] => [6,1,2,7,3,5,4] => [6,5,3,2,1,7,4] => ? = 4 - 1
{{1,2,3,5,7},{4,6}}
=> [2,3,5,6,7,4,1] => [7,1,2,6,3,4,5] => [6,4,7,5,3,2,1] => ? = 3 - 1
{{1,2,3,5},{4,6,7}}
=> [2,3,5,6,1,7,4] => [5,1,2,7,3,4,6] => [5,3,2,1,7,6,4] => ? = 4 - 1
{{1,2,3,5},{4,6},{7}}
=> [2,3,5,6,1,4,7] => [5,1,2,6,3,4,7] => [5,3,2,1,6,4,7] => ? = 3 - 1
{{1,2,3,5,7},{4},{6}}
=> [2,3,5,4,7,6,1] => [7,1,2,4,3,6,5] => [4,6,7,5,3,2,1] => ? = 3 - 1
{{1,2,3,5},{4,7},{6}}
=> [2,3,5,7,1,6,4] => [5,1,2,7,3,6,4] => [5,3,2,1,6,7,4] => ? = 3 - 1
{{1,2,3,5},{4},{6,7}}
=> [2,3,5,4,1,7,6] => [5,1,2,4,3,7,6] => [4,5,3,2,1,7,6] => ? = 4 - 1
{{1,2,3,5},{4},{6},{7}}
=> [2,3,5,4,1,6,7] => [5,1,2,4,3,6,7] => [4,5,3,2,1,6,7] => ? = 3 - 1
{{1,2,3,6,7},{4,5}}
=> [2,3,6,5,4,7,1] => [7,1,2,5,4,3,6] => [5,4,7,6,3,2,1] => ? = 5 - 1
{{1,2,3,6},{4,5,7}}
=> [2,3,6,5,7,1,4] => [6,1,2,7,4,3,5] => [6,3,2,1,7,5,4] => ? = 4 - 1
{{1,2,3,7},{4,5,6}}
=> [2,3,7,5,6,4,1] => [7,1,2,6,4,5,3] => [6,5,4,7,3,2,1] => ? = 5 - 1
{{1,2,3},{4,5,6,7}}
=> [2,3,1,5,6,7,4] => [3,1,2,7,4,5,6] => [3,2,1,7,6,5,4] => ? = 6 - 1
{{1,2,3},{4,5,6},{7}}
=> [2,3,1,5,6,4,7] => [3,1,2,6,4,5,7] => [3,2,1,6,5,4,7] => ? = 5 - 1
{{1,2,3,7},{4,5},{6}}
=> [2,3,7,5,4,6,1] => [7,1,2,5,4,6,3] => [5,4,6,7,3,2,1] => ? = 4 - 1
{{1,2,3},{4,5,7},{6}}
=> [2,3,1,5,7,6,4] => [3,1,2,7,4,6,5] => [3,2,1,6,7,5,4] => ? = 4 - 1
{{1,2,3},{4,5},{6,7}}
=> [2,3,1,5,4,7,6] => [3,1,2,5,4,7,6] => [3,2,1,5,4,7,6] => ? = 5 - 1
{{1,2,3},{4,5},{6},{7}}
=> [2,3,1,5,4,6,7] => [3,1,2,5,4,6,7] => [3,2,1,5,4,6,7] => ? = 4 - 1
{{1,2,3,6,7},{4},{5}}
=> [2,3,6,4,5,7,1] => [7,1,2,4,5,3,6] => [4,5,7,6,3,2,1] => ? = 4 - 1
{{1,2,3,6},{4,7},{5}}
=> [2,3,6,7,5,1,4] => [6,1,2,7,5,3,4] => [5,6,3,2,1,7,4] => ? = 3 - 1
{{1,2,3,6},{4},{5,7}}
=> [2,3,6,4,7,1,5] => [6,1,2,4,7,3,5] => [4,6,3,2,1,7,5] => ? = 3 - 1
{{1,2,3,7},{4,6},{5}}
=> [2,3,7,6,5,4,1] => [7,1,2,6,5,4,3] => [5,6,4,7,3,2,1] => ? = 3 - 1
{{1,2,3},{4,6,7},{5}}
=> [2,3,1,6,5,7,4] => [3,1,2,7,5,4,6] => [3,2,1,5,7,6,4] => ? = 4 - 1
{{1,2,3},{4,6},{5,7}}
=> [2,3,1,6,7,4,5] => [3,1,2,6,7,4,5] => [3,2,1,6,4,7,5] => ? = 3 - 1
{{1,2,3},{4,6},{5},{7}}
=> [2,3,1,6,5,4,7] => [3,1,2,6,5,4,7] => [3,2,1,5,6,4,7] => ? = 3 - 1
{{1,2,3,7},{4},{5,6}}
=> [2,3,7,4,6,5,1] => [7,1,2,4,6,5,3] => [4,6,5,7,3,2,1] => ? = 4 - 1
{{1,2,3},{4,7},{5,6}}
=> [2,3,1,7,6,5,4] => [3,1,2,7,6,5,4] => [3,2,1,6,5,7,4] => ? = 4 - 1
{{1,2,3},{4},{5,6,7}}
=> [2,3,1,4,6,7,5] => [3,1,2,4,7,5,6] => [3,2,1,4,7,6,5] => ? = 5 - 1
{{1,2,3},{4},{5,6},{7}}
=> [2,3,1,4,6,5,7] => [3,1,2,4,6,5,7] => [3,2,1,4,6,5,7] => ? = 4 - 1
{{1,2,3,7},{4},{5},{6}}
=> [2,3,7,4,5,6,1] => [7,1,2,4,5,6,3] => [4,5,6,7,3,2,1] => ? = 3 - 1
{{1,2,3},{4,7},{5},{6}}
=> [2,3,1,7,5,6,4] => [3,1,2,7,5,6,4] => [3,2,1,5,6,7,4] => ? = 3 - 1
{{1,2,3},{4},{5,7},{6}}
=> [2,3,1,4,7,6,5] => [3,1,2,4,7,6,5] => [3,2,1,4,6,7,5] => ? = 3 - 1
{{1,2,3},{4},{5},{6,7}}
=> [2,3,1,4,5,7,6] => [3,1,2,4,5,7,6] => [3,2,1,4,5,7,6] => ? = 4 - 1
{{1,2,4,5,6,7},{3}}
=> [2,4,3,5,6,7,1] => [7,1,3,2,4,5,6] => [3,7,6,5,4,2,1] => ? = 5 - 1
{{1,2,4,5,6},{3,7}}
=> [2,4,7,5,6,1,3] => [6,1,7,2,4,5,3] => [6,5,4,2,1,7,3] => ? = 4 - 1
{{1,2,4,5,7},{3,6}}
=> [2,4,6,5,7,3,1] => [7,1,6,2,4,3,5] => [6,3,7,5,4,2,1] => ? = 3 - 1
{{1,2,4,5},{3,6,7}}
=> [2,4,6,5,1,7,3] => [5,1,7,2,4,3,6] => [5,4,2,1,7,6,3] => ? = 4 - 1
{{1,2,4,5},{3,6},{7}}
=> [2,4,6,5,1,3,7] => [5,1,6,2,4,3,7] => [5,4,2,1,6,3,7] => ? = 3 - 1
{{1,2,4,5,7},{3},{6}}
=> [2,4,3,5,7,6,1] => [7,1,3,2,4,6,5] => [3,6,7,5,4,2,1] => ? = 3 - 1
{{1,2,4,5},{3,7},{6}}
=> [2,4,7,5,1,6,3] => [5,1,7,2,4,6,3] => [5,4,2,1,6,7,3] => ? = 3 - 1
{{1,2,4,5},{3},{6,7}}
=> [2,4,3,5,1,7,6] => [5,1,3,2,4,7,6] => [3,5,4,2,1,7,6] => ? = 4 - 1
Description
The number of adjacencies of a permutation.
An adjacency of a permutation $\pi$ is an index $i$ such that $\pi(i)-1 = \pi(i+1)$. Adjacencies are also known as ''small descents''.
This can be also described as an occurrence of the bivincular pattern ([2,1], {((0,1),(1,0),(1,1),(1,2),(2,1)}), i.e., the middle row and the middle column are shaded, see [3].
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