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Your data matches 8 different statistics following compositions of up to 3 maps.
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Matching statistic: St000561
(load all 33 compositions to match this statistic)
(load all 33 compositions to match this statistic)
St000561: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> 0
{{1},{2}}
=> 0
{{1,2,3}}
=> 1
{{1,2},{3}}
=> 0
{{1,3},{2}}
=> 0
{{1},{2,3}}
=> 0
{{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> 4
{{1,2,3},{4}}
=> 1
{{1,2,4},{3}}
=> 1
{{1,2},{3,4}}
=> 0
{{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> 1
{{1,3},{2,4}}
=> 0
{{1,3},{2},{4}}
=> 0
{{1,4},{2,3}}
=> 0
{{1},{2,3,4}}
=> 1
{{1},{2,3},{4}}
=> 0
{{1,4},{2},{3}}
=> 0
{{1},{2,4},{3}}
=> 0
{{1},{2},{3,4}}
=> 0
{{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> 10
{{1,2,3,4},{5}}
=> 4
{{1,2,3,5},{4}}
=> 4
{{1,2,3},{4,5}}
=> 1
{{1,2,3},{4},{5}}
=> 1
{{1,2,4,5},{3}}
=> 4
{{1,2,4},{3,5}}
=> 1
{{1,2,4},{3},{5}}
=> 1
{{1,2,5},{3,4}}
=> 1
{{1,2},{3,4,5}}
=> 1
{{1,2},{3,4},{5}}
=> 0
{{1,2,5},{3},{4}}
=> 1
{{1,2},{3,5},{4}}
=> 0
{{1,2},{3},{4,5}}
=> 0
{{1,2},{3},{4},{5}}
=> 0
{{1,3,4,5},{2}}
=> 4
{{1,3,4},{2,5}}
=> 1
{{1,3,4},{2},{5}}
=> 1
{{1,3,5},{2,4}}
=> 1
{{1,3},{2,4,5}}
=> 1
{{1,3},{2,4},{5}}
=> 0
{{1,3,5},{2},{4}}
=> 1
{{1,3},{2,5},{4}}
=> 0
{{1,3},{2},{4,5}}
=> 0
{{1,3},{2},{4},{5}}
=> 0
{{1,4,5},{2,3}}
=> 1
{{1,4},{2,3,5}}
=> 1
{{1,4},{2,3},{5}}
=> 0
Description
The number of occurrences of the pattern {{1,2,3}} in a set partition.
Matching statistic: St000002
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000002: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000002: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> [[1,2]]
=> [1,2] => 0
{{1},{2}}
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
{{1,2,3}}
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 1
{{1,2},{3}}
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
{{1,3},{2}}
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
{{1},{2,3}}
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
{{1},{2},{3}}
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
{{1,2,3,4}}
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 4
{{1,2,3},{4}}
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
{{1,2,4},{3}}
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
{{1,2},{3,4}}
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
{{1,2},{3},{4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
{{1,3,4},{2}}
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
{{1,3},{2,4}}
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
{{1,3},{2},{4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
{{1,4},{2,3}}
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
{{1},{2,3,4}}
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
{{1,4},{2},{3}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
{{1},{2,4},{3}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
{{1},{2},{3,4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
{{1,2,3,4,5}}
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 10
{{1,2,3,4},{5}}
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 4
{{1,2,3,5},{4}}
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 4
{{1,2,3},{4,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
{{1,2,4,5},{3}}
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 4
{{1,2,4},{3,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
{{1,2,5},{3,4}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1
{{1,2},{3,4,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 0
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 0
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0
{{1,3,4,5},{2}}
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 4
{{1,3,4},{2,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
{{1,3,5},{2,4}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1
{{1,3},{2,4,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 0
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 0
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0
{{1,4,5},{2,3}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1
{{1,4},{2,3,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 0
{{1},{2,3,5,6,7,8},{4}}
=> ?
=> ?
=> ? => ? = 20
{{1},{2,3,4,5,6,7},{8}}
=> ?
=> ?
=> ? => ? = 20
{{1,4,5,6,7,8},{2},{3}}
=> ?
=> ?
=> ? => ? = 20
{{1,3,5,6,7,8},{2},{4}}
=> ?
=> ?
=> ? => ? = 20
{{1,2,5,6,7,8},{3,4}}
=> ?
=> ?
=> ? => ? = 20
{{1,2,3,4,7,8},{5,6}}
=> ?
=> ?
=> ? => ? = 20
{{1,3,4,5,7,8},{2,6}}
=> ?
=> ?
=> ? => ? = 20
{{1,3,4,5,6,8},{2,7}}
=> ?
=> ?
=> ? => ? = 20
{{1,3,4,5,6,7},{2,8}}
=> ?
=> ?
=> ? => ? = 20
{{1,2,3,5,6,7},{4,8}}
=> ?
=> ?
=> ? => ? = 20
{{1,2,3,4,6,7},{5,8}}
=> ?
=> ?
=> ? => ? = 20
{{1,2,3,4,5,7},{6,8}}
=> ?
=> ?
=> ? => ? = 20
{{1,7},{2,3,4,5,6,8}}
=> ?
=> ?
=> ? => ? = 20
Description
The number of occurrences of the pattern 123 in a permutation.
Matching statistic: St000436
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000436: Permutations ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 67%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000436: Permutations ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 67%
Values
{{1,2}}
=> [2] => [1,1,0,0]
=> [2,1] => 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [1,2] => 0
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [3,2,1] => 1
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 0
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 10
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
{{1,4},{2,3},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,2,3,4,5,6,7}}
=> [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => ? = 35
{{1,2,3,4,5,6},{7}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 20
{{1,2,3,4,5,7},{6}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 20
{{1,2,3,4,5},{6,7}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,4,3,2,1,7,6] => ? = 10
{{1,2,3,4,5},{6},{7}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,4,3,2,1,6,7] => ? = 10
{{1,2,3,4,6,7},{5}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 20
{{1,2,3,4,6},{5,7}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,4,3,2,1,7,6] => ? = 10
{{1,2,3,4,6},{5},{7}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,4,3,2,1,6,7] => ? = 10
{{1,2,3,4,7},{5,6}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,4,3,2,1,7,6] => ? = 10
{{1,2,3,4},{5,6,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 5
{{1,2,3,4},{5,6},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,3,4,7},{5},{6}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,4,3,2,1,6,7] => ? = 10
{{1,2,3,4},{5,7},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,3,4},{5},{6,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 4
{{1,2,3,4},{5},{6},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7] => ? = 4
{{1,2,3,5,6,7},{4}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 20
{{1,2,3,5,6},{4,7}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,4,3,2,1,7,6] => ? = 10
{{1,2,3,5,6},{4},{7}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,4,3,2,1,6,7] => ? = 10
{{1,2,3,5,7},{4,6}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,4,3,2,1,7,6] => ? = 10
{{1,2,3,5},{4,6,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 5
{{1,2,3,5},{4,6},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,3,5,7},{4},{6}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,4,3,2,1,6,7] => ? = 10
{{1,2,3,5},{4,7},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,3,5},{4},{6,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 4
{{1,2,3,5},{4},{6},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7] => ? = 4
{{1,2,3,6,7},{4,5}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,4,3,2,1,7,6] => ? = 10
{{1,2,3,6},{4,5,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 5
{{1,2,3,6},{4,5},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,3,7},{4,5,6}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 5
{{1,2,3},{4,5,6,7}}
=> [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,2,1,7,6,5,4] => ? = 5
{{1,2,3},{4,5,6},{7}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 2
{{1,2,3,7},{4,5},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,3},{4,5,7},{6}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 2
{{1,2,3},{4,5},{6,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 1
{{1,2,3},{4,5},{6},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,2,1,5,4,6,7] => ? = 1
{{1,2,3,6,7},{4},{5}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,4,3,2,1,6,7] => ? = 10
{{1,2,3,6},{4,7},{5}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,3,6},{4},{5,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 4
{{1,2,3,6},{4},{5},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7] => ? = 4
{{1,2,3,7},{4,6},{5}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,3},{4,6,7},{5}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 2
{{1,2,3},{4,6},{5,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 1
{{1,2,3},{4,6},{5},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,2,1,5,4,6,7] => ? = 1
{{1,2,3,7},{4},{5,6}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 4
{{1,2,3},{4,7},{5,6}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 1
{{1,2,3},{4},{5,6,7}}
=> [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,2,1,4,7,6,5] => ? = 2
{{1,2,3},{4},{5,6},{7}}
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [3,2,1,4,6,5,7] => ? = 1
{{1,2,3,7},{4},{5},{6}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7] => ? = 4
{{1,2,3},{4,7},{5},{6}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,2,1,5,4,6,7] => ? = 1
{{1,2,3},{4},{5,7},{6}}
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [3,2,1,4,6,5,7] => ? = 1
Description
The number of occurrences of the pattern 231 or of the pattern 321 in a permutation.
Matching statistic: St000437
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000437: Permutations ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 67%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000437: Permutations ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 67%
Values
{{1,2}}
=> [2] => [1,1,0,0]
=> [2,1] => 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [1,2] => 0
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [3,2,1] => 1
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 0
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 10
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
{{1,4},{2,3},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,2,3,4,5,6,7}}
=> [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => ? = 35
{{1,2,3,4,5,6},{7}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 20
{{1,2,3,4,5,7},{6}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 20
{{1,2,3,4,5},{6,7}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,4,3,2,1,7,6] => ? = 10
{{1,2,3,4,5},{6},{7}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,4,3,2,1,6,7] => ? = 10
{{1,2,3,4,6,7},{5}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 20
{{1,2,3,4,6},{5,7}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,4,3,2,1,7,6] => ? = 10
{{1,2,3,4,6},{5},{7}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,4,3,2,1,6,7] => ? = 10
{{1,2,3,4,7},{5,6}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,4,3,2,1,7,6] => ? = 10
{{1,2,3,4},{5,6,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 5
{{1,2,3,4},{5,6},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,3,4,7},{5},{6}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,4,3,2,1,6,7] => ? = 10
{{1,2,3,4},{5,7},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,3,4},{5},{6,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 4
{{1,2,3,4},{5},{6},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7] => ? = 4
{{1,2,3,5,6,7},{4}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 20
{{1,2,3,5,6},{4,7}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,4,3,2,1,7,6] => ? = 10
{{1,2,3,5,6},{4},{7}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,4,3,2,1,6,7] => ? = 10
{{1,2,3,5,7},{4,6}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,4,3,2,1,7,6] => ? = 10
{{1,2,3,5},{4,6,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 5
{{1,2,3,5},{4,6},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,3,5,7},{4},{6}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,4,3,2,1,6,7] => ? = 10
{{1,2,3,5},{4,7},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,3,5},{4},{6,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 4
{{1,2,3,5},{4},{6},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7] => ? = 4
{{1,2,3,6,7},{4,5}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,4,3,2,1,7,6] => ? = 10
{{1,2,3,6},{4,5,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 5
{{1,2,3,6},{4,5},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,3,7},{4,5,6}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 5
{{1,2,3},{4,5,6,7}}
=> [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,2,1,7,6,5,4] => ? = 5
{{1,2,3},{4,5,6},{7}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 2
{{1,2,3,7},{4,5},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,3},{4,5,7},{6}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 2
{{1,2,3},{4,5},{6,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 1
{{1,2,3},{4,5},{6},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,2,1,5,4,6,7] => ? = 1
{{1,2,3,6,7},{4},{5}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,4,3,2,1,6,7] => ? = 10
{{1,2,3,6},{4,7},{5}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,3,6},{4},{5,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 4
{{1,2,3,6},{4},{5},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7] => ? = 4
{{1,2,3,7},{4,6},{5}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,3},{4,6,7},{5}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 2
{{1,2,3},{4,6},{5,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 1
{{1,2,3},{4,6},{5},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,2,1,5,4,6,7] => ? = 1
{{1,2,3,7},{4},{5,6}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 4
{{1,2,3},{4,7},{5,6}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 1
{{1,2,3},{4},{5,6,7}}
=> [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,2,1,4,7,6,5] => ? = 2
{{1,2,3},{4},{5,6},{7}}
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [3,2,1,4,6,5,7] => ? = 1
{{1,2,3,7},{4},{5},{6}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7] => ? = 4
{{1,2,3},{4,7},{5},{6}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,2,1,5,4,6,7] => ? = 1
{{1,2,3},{4},{5,7},{6}}
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [3,2,1,4,6,5,7] => ? = 1
Description
The number of occurrences of the pattern 312 or of the pattern 321 in a permutation.
Matching statistic: St001411
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001411: Permutations ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 67%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001411: Permutations ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 67%
Values
{{1,2}}
=> [2] => [1,1,0,0]
=> [2,1] => 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [1,2] => 0
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [3,2,1] => 1
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 0
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 10
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
{{1,4},{2,3},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,2,3,4,5,6,7}}
=> [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => ? = 35
{{1,2,3,4,5,6},{7}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 20
{{1,2,3,4,5,7},{6}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 20
{{1,2,3,4,5},{6,7}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,4,3,2,1,7,6] => ? = 10
{{1,2,3,4,5},{6},{7}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,4,3,2,1,6,7] => ? = 10
{{1,2,3,4,6,7},{5}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 20
{{1,2,3,4,6},{5,7}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,4,3,2,1,7,6] => ? = 10
{{1,2,3,4,6},{5},{7}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,4,3,2,1,6,7] => ? = 10
{{1,2,3,4,7},{5,6}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,4,3,2,1,7,6] => ? = 10
{{1,2,3,4},{5,6,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 5
{{1,2,3,4},{5,6},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,3,4,7},{5},{6}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,4,3,2,1,6,7] => ? = 10
{{1,2,3,4},{5,7},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,3,4},{5},{6,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 4
{{1,2,3,4},{5},{6},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7] => ? = 4
{{1,2,3,5,6,7},{4}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 20
{{1,2,3,5,6},{4,7}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,4,3,2,1,7,6] => ? = 10
{{1,2,3,5,6},{4},{7}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,4,3,2,1,6,7] => ? = 10
{{1,2,3,5,7},{4,6}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,4,3,2,1,7,6] => ? = 10
{{1,2,3,5},{4,6,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 5
{{1,2,3,5},{4,6},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,3,5,7},{4},{6}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,4,3,2,1,6,7] => ? = 10
{{1,2,3,5},{4,7},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,3,5},{4},{6,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 4
{{1,2,3,5},{4},{6},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7] => ? = 4
{{1,2,3,6,7},{4,5}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,4,3,2,1,7,6] => ? = 10
{{1,2,3,6},{4,5,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 5
{{1,2,3,6},{4,5},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,3,7},{4,5,6}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 5
{{1,2,3},{4,5,6,7}}
=> [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,2,1,7,6,5,4] => ? = 5
{{1,2,3},{4,5,6},{7}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 2
{{1,2,3,7},{4,5},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,3},{4,5,7},{6}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 2
{{1,2,3},{4,5},{6,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 1
{{1,2,3},{4,5},{6},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,2,1,5,4,6,7] => ? = 1
{{1,2,3,6,7},{4},{5}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,4,3,2,1,6,7] => ? = 10
{{1,2,3,6},{4,7},{5}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,3,6},{4},{5,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 4
{{1,2,3,6},{4},{5},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7] => ? = 4
{{1,2,3,7},{4,6},{5}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,3},{4,6,7},{5}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 2
{{1,2,3},{4,6},{5,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 1
{{1,2,3},{4,6},{5},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,2,1,5,4,6,7] => ? = 1
{{1,2,3,7},{4},{5,6}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 4
{{1,2,3},{4,7},{5,6}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 1
{{1,2,3},{4},{5,6,7}}
=> [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,2,1,4,7,6,5] => ? = 2
{{1,2,3},{4},{5,6},{7}}
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [3,2,1,4,6,5,7] => ? = 1
{{1,2,3,7},{4},{5},{6}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7] => ? = 4
{{1,2,3},{4,7},{5},{6}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,2,1,5,4,6,7] => ? = 1
{{1,2,3},{4},{5,7},{6}}
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [3,2,1,4,6,5,7] => ? = 1
Description
The number of patterns 321 or 3412 in a permutation.
A permutation is '''boolean''' if its principal order ideal in the (strong) Bruhat order is boolean.
It is shown in [1, Theorem 5.3] that a permutation is boolean if and only if it avoids the two patterns 321 and 3412.
Matching statistic: St000119
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000119: Permutations ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 89%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000119: Permutations ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 89%
Values
{{1,2}}
=> [2] => [1,1,0,0]
=> [2,1] => 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [1,2] => 0
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [3,2,1] => 1
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 0
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 10
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
{{1,4},{2,3},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
{{1,2,3,4},{5,6,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 5
{{1,2,3,4},{5,6},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,3,4},{5,7},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,3,4},{5},{6,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 4
{{1,2,3,5},{4,6,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 5
{{1,2,3,5},{4,6},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,3,5},{4,7},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,3,5},{4},{6,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 4
{{1,2,3,6},{4,5,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 5
{{1,2,3,6},{4,5},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,3,7},{4,5,6}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 5
{{1,2,3},{4,5,6,7}}
=> [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,2,1,7,6,5,4] => ? = 5
{{1,2,3},{4,5,6},{7}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 2
{{1,2,3,7},{4,5},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,3},{4,5,7},{6}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 2
{{1,2,3},{4,5},{6,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 1
{{1,2,3},{4,5},{6},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,2,1,5,4,6,7] => ? = 1
{{1,2,3,6},{4,7},{5}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,3,6},{4},{5,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 4
{{1,2,3,7},{4,6},{5}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,3},{4,6,7},{5}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 2
{{1,2,3},{4,6},{5,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 1
{{1,2,3},{4,6},{5},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,2,1,5,4,6,7] => ? = 1
{{1,2,3,7},{4},{5,6}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 4
{{1,2,3},{4,7},{5,6}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 1
{{1,2,3},{4},{5,6,7}}
=> [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,2,1,4,7,6,5] => ? = 2
{{1,2,3},{4},{5,6},{7}}
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [3,2,1,4,6,5,7] => ? = 1
{{1,2,3},{4,7},{5},{6}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,2,1,5,4,6,7] => ? = 1
{{1,2,3},{4},{5,7},{6}}
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [3,2,1,4,6,5,7] => ? = 1
{{1,2,3},{4},{5},{6,7}}
=> [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [3,2,1,4,5,7,6] => ? = 1
{{1,2,4,5},{3,6,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 5
{{1,2,4,5},{3,6},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,4,5},{3,7},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,4,5},{3},{6,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 4
{{1,2,4,6},{3,5,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 5
{{1,2,4,6},{3,5},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,4,7},{3,5,6}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 5
{{1,2,4},{3,5,6,7}}
=> [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,2,1,7,6,5,4] => ? = 5
{{1,2,4},{3,5,6},{7}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 2
{{1,2,4,7},{3,5},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,4},{3,5,7},{6}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 2
{{1,2,4},{3,5},{6,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 1
{{1,2,4},{3,5},{6},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,2,1,5,4,6,7] => ? = 1
{{1,2,4,6},{3,7},{5}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,4,6},{3},{5,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 4
{{1,2,4,7},{3,6},{5}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 4
{{1,2,4},{3,6,7},{5}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 2
{{1,2,4},{3,6},{5,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 1
{{1,2,4},{3,6},{5},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,2,1,5,4,6,7] => ? = 1
{{1,2,4,7},{3},{5,6}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 4
Description
The number of occurrences of the pattern 321 in a permutation.
Matching statistic: St000423
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000423: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 78%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000423: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 78%
Values
{{1,2}}
=> [2] => [1,1,0,0]
=> [1,2] => 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [2,1] => 0
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [1,2,3] => 1
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 0
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 0
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [2,3,1] => 0
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [3,2,1] => 0
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 0
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 0
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 0
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 0
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 0
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 0
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 0
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 10
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 0
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 0
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 0
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 0
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 0
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 0
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1
{{1,4},{2,3},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 0
{{1,2,3,4,5,6},{7}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 20
{{1,2,3,4,5,7},{6}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 20
{{1,2,3,4,5},{6,7}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => ? = 10
{{1,2,3,4,5},{6},{7}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,2,3,4,5] => ? = 10
{{1,2,3,4,6,7},{5}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 20
{{1,2,3,4,6},{5,7}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => ? = 10
{{1,2,3,4,6},{5},{7}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,2,3,4,5] => ? = 10
{{1,2,3,4,7},{5,6}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => ? = 10
{{1,2,3,4},{5,6,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [5,6,7,1,2,3,4] => ? = 5
{{1,2,3,4},{5,6},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [7,5,6,1,2,3,4] => ? = 4
{{1,2,3,4,7},{5},{6}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,2,3,4,5] => ? = 10
{{1,2,3,4},{5,7},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [7,5,6,1,2,3,4] => ? = 4
{{1,2,3,4},{5},{6,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [6,7,5,1,2,3,4] => ? = 4
{{1,2,3,4},{5},{6},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5,1,2,3,4] => ? = 4
{{1,2,3,5,6,7},{4}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 20
{{1,2,3,5,6},{4,7}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => ? = 10
{{1,2,3,5,6},{4},{7}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,2,3,4,5] => ? = 10
{{1,2,3,5,7},{4,6}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => ? = 10
{{1,2,3,5},{4,6,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [5,6,7,1,2,3,4] => ? = 5
{{1,2,3,5},{4,6},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [7,5,6,1,2,3,4] => ? = 4
{{1,2,3,5,7},{4},{6}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,2,3,4,5] => ? = 10
{{1,2,3,5},{4,7},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [7,5,6,1,2,3,4] => ? = 4
{{1,2,3,5},{4},{6,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [6,7,5,1,2,3,4] => ? = 4
{{1,2,3,5},{4},{6},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5,1,2,3,4] => ? = 4
{{1,2,3,6,7},{4,5}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => ? = 10
{{1,2,3,6},{4,5,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [5,6,7,1,2,3,4] => ? = 5
{{1,2,3,6},{4,5},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [7,5,6,1,2,3,4] => ? = 4
{{1,2,3,7},{4,5,6}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [5,6,7,1,2,3,4] => ? = 5
{{1,2,3},{4,5,6,7}}
=> [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,1,2,3] => ? = 5
{{1,2,3},{4,5,6},{7}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,1,2,3] => ? = 2
{{1,2,3,7},{4,5},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [7,5,6,1,2,3,4] => ? = 4
{{1,2,3},{4,5,7},{6}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,1,2,3] => ? = 2
{{1,2,3},{4,5},{6,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,1,2,3] => ? = 1
{{1,2,3},{4,5},{6},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,1,2,3] => ? = 1
{{1,2,3,6,7},{4},{5}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,2,3,4,5] => ? = 10
{{1,2,3,6},{4,7},{5}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [7,5,6,1,2,3,4] => ? = 4
{{1,2,3,6},{4},{5,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [6,7,5,1,2,3,4] => ? = 4
{{1,2,3,6},{4},{5},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5,1,2,3,4] => ? = 4
{{1,2,3,7},{4,6},{5}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [7,5,6,1,2,3,4] => ? = 4
{{1,2,3},{4,6,7},{5}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,1,2,3] => ? = 2
{{1,2,3},{4,6},{5,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,1,2,3] => ? = 1
{{1,2,3},{4,6},{5},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,1,2,3] => ? = 1
{{1,2,3,7},{4},{5,6}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [6,7,5,1,2,3,4] => ? = 4
{{1,2,3},{4,7},{5,6}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,1,2,3] => ? = 1
{{1,2,3},{4},{5,6,7}}
=> [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,1,2,3] => ? = 2
{{1,2,3},{4},{5,6},{7}}
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,1,2,3] => ? = 1
{{1,2,3,7},{4},{5},{6}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5,1,2,3,4] => ? = 4
{{1,2,3},{4,7},{5},{6}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,1,2,3] => ? = 1
{{1,2,3},{4},{5,7},{6}}
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,1,2,3] => ? = 1
{{1,2,3},{4},{5},{6,7}}
=> [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,1,2,3] => ? = 1
Description
The number of occurrences of the pattern 123 or of the pattern 132 in a permutation.
Matching statistic: St000428
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000428: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 78%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000428: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 78%
Values
{{1,2}}
=> [2] => [1,1,0,0]
=> [1,2] => 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [2,1] => 0
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [1,2,3] => 1
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 0
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 0
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [2,3,1] => 0
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [3,2,1] => 0
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 0
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 0
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 0
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 0
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 0
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 0
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 0
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 10
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 0
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 0
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 0
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 0
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 0
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 0
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1
{{1,4},{2,3},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 0
{{1,2,3,4,5,6},{7}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 20
{{1,2,3,4,5,7},{6}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 20
{{1,2,3,4,5},{6,7}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => ? = 10
{{1,2,3,4,5},{6},{7}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,2,3,4,5] => ? = 10
{{1,2,3,4,6,7},{5}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 20
{{1,2,3,4,6},{5,7}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => ? = 10
{{1,2,3,4,6},{5},{7}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,2,3,4,5] => ? = 10
{{1,2,3,4,7},{5,6}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => ? = 10
{{1,2,3,4},{5,6,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [5,6,7,1,2,3,4] => ? = 5
{{1,2,3,4},{5,6},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [7,5,6,1,2,3,4] => ? = 4
{{1,2,3,4,7},{5},{6}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,2,3,4,5] => ? = 10
{{1,2,3,4},{5,7},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [7,5,6,1,2,3,4] => ? = 4
{{1,2,3,4},{5},{6,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [6,7,5,1,2,3,4] => ? = 4
{{1,2,3,4},{5},{6},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5,1,2,3,4] => ? = 4
{{1,2,3,5,6,7},{4}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 20
{{1,2,3,5,6},{4,7}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => ? = 10
{{1,2,3,5,6},{4},{7}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,2,3,4,5] => ? = 10
{{1,2,3,5,7},{4,6}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => ? = 10
{{1,2,3,5},{4,6,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [5,6,7,1,2,3,4] => ? = 5
{{1,2,3,5},{4,6},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [7,5,6,1,2,3,4] => ? = 4
{{1,2,3,5,7},{4},{6}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,2,3,4,5] => ? = 10
{{1,2,3,5},{4,7},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [7,5,6,1,2,3,4] => ? = 4
{{1,2,3,5},{4},{6,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [6,7,5,1,2,3,4] => ? = 4
{{1,2,3,5},{4},{6},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5,1,2,3,4] => ? = 4
{{1,2,3,6,7},{4,5}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => ? = 10
{{1,2,3,6},{4,5,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [5,6,7,1,2,3,4] => ? = 5
{{1,2,3,6},{4,5},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [7,5,6,1,2,3,4] => ? = 4
{{1,2,3,7},{4,5,6}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [5,6,7,1,2,3,4] => ? = 5
{{1,2,3},{4,5,6,7}}
=> [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,1,2,3] => ? = 5
{{1,2,3},{4,5,6},{7}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,1,2,3] => ? = 2
{{1,2,3,7},{4,5},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [7,5,6,1,2,3,4] => ? = 4
{{1,2,3},{4,5,7},{6}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,1,2,3] => ? = 2
{{1,2,3},{4,5},{6,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,1,2,3] => ? = 1
{{1,2,3},{4,5},{6},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,1,2,3] => ? = 1
{{1,2,3,6,7},{4},{5}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,2,3,4,5] => ? = 10
{{1,2,3,6},{4,7},{5}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [7,5,6,1,2,3,4] => ? = 4
{{1,2,3,6},{4},{5,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [6,7,5,1,2,3,4] => ? = 4
{{1,2,3,6},{4},{5},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5,1,2,3,4] => ? = 4
{{1,2,3,7},{4,6},{5}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [7,5,6,1,2,3,4] => ? = 4
{{1,2,3},{4,6,7},{5}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,1,2,3] => ? = 2
{{1,2,3},{4,6},{5,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,1,2,3] => ? = 1
{{1,2,3},{4,6},{5},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,1,2,3] => ? = 1
{{1,2,3,7},{4},{5,6}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [6,7,5,1,2,3,4] => ? = 4
{{1,2,3},{4,7},{5,6}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,1,2,3] => ? = 1
{{1,2,3},{4},{5,6,7}}
=> [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,1,2,3] => ? = 2
{{1,2,3},{4},{5,6},{7}}
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,1,2,3] => ? = 1
{{1,2,3,7},{4},{5},{6}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5,1,2,3,4] => ? = 4
{{1,2,3},{4,7},{5},{6}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,1,2,3] => ? = 1
{{1,2,3},{4},{5,7},{6}}
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,1,2,3] => ? = 1
{{1,2,3},{4},{5},{6,7}}
=> [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,1,2,3] => ? = 1
Description
The number of occurrences of the pattern 123 or of the pattern 213 in a permutation.
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