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Your data matches 15 different statistics following compositions of up to 3 maps.
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Matching statistic: St000445
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> [1,1,0,0]
=> 0
{{1,2}}
=> [2] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 0
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 2
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 2
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 0
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 2
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 3
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 2
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 0
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 2
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 3
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 0
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 0
Description
The number of rises of length 1 of a Dyck path.
Matching statistic: St001484
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001484: Integer partitions ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001484: Integer partitions ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> []
=> 0
{{1,2}}
=> [2] => [1,1,0,0]
=> []
=> 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [1]
=> 1
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> []
=> 0
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [2]
=> 1
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [2]
=> 1
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [1,1]
=> 0
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [2,1]
=> 2
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> []
=> 0
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 0
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 2
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 0
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 2
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 0
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 2
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 0
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 2
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 2
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 0
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 2
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 3
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 2
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 0
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 2
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 3
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 0
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 0
{{1},{2},{3},{4},{5},{6,8},{7}}
=> [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,5,4,3,2,1]
=> ? = 5
{{1},{2},{3},{4},{5,6},{7},{8}}
=> [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,4,3,2,1]
=> ? = 5
{{1},{2},{3},{4},{5,7},{6},{8}}
=> [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,4,3,2,1]
=> ? = 5
{{1},{2},{3},{4},{5,8},{6},{7}}
=> [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,4,3,2,1]
=> ? = 5
{{1},{2},{3},{4},{5,7,8},{6}}
=> [1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,4,4,3,2,1]
=> ? = 4
{{1},{2},{3},{4},{5,6,7},{8}}
=> [1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,4,4,3,2,1]
=> ? = 4
{{1},{2},{3},{4},{5,6,8},{7}}
=> [1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,4,4,3,2,1]
=> ? = 4
{{1},{2},{3},{4,8},{5},{6},{7}}
=> [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,3,2,1]
=> ? = 5
{{1},{2,3},{4,5},{6,7},{8}}
=> [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,3,1,1]
=> ? = 1
{{1},{2,3},{4,5},{6,8},{7}}
=> [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,3,1,1]
=> ? = 1
{{1},{2,3},{4,5},{6,7,8}}
=> [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [5,5,5,3,3,1,1]
=> ? = 0
{{1},{2,3},{4,7},{5,6},{8}}
=> [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,3,1,1]
=> ? = 1
{{1},{2,3},{4,8},{5,6},{7}}
=> [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,3,1,1]
=> ? = 1
{{1},{2,3},{4,8},{5,7},{6}}
=> [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,3,1,1]
=> ? = 1
{{1},{2,3},{4,8},{5,6,7}}
=> [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [5,5,5,3,3,1,1]
=> ? = 0
{{1},{2,8},{3},{4},{5},{6},{7}}
=> [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,1,1]
=> ? = 5
{{1},{2,5},{3,4},{6,7},{8}}
=> [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,3,1,1]
=> ? = 1
{{1},{2,5},{3,4},{6,8},{7}}
=> [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,3,1,1]
=> ? = 1
{{1},{2,5},{3,4},{6,7,8}}
=> [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [5,5,5,3,3,1,1]
=> ? = 0
{{1},{2,7},{3,4},{5,6},{8}}
=> [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,3,1,1]
=> ? = 1
{{1},{2,8},{3,4},{5,6},{7}}
=> [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,3,1,1]
=> ? = 1
{{1},{2,8},{3,4},{5,7},{6}}
=> [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,3,1,1]
=> ? = 1
{{1},{2,8},{3,4},{5,6,7}}
=> [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [5,5,5,3,3,1,1]
=> ? = 0
{{1},{2,7},{3,6},{4,5},{8}}
=> [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,3,1,1]
=> ? = 1
{{1},{2,8},{3,6},{4,5},{7}}
=> [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,3,1,1]
=> ? = 1
{{1},{2,8},{3,7},{4,5},{6}}
=> [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,3,1,1]
=> ? = 1
{{1},{2,8},{3,7},{4,6},{5}}
=> [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,3,1,1]
=> ? = 1
{{1},{2,8},{3,7},{4,5,6}}
=> [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [5,5,5,3,3,1,1]
=> ? = 0
{{1},{2,6},{3,4},{5,7,8}}
=> [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [5,5,5,3,3,1,1]
=> ? = 0
{{1},{2,7},{3,4},{5,6,8}}
=> [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [5,5,5,3,3,1,1]
=> ? = 0
{{1},{2,4},{3,7},{5,6,8}}
=> [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [5,5,5,3,3,1,1]
=> ? = 0
{{1},{2,4},{3,6},{5,7,8}}
=> [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [5,5,5,3,3,1,1]
=> ? = 0
{{1},{2,4},{3,8},{5,6,7}}
=> [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [5,5,5,3,3,1,1]
=> ? = 0
{{1},{2},{3},{4,9},{5},{6},{7},{8}}
=> [1,1,1,2,1,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,3,3,2,1]
=> ? = 6
{{1,9},{2},{3},{4},{5},{6},{7},{8}}
=> [2,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,3,2]
=> ? = 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]
=> [5,5,5,3,3,1,1]
=> ? = 0
{{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]
=> [5,5,5,3,3,1,1]
=> ? = 0
{{1},{2,6},{3,5},{4,7,8}}
=> [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [5,5,5,3,3,1,1]
=> ? = 0
{{1},{2,6},{3,7},{4,5,8}}
=> [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [5,5,5,3,3,1,1]
=> ? = 0
{{1},{2,8},{3,6},{4,5,7}}
=> [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [5,5,5,3,3,1,1]
=> ? = 0
{{1},{2,8},{3,5},{4,6,7}}
=> [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [5,5,5,3,3,1,1]
=> ? = 0
{{1},{2,6},{3,7},{4,8},{5}}
=> [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,3,1,1]
=> ? = 1
{{1},{2,7},{3,8},{4,5,6}}
=> [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [5,5,5,3,3,1,1]
=> ? = 0
{{1},{2,5},{3,8},{4,6,7}}
=> [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [5,5,5,3,3,1,1]
=> ? = 0
{{1},{2,4},{3,5},{6,7},{8}}
=> [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,3,1,1]
=> ? = 1
{{1},{2,5},{3,6},{4,7},{8}}
=> [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,3,1,1]
=> ? = 1
{{1},{2,3},{4,7},{5,6,8}}
=> [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [5,5,5,3,3,1,1]
=> ? = 0
{{1},{2,6},{3,8},{4,5,7}}
=> [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [5,5,5,3,3,1,1]
=> ? = 0
{{1},{2,4},{3,6},{5,8},{7}}
=> [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,3,1,1]
=> ? = 1
{{1},{2,6},{3,5},{4,8},{7}}
=> [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,3,1,1]
=> ? = 1
Description
The number of singletons of an integer partition.
A singleton in an integer partition is a part that appear precisely once.
Matching statistic: St000932
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 95%●distinct values known / distinct values provided: 80%
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 95%●distinct values known / distinct values provided: 80%
Values
{{1}}
=> [1] => [1] => [1,0]
=> ? = 0
{{1,2}}
=> [2] => [2] => [1,1,0,0]
=> 0
{{1},{2}}
=> [1,1] => [1,1] => [1,0,1,0]
=> 1
{{1,2,3}}
=> [3] => [3] => [1,1,1,0,0,0]
=> 0
{{1,2},{3}}
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
{{1,3},{2}}
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
{{1},{2,3}}
=> [1,2] => [2,1] => [1,1,0,0,1,0]
=> 0
{{1},{2},{3}}
=> [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2
{{1,2,3,4}}
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
{{1,2,3},{4}}
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1,2,4},{3}}
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1,2},{3,4}}
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
{{1,3,4},{2}}
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1,3},{2,4}}
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
{{1,4},{2,3}}
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
{{1},{2,3,4}}
=> [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
{{1},{2,3},{4}}
=> [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
{{1},{2,4},{3}}
=> [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
{{1},{2},{3,4}}
=> [1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3
{{1,2,3,4,5}}
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
{{1,2,3,4},{5}}
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,3,5},{4}}
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,3},{4,5}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
{{1,2,4,5},{3}}
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,4},{3,5}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
{{1,2,5},{3,4}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
{{1,2},{3,4,5}}
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 3
{{1,3,4,5},{2}}
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,3,4},{2,5}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
{{1,3,5},{2,4}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
{{1,3},{2,4,5}}
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 3
{{1,4,5},{2,3}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
{{1,4},{2,3,5}}
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
{{1,4},{2,3},{5}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1},{2},{3},{4},{5},{6,7,8}}
=> [1,1,1,1,1,3] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4
{{1},{2},{3},{4},{5,6,7,8}}
=> [1,1,1,1,4] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
{{1},{2,3},{4,5},{6,7,8}}
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
{{1},{2,3},{4,8},{5,6,7}}
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
{{1},{2,5},{3,4},{6,7,8}}
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
{{1},{2,8},{3,4},{5,6,7}}
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
{{1},{2,8},{3,7},{4,5,6}}
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
{{1},{2,3,4,5,6,7,8}}
=> [1,7] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
{{1,2,3,4,5,6,7,8}}
=> [8] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
{{1},{2,6},{3,4},{5,7,8}}
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
{{1},{2,7},{3,4},{5,6,8}}
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
{{1},{2,4},{3,7},{5,6,8}}
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
{{1},{2,4},{3,6},{5,7,8}}
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
{{1},{2,4},{3,8},{5,6,7}}
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
{{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> [1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8
{{1},{2},{3},{4},{5},{6},{7},{8,9}}
=> [1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
{{1},{2},{3},{4,9},{5},{6},{7},{8}}
=> [1,1,1,2,1,1,1,1] => [1,1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 6
{{1,9},{2},{3},{4},{5},{6},{7},{8}}
=> [2,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 7
{{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> [1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
{{1},{2},{3},{4},{5},{6},{7},{8},{9,10}}
=> [1,1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
{{1},{2},{3},{4},{5},{6},{7,8,9}}
=> [1,1,1,1,1,1,3] => [3,1,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5
{{1,2,3,4,5,6,7,8},{9}}
=> [8,1] => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2,3,4,5,6,7,8,9}}
=> [1,8] => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 0
{{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1] => [1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2,3,4,5,6,7,8,9,10}}
=> [1,9] => [9,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 0
{{1,2,3,4,5,6,7,8,9,10},{11}}
=> [10,1] => [1,10] => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2,3,4,5,6,7,8,9,10,11}}
=> [1,10] => [10,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 0
{{1},{2,5},{3,7},{4,6,8}}
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
{{1},{2,5},{3,6},{4,7,8}}
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
{{1},{2,6},{3,5},{4,7,8}}
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
{{1},{2,6},{3,7},{4,5,8}}
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
{{1},{2,8},{3,6},{4,5,7}}
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
{{1},{2,8},{3,5},{4,6,7}}
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
{{1},{2,7},{3,8},{4,5,6}}
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
{{1},{2,5},{3,8},{4,6,7}}
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
{{1},{2,3},{4,7},{5,6,8}}
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
{{1},{2,6},{3,8},{4,5,7}}
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
{{1,2},{3},{4},{5},{6},{7},{8},{9}}
=> [2,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 7
{{1,3},{2},{4},{5},{6},{7},{8},{9}}
=> [2,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 7
{{1,4},{2},{3},{5},{6},{7},{8},{9}}
=> [2,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 7
{{1,5},{2},{3},{4},{6},{7},{8},{9}}
=> [2,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 7
{{1,6},{2},{3},{4},{5},{7},{8},{9}}
=> [2,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 7
{{1,7},{2},{3},{4},{5},{6},{8},{9}}
=> [2,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 7
{{1,8},{2},{3},{4},{5},{6},{7},{9}}
=> [2,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 7
{{1,2,3,4,5,6,7,8,9}}
=> [9] => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0
{{1,2,3,4,5,6,7},{8,9}}
=> [7,2] => [2,7] => [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
{{1,2,3,4,5,6,7,8,9,10}}
=> [10] => [10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 0
{{1},{2,3},{4,6},{5,7,8}}
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
{{1},{2,7},{3,6},{4,5,8}}
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
Description
The number of occurrences of the pattern UDU in a Dyck path.
The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
Matching statistic: St000382
Mp00221: Set partitions —conjugate⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> {{1}}
=> {{1}}
=> [1] => 1 = 0 + 1
{{1,2}}
=> {{1},{2}}
=> {{1},{2}}
=> [1,1] => 1 = 0 + 1
{{1},{2}}
=> {{1,2}}
=> {{1,2}}
=> [2] => 2 = 1 + 1
{{1,2,3}}
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> [1,1,1] => 1 = 0 + 1
{{1,2},{3}}
=> {{1,2},{3}}
=> {{1,2},{3}}
=> [2,1] => 2 = 1 + 1
{{1,3},{2}}
=> {{1},{2,3}}
=> {{1,3},{2}}
=> [2,1] => 2 = 1 + 1
{{1},{2,3}}
=> {{1,3},{2}}
=> {{1},{2,3}}
=> [1,2] => 1 = 0 + 1
{{1},{2},{3}}
=> {{1,2,3}}
=> {{1,2,3}}
=> [3] => 3 = 2 + 1
{{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [1,1,1,1] => 1 = 0 + 1
{{1,2,3},{4}}
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> [2,1,1] => 2 = 1 + 1
{{1,2,4},{3}}
=> {{1},{2,3},{4}}
=> {{1,3},{2},{4}}
=> [2,1,1] => 2 = 1 + 1
{{1,2},{3,4}}
=> {{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> [1,2,1] => 1 = 0 + 1
{{1,2},{3},{4}}
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> [3,1] => 3 = 2 + 1
{{1,3,4},{2}}
=> {{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> [2,1,1] => 2 = 1 + 1
{{1,3},{2,4}}
=> {{1,3},{2,4}}
=> {{1},{2,3,4}}
=> [1,3] => 1 = 0 + 1
{{1,3},{2},{4}}
=> {{1,2},{3,4}}
=> {{1,2,4},{3}}
=> [3,1] => 3 = 2 + 1
{{1,4},{2,3}}
=> {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> [1,2,1] => 1 = 0 + 1
{{1},{2,3,4}}
=> {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> [1,1,2] => 1 = 0 + 1
{{1},{2,3},{4}}
=> {{1,2,4},{3}}
=> {{1,2},{3,4}}
=> [2,2] => 2 = 1 + 1
{{1,4},{2},{3}}
=> {{1},{2,3,4}}
=> {{1,3,4},{2}}
=> [3,1] => 3 = 2 + 1
{{1},{2,4},{3}}
=> {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> [2,2] => 2 = 1 + 1
{{1},{2},{3,4}}
=> {{1,3,4},{2}}
=> {{1,4},{2,3}}
=> [2,2] => 2 = 1 + 1
{{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> [4] => 4 = 3 + 1
{{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => 1 = 0 + 1
{{1,2,3,4},{5}}
=> {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> [2,1,1,1] => 2 = 1 + 1
{{1,2,3,5},{4}}
=> {{1},{2,3},{4},{5}}
=> {{1,3},{2},{4},{5}}
=> [2,1,1,1] => 2 = 1 + 1
{{1,2,3},{4,5}}
=> {{1,3},{2},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> [1,2,1,1] => 1 = 0 + 1
{{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> [3,1,1] => 3 = 2 + 1
{{1,2,4,5},{3}}
=> {{1},{2},{3,4},{5}}
=> {{1,4},{2},{3},{5}}
=> [2,1,1,1] => 2 = 1 + 1
{{1,2,4},{3,5}}
=> {{1,3},{2,4},{5}}
=> {{1},{2,3,4},{5}}
=> [1,3,1] => 1 = 0 + 1
{{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> [3,1,1] => 3 = 2 + 1
{{1,2,5},{3,4}}
=> {{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> [1,2,1,1] => 1 = 0 + 1
{{1,2},{3,4,5}}
=> {{1,4},{2},{3},{5}}
=> {{1},{2},{3,4},{5}}
=> [1,1,2,1] => 1 = 0 + 1
{{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> [2,2,1] => 2 = 1 + 1
{{1,2,5},{3},{4}}
=> {{1},{2,3,4},{5}}
=> {{1,3,4},{2},{5}}
=> [3,1,1] => 3 = 2 + 1
{{1,2},{3,5},{4}}
=> {{1,4},{2,3},{5}}
=> {{1,3},{2,4},{5}}
=> [2,2,1] => 2 = 1 + 1
{{1,2},{3},{4,5}}
=> {{1,3,4},{2},{5}}
=> {{1,4},{2,3},{5}}
=> [2,2,1] => 2 = 1 + 1
{{1,2},{3},{4},{5}}
=> {{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> [4,1] => 4 = 3 + 1
{{1,3,4,5},{2}}
=> {{1},{2},{3},{4,5}}
=> {{1,5},{2},{3},{4}}
=> [2,1,1,1] => 2 = 1 + 1
{{1,3,4},{2,5}}
=> {{1,4},{2,5},{3}}
=> {{1},{2},{3,4,5}}
=> [1,1,3] => 1 = 0 + 1
{{1,3,4},{2},{5}}
=> {{1,2},{3},{4,5}}
=> {{1,2,5},{3},{4}}
=> [3,1,1] => 3 = 2 + 1
{{1,3,5},{2,4}}
=> {{1},{2,4},{3,5}}
=> {{1},{2,4,5},{3}}
=> [1,3,1] => 1 = 0 + 1
{{1,3},{2,4,5}}
=> {{1,4},{2},{3,5}}
=> {{1},{2,5},{3,4}}
=> [1,2,2] => 1 = 0 + 1
{{1,3},{2,4},{5}}
=> {{1,2,4},{3,5}}
=> {{1,2},{3,4,5}}
=> [2,3] => 2 = 1 + 1
{{1,3,5},{2},{4}}
=> {{1},{2,3},{4,5}}
=> {{1,3,5},{2},{4}}
=> [3,1,1] => 3 = 2 + 1
{{1,3},{2,5},{4}}
=> {{1,4},{2,3,5}}
=> {{1,3},{2,4,5}}
=> [2,3] => 2 = 1 + 1
{{1,3},{2},{4,5}}
=> {{1,3},{2},{4,5}}
=> {{1,5},{2,3},{4}}
=> [2,2,1] => 2 = 1 + 1
{{1,3},{2},{4},{5}}
=> {{1,2,3},{4,5}}
=> {{1,2,3,5},{4}}
=> [4,1] => 4 = 3 + 1
{{1,4,5},{2,3}}
=> {{1},{2},{3,5},{4}}
=> {{1},{2,5},{3},{4}}
=> [1,2,1,1] => 1 = 0 + 1
{{1,4},{2,3,5}}
=> {{1,3},{2,5},{4}}
=> {{1},{2,3},{4,5}}
=> [1,2,2] => 1 = 0 + 1
{{1},{2},{3},{4},{5},{6,7,8}}
=> {{1,4,5,6,7,8},{2},{3}}
=> {{1,5,6,7,8},{2},{3,4}}
=> ? => ? = 4 + 1
{{1},{2},{3},{4},{5,7,8},{6}}
=> {{1,5,6,7,8},{2},{3,4}}
=> ?
=> ? => ? = 4 + 1
{{1},{2},{3},{4},{5,6,7},{8}}
=> {{1,2,5,6,7,8},{3},{4}}
=> {{1,2,6,7,8},{3},{4,5}}
=> ? => ? = 4 + 1
{{1},{2},{3},{4},{5,6,8},{7}}
=> {{1,5,6,7,8},{2,3},{4}}
=> {{1,3,6,7,8},{2},{4,5}}
=> ? => ? = 4 + 1
{{1},{2,3},{4,8},{5,6},{7}}
=> ?
=> ?
=> ? => ? = 1 + 1
{{1},{2,3},{4,8},{5,6,7}}
=> {{1,6,8},{2,5},{3},{4},{7}}
=> ?
=> ? => ? = 0 + 1
{{1},{2,5},{3,4},{6,7,8}}
=> {{1,4,8},{2},{3},{5,7},{6}}
=> ?
=> ? => ? = 0 + 1
{{1},{2,8},{3,4},{5,6},{7}}
=> ?
=> ?
=> ? => ? = 1 + 1
{{1},{2,8},{3,7},{4,5,6}}
=> {{1,8},{2,7},{3,6},{4},{5}}
=> ?
=> ? => ? = 0 + 1
{{1},{2,3,4,5,7,8},{6}}
=> {{1,8},{2},{3,4},{5},{6},{7}}
=> {{1,4},{2},{3},{5},{6},{7,8}}
=> ? => ? = 1 + 1
{{1,2},{3,5,6,7,8},{4}}
=> {{1,7},{2},{3},{4},{5,6},{8}}
=> {{1,6},{2},{3},{4},{5,7},{8}}
=> ? => ? = 1 + 1
{{1,2},{3,4,6,7,8},{5}}
=> {{1,7},{2},{3},{4,5},{6},{8}}
=> ?
=> ? => ? = 1 + 1
{{1,2},{3,4,5,7,8},{6}}
=> {{1,7},{2},{3,4},{5},{6},{8}}
=> ?
=> ? => ? = 1 + 1
{{1,2},{3,4,5,6,8},{7}}
=> {{1,7},{2,3},{4},{5},{6},{8}}
=> {{1,3},{2},{4},{5},{6,7},{8}}
=> ? => ? = 1 + 1
{{1,3,4,5,7,8},{2},{6}}
=> {{1},{2},{3,4},{5},{6},{7,8}}
=> {{1,4,8},{2},{3},{5},{6},{7}}
=> ? => ? = 2 + 1
{{1,2,4,6,7,8},{3},{5}}
=> {{1},{2},{3},{4,5},{6,7},{8}}
=> ?
=> ? => ? = 2 + 1
{{1,2,4,5,7,8},{3},{6}}
=> {{1},{2},{3,4},{5},{6,7},{8}}
=> ?
=> ? => ? = 2 + 1
{{1,8},{2,4,5,6,7},{3}}
=> {{1},{2,8},{3},{4},{5},{6,7}}
=> ?
=> ? => ? = 1 + 1
{{1,2,4,5,6,7,8},{3}}
=> {{1},{2},{3},{4},{5},{6,7},{8}}
=> ?
=> ? => ? = 1 + 1
{{1,2,5,6,7,8},{3,4}}
=> {{1},{2},{3},{4},{5,7},{6},{8}}
=> {{1},{2,7},{3},{4},{5},{6},{8}}
=> ? => ? = 0 + 1
{{1,2,3,6,7,8},{4},{5}}
=> {{1},{2},{3},{4,5,6},{7},{8}}
=> ?
=> ? => ? = 2 + 1
{{1,8},{2,3,5,6,7},{4}}
=> {{1},{2,8},{3},{4},{5,6},{7}}
=> ?
=> ? => ? = 1 + 1
{{1,2,3,5,6,8},{4},{7}}
=> {{1},{2,3},{4},{5,6},{7},{8}}
=> ?
=> ? => ? = 2 + 1
{{1,2,3,6,7,8},{4,5}}
=> {{1},{2},{3},{4,6},{5},{7},{8}}
=> ?
=> ? => ? = 0 + 1
{{1,8},{2,3,4,6,7},{5}}
=> {{1},{2,8},{3},{4,5},{6},{7}}
=> ?
=> ? => ? = 1 + 1
{{1,2,3,4,6,8},{5},{7}}
=> {{1},{2,3},{4,5},{6},{7},{8}}
=> ?
=> ? => ? = 2 + 1
{{1,2,3,4,6,7,8},{5}}
=> {{1},{2},{3},{4,5},{6},{7},{8}}
=> ?
=> ? => ? = 1 + 1
{{1,2,3,4,5,6},{7,8}}
=> {{1,3},{2},{4},{5},{6},{7},{8}}
=> {{1},{2,3},{4},{5},{6},{7},{8}}
=> ? => ? = 0 + 1
{{1,2,3,4,7,8},{5,6}}
=> {{1},{2},{3,5},{4},{6},{7},{8}}
=> ?
=> ? => ? = 0 + 1
{{1,2,3,4,5,7,8},{6}}
=> {{1},{2},{3,4},{5},{6},{7},{8}}
=> ?
=> ? => ? = 1 + 1
{{1,2,3,4,5,8},{6,7}}
=> {{1},{2,4},{3},{5},{6},{7},{8}}
=> ?
=> ? => ? = 0 + 1
{{1,2,3,4,5,6,8},{7}}
=> {{1},{2,3},{4},{5},{6},{7},{8}}
=> ?
=> ? => ? = 1 + 1
{{1,2,4,6,7,8},{3,5}}
=> {{1},{2},{3},{4,6},{5,7},{8}}
=> ?
=> ? => ? = 0 + 1
{{1,3,4,5,7,8},{2,6}}
=> {{1},{2},{3,7},{4,8},{5},{6}}
=> {{1},{2},{3},{4,7,8},{5},{6}}
=> ? => ? = 0 + 1
{{1,2,4,5,7,8},{3,6}}
=> {{1},{2},{3,6},{4,7},{5},{8}}
=> ?
=> ? => ? = 0 + 1
{{1,2,3,5,7,8},{4,6}}
=> {{1},{2},{3,5},{4,6},{7},{8}}
=> {{1},{2,5,6},{3},{4},{7},{8}}
=> ? => ? = 0 + 1
{{1,2,4,5,6,8},{3,7}}
=> {{1},{2,6},{3,7},{4},{5},{8}}
=> {{1},{2},{3},{4,6,7},{5},{8}}
=> ? => ? = 0 + 1
{{1,2,3,5,6,8},{4,7}}
=> {{1},{2,5},{3,6},{4},{7},{8}}
=> {{1},{2},{3,5,6},{4},{7},{8}}
=> ? => ? = 0 + 1
{{1,2,3,4,6,8},{5,7}}
=> {{1},{2,4},{3,5},{6},{7},{8}}
=> ?
=> ? => ? = 0 + 1
{{1,2,4,5,6,7},{3,8}}
=> {{1,6},{2,7},{3},{4},{5},{8}}
=> ?
=> ? => ? = 0 + 1
{{1,2,3,5,6,7},{4,8}}
=> {{1,5},{2,6},{3},{4},{7},{8}}
=> {{1},{2},{3},{4,5,6},{7},{8}}
=> ? => ? = 0 + 1
{{1},{2},{3},{4,9},{5},{6},{7},{8}}
=> {{1,7,8,9},{2,3,4,5,6}}
=> {{1,3,4,5,6,8,9},{2,7}}
=> ? => ? = 6 + 1
{{1},{2},{3},{4},{5},{6},{7,8,9}}
=> {{1,4,5,6,7,8,9},{2},{3}}
=> {{1,5,6,7,8,9},{2},{3,4}}
=> ? => ? = 5 + 1
{{1,2,3,4,5,6,7,8,9,10},{11}}
=> {{1,2},{3},{4},{5},{6},{7},{8},{9},{10},{11}}
=> {{1,2},{3},{4},{5},{6},{7},{8},{9},{10},{11}}
=> [2,1,1,1,1,1,1,1,1,1] => ? = 1 + 1
{{1},{2,6},{3,7},{4,8},{5}}
=> {{1,4,5,8},{2,6},{3,7}}
=> {{1,5},{2},{3,4,6,7,8}}
=> ? => ? = 1 + 1
{{1},{2,4},{3,6},{5,8},{7}}
=> {{1,4,8},{2,3,6},{5,7}}
=> ?
=> ? => ? = 1 + 1
{{1},{2,8},{3,5},{4,7},{6}}
=> {{1,8},{2,5,7},{3,4,6}}
=> ?
=> ? => ? = 1 + 1
{{1},{2,4},{3,8},{5,7},{6}}
=> {{1,6,8},{2,5,7},{3,4}}
=> {{1,4},{2,5,7,8},{3,6}}
=> ? => ? = 1 + 1
{{1,2,3,4,5,6,7},{8,9}}
=> {{1,3},{2},{4},{5},{6},{7},{8},{9}}
=> {{1},{2,3},{4},{5},{6},{7},{8},{9}}
=> ? => ? = 0 + 1
{{1},{2,3},{4,6},{5,7,8}}
=> {{1,4,6,8},{2},{3,5},{7}}
=> {{1},{2,5,6,8},{3,4},{7}}
=> ? => ? = 0 + 1
Description
The first part of an integer composition.
Matching statistic: St000504
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00221: Set partitions —conjugate⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000504: Set partitions ⟶ ℤResult quality: 80% ●values known / values provided: 87%●distinct values known / distinct values provided: 80%
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000504: Set partitions ⟶ ℤResult quality: 80% ●values known / values provided: 87%●distinct values known / distinct values provided: 80%
Values
{{1}}
=> {{1}}
=> {{1}}
=> ? = 0 + 1
{{1,2}}
=> {{1},{2}}
=> {{1},{2}}
=> 1 = 0 + 1
{{1},{2}}
=> {{1,2}}
=> {{1,2}}
=> 2 = 1 + 1
{{1,2,3}}
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 1 = 0 + 1
{{1,2},{3}}
=> {{1,2},{3}}
=> {{1,2},{3}}
=> 2 = 1 + 1
{{1,3},{2}}
=> {{1},{2,3}}
=> {{1,3},{2}}
=> 2 = 1 + 1
{{1},{2,3}}
=> {{1,3},{2}}
=> {{1},{2,3}}
=> 1 = 0 + 1
{{1},{2},{3}}
=> {{1,2,3}}
=> {{1,2,3}}
=> 3 = 2 + 1
{{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 1 = 0 + 1
{{1,2,3},{4}}
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> 2 = 1 + 1
{{1,2,4},{3}}
=> {{1},{2,3},{4}}
=> {{1,3},{2},{4}}
=> 2 = 1 + 1
{{1,2},{3,4}}
=> {{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> 1 = 0 + 1
{{1,2},{3},{4}}
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 3 = 2 + 1
{{1,3,4},{2}}
=> {{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> 2 = 1 + 1
{{1,3},{2,4}}
=> {{1,3},{2,4}}
=> {{1},{2,3,4}}
=> 1 = 0 + 1
{{1,3},{2},{4}}
=> {{1,2},{3,4}}
=> {{1,2,4},{3}}
=> 3 = 2 + 1
{{1,4},{2,3}}
=> {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> 1 = 0 + 1
{{1},{2,3,4}}
=> {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> 1 = 0 + 1
{{1},{2,3},{4}}
=> {{1,2,4},{3}}
=> {{1,2},{3,4}}
=> 2 = 1 + 1
{{1,4},{2},{3}}
=> {{1},{2,3,4}}
=> {{1,3,4},{2}}
=> 3 = 2 + 1
{{1},{2,4},{3}}
=> {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> 2 = 1 + 1
{{1},{2},{3,4}}
=> {{1,3,4},{2}}
=> {{1,4},{2,3}}
=> 2 = 1 + 1
{{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 4 = 3 + 1
{{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 1 = 0 + 1
{{1,2,3,4},{5}}
=> {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> 2 = 1 + 1
{{1,2,3,5},{4}}
=> {{1},{2,3},{4},{5}}
=> {{1,3},{2},{4},{5}}
=> 2 = 1 + 1
{{1,2,3},{4,5}}
=> {{1,3},{2},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> 1 = 0 + 1
{{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 3 = 2 + 1
{{1,2,4,5},{3}}
=> {{1},{2},{3,4},{5}}
=> {{1,4},{2},{3},{5}}
=> 2 = 1 + 1
{{1,2,4},{3,5}}
=> {{1,3},{2,4},{5}}
=> {{1},{2,3,4},{5}}
=> 1 = 0 + 1
{{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> 3 = 2 + 1
{{1,2,5},{3,4}}
=> {{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> 1 = 0 + 1
{{1,2},{3,4,5}}
=> {{1,4},{2},{3},{5}}
=> {{1},{2},{3,4},{5}}
=> 1 = 0 + 1
{{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> 2 = 1 + 1
{{1,2,5},{3},{4}}
=> {{1},{2,3,4},{5}}
=> {{1,3,4},{2},{5}}
=> 3 = 2 + 1
{{1,2},{3,5},{4}}
=> {{1,4},{2,3},{5}}
=> {{1,3},{2,4},{5}}
=> 2 = 1 + 1
{{1,2},{3},{4,5}}
=> {{1,3,4},{2},{5}}
=> {{1,4},{2,3},{5}}
=> 2 = 1 + 1
{{1,2},{3},{4},{5}}
=> {{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> 4 = 3 + 1
{{1,3,4,5},{2}}
=> {{1},{2},{3},{4,5}}
=> {{1,5},{2},{3},{4}}
=> 2 = 1 + 1
{{1,3,4},{2,5}}
=> {{1,4},{2,5},{3}}
=> {{1},{2},{3,4,5}}
=> 1 = 0 + 1
{{1,3,4},{2},{5}}
=> {{1,2},{3},{4,5}}
=> {{1,2,5},{3},{4}}
=> 3 = 2 + 1
{{1,3,5},{2,4}}
=> {{1},{2,4},{3,5}}
=> {{1},{2,4,5},{3}}
=> 1 = 0 + 1
{{1,3},{2,4,5}}
=> {{1,4},{2},{3,5}}
=> {{1},{2,5},{3,4}}
=> 1 = 0 + 1
{{1,3},{2,4},{5}}
=> {{1,2,4},{3,5}}
=> {{1,2},{3,4,5}}
=> 2 = 1 + 1
{{1,3,5},{2},{4}}
=> {{1},{2,3},{4,5}}
=> {{1,3,5},{2},{4}}
=> 3 = 2 + 1
{{1,3},{2,5},{4}}
=> {{1,4},{2,3,5}}
=> {{1,3},{2,4,5}}
=> 2 = 1 + 1
{{1,3},{2},{4,5}}
=> {{1,3},{2},{4,5}}
=> {{1,5},{2,3},{4}}
=> 2 = 1 + 1
{{1,3},{2},{4},{5}}
=> {{1,2,3},{4,5}}
=> {{1,2,3,5},{4}}
=> 4 = 3 + 1
{{1,4,5},{2,3}}
=> {{1},{2},{3,5},{4}}
=> {{1},{2,5},{3},{4}}
=> 1 = 0 + 1
{{1,4},{2,3,5}}
=> {{1,3},{2,5},{4}}
=> {{1},{2,3},{4,5}}
=> 1 = 0 + 1
{{1,4},{2,3},{5}}
=> {{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> 2 = 1 + 1
{{1},{2},{3},{4},{5},{6,7,8}}
=> {{1,4,5,6,7,8},{2},{3}}
=> {{1,5,6,7,8},{2},{3,4}}
=> ? = 4 + 1
{{1},{2},{3},{4},{5,7,8},{6}}
=> {{1,5,6,7,8},{2},{3,4}}
=> ?
=> ? = 4 + 1
{{1},{2},{3},{4},{5,6,7},{8}}
=> {{1,2,5,6,7,8},{3},{4}}
=> {{1,2,6,7,8},{3},{4,5}}
=> ? = 4 + 1
{{1},{2},{3},{4},{5,6,8},{7}}
=> {{1,5,6,7,8},{2,3},{4}}
=> {{1,3,6,7,8},{2},{4,5}}
=> ? = 4 + 1
{{1},{2},{3},{4},{5,6,7,8}}
=> {{1,5,6,7,8},{2},{3},{4}}
=> {{1,6,7,8},{2},{3},{4,5}}
=> ? = 3 + 1
{{1},{2,3},{4,5},{6,7},{8}}
=> {{1,2,4,6,8},{3},{5},{7}}
=> {{1,2},{3,4,6,8},{5},{7}}
=> ? = 1 + 1
{{1},{2,3},{4,5},{6,8},{7}}
=> {{1,4,6,8},{2,3},{5},{7}}
=> {{1,3},{2,4,6,8},{5},{7}}
=> ? = 1 + 1
{{1},{2,3},{4,5},{6,7,8}}
=> {{1,4,6,8},{2},{3},{5},{7}}
=> {{1},{2,6,8},{3,4},{5},{7}}
=> ? = 0 + 1
{{1},{2,3},{4,7},{5,6},{8}}
=> {{1,2,6,8},{3,5},{4},{7}}
=> {{1,2},{3,5,8},{4,6},{7}}
=> ? = 1 + 1
{{1},{2,3},{4,8},{5,6},{7}}
=> ?
=> ?
=> ? = 1 + 1
{{1},{2,3},{4,8},{5,7},{6}}
=> {{1,6,8},{2,5},{3,4},{7}}
=> {{1,4},{2,5,8},{3,6},{7}}
=> ? = 1 + 1
{{1},{2,3},{4,8},{5,6,7}}
=> {{1,6,8},{2,5},{3},{4},{7}}
=> ?
=> ? = 0 + 1
{{1},{2,4,5,6,7,8},{3}}
=> {{1,8},{2},{3},{4},{5},{6,7}}
=> {{1,7},{2},{3},{4},{5},{6,8}}
=> ? = 1 + 1
{{1},{2,5},{3,4},{6,7},{8}}
=> {{1,2,4,8},{3},{5,7},{6}}
=> {{1,2},{3,4,7},{5,8},{6}}
=> ? = 1 + 1
{{1},{2,5},{3,4},{6,8},{7}}
=> {{1,4,8},{2,3},{5,7},{6}}
=> {{1,3},{2,4,7},{5,8},{6}}
=> ? = 1 + 1
{{1},{2,5},{3,4},{6,7,8}}
=> {{1,4,8},{2},{3},{5,7},{6}}
=> ?
=> ? = 0 + 1
{{1},{2,7},{3,4},{5,6},{8}}
=> {{1,2,8},{3,5,7},{4},{6}}
=> {{1,2},{3,5,7},{4},{6,8}}
=> ? = 1 + 1
{{1},{2,8},{3,4},{5,6},{7}}
=> ?
=> ?
=> ? = 1 + 1
{{1},{2,8},{3,4},{5,7},{6}}
=> {{1,8},{2,5,7},{3,4},{6}}
=> {{1,4},{2,5,7},{3},{6,8}}
=> ? = 1 + 1
{{1},{2,8},{3,4},{5,6,7}}
=> {{1,8},{2,5,7},{3},{4},{6}}
=> {{1},{2,7},{3,5},{4},{6,8}}
=> ? = 0 + 1
{{1},{2,3,5,6,7,8},{4}}
=> {{1,8},{2},{3},{4},{5,6},{7}}
=> {{1,6},{2},{3},{4},{5},{7,8}}
=> ? = 1 + 1
{{1},{2,7},{3,6},{4,5},{8}}
=> {{1,2,8},{3,7},{4,6},{5}}
=> {{1,2},{3,6},{4,7},{5,8}}
=> ? = 1 + 1
{{1},{2,8},{3,6},{4,5},{7}}
=> {{1,8},{2,3,7},{4,6},{5}}
=> {{1,3},{2,6},{4,7},{5,8}}
=> ? = 1 + 1
{{1},{2,8},{3,7},{4,5},{6}}
=> {{1,8},{2,7},{3,4,6},{5}}
=> {{1,4},{2,6},{3,7},{5,8}}
=> ? = 1 + 1
{{1},{2,8},{3,7},{4,6},{5}}
=> {{1,8},{2,7},{3,6},{4,5}}
=> {{1,5},{2,6},{3,7},{4,8}}
=> ? = 1 + 1
{{1},{2,3,4,6,7,8},{5}}
=> {{1,8},{2},{3},{4,5},{6},{7}}
=> {{1,5},{2},{3},{4},{6},{7,8}}
=> ? = 1 + 1
{{1},{2,8},{3,7},{4,5,6}}
=> {{1,8},{2,7},{3,6},{4},{5}}
=> ?
=> ? = 0 + 1
{{1},{2,3,4,5,7,8},{6}}
=> {{1,8},{2},{3,4},{5},{6},{7}}
=> {{1,4},{2},{3},{5},{6},{7,8}}
=> ? = 1 + 1
{{1},{2,3,4,5,6,7},{8}}
=> {{1,2,8},{3},{4},{5},{6},{7}}
=> {{1,2},{3},{4},{5},{6},{7,8}}
=> ? = 1 + 1
{{1},{2,3,4,5,6,8},{7}}
=> {{1,8},{2,3},{4},{5},{6},{7}}
=> {{1,3},{2},{4},{5},{6},{7,8}}
=> ? = 1 + 1
{{1},{2,3,4,5,6,7,8}}
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 0 + 1
{{1,2},{3,5,6,7,8},{4}}
=> {{1,7},{2},{3},{4},{5,6},{8}}
=> {{1,6},{2},{3},{4},{5,7},{8}}
=> ? = 1 + 1
{{1,2},{3,4,6,7,8},{5}}
=> {{1,7},{2},{3},{4,5},{6},{8}}
=> ?
=> ? = 1 + 1
{{1,2},{3,4,5,7,8},{6}}
=> {{1,7},{2},{3,4},{5},{6},{8}}
=> ?
=> ? = 1 + 1
{{1,2},{3,4,5,6,7},{8}}
=> {{1,2,7},{3},{4},{5},{6},{8}}
=> {{1,2},{3},{4},{5},{6,7},{8}}
=> ? = 1 + 1
{{1,2},{3,4,5,6,8},{7}}
=> {{1,7},{2,3},{4},{5},{6},{8}}
=> {{1,3},{2},{4},{5},{6,7},{8}}
=> ? = 1 + 1
{{1,4,5,6,7,8},{2},{3}}
=> {{1},{2},{3},{4},{5},{6,7,8}}
=> {{1,7,8},{2},{3},{4},{5},{6}}
=> ? = 2 + 1
{{1,3,5,6,7,8},{2},{4}}
=> {{1},{2},{3},{4},{5,6},{7,8}}
=> {{1,6,8},{2},{3},{4},{5},{7}}
=> ? = 2 + 1
{{1,3,4,6,7,8},{2},{5}}
=> {{1},{2},{3},{4,5},{6},{7,8}}
=> {{1,5,8},{2},{3},{4},{6},{7}}
=> ? = 2 + 1
{{1,3,4,5,7,8},{2},{6}}
=> {{1},{2},{3,4},{5},{6},{7,8}}
=> {{1,4,8},{2},{3},{5},{6},{7}}
=> ? = 2 + 1
{{1,3,4,5,6,7},{2},{8}}
=> {{1,2},{3},{4},{5},{6},{7,8}}
=> {{1,2,8},{3},{4},{5},{6},{7}}
=> ? = 2 + 1
{{1,3,4,5,6,8},{2},{7}}
=> {{1},{2,3},{4},{5},{6},{7,8}}
=> {{1,3,8},{2},{4},{5},{6},{7}}
=> ? = 2 + 1
{{1,3,4,5,6,7,8},{2}}
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? = 1 + 1
{{1,4,5,6,7,8},{2,3}}
=> {{1},{2},{3},{4},{5},{6,8},{7}}
=> {{1},{2,8},{3},{4},{5},{6},{7}}
=> ? = 0 + 1
{{1,2,5,6,7,8},{3},{4}}
=> {{1},{2},{3},{4},{5,6,7},{8}}
=> {{1,6,7},{2},{3},{4},{5},{8}}
=> ? = 2 + 1
{{1,2,4,6,7,8},{3},{5}}
=> {{1},{2},{3},{4,5},{6,7},{8}}
=> ?
=> ? = 2 + 1
{{1,2,4,5,7,8},{3},{6}}
=> {{1},{2},{3,4},{5},{6,7},{8}}
=> ?
=> ? = 2 + 1
{{1,2,4,5,6,7},{3},{8}}
=> {{1,2},{3},{4},{5},{6,7},{8}}
=> {{1,2,7},{3},{4},{5},{6},{8}}
=> ? = 2 + 1
{{1,8},{2,4,5,6,7},{3}}
=> {{1},{2,8},{3},{4},{5},{6,7}}
=> ?
=> ? = 1 + 1
Description
The cardinality of the first block of a set partition.
The number of partitions of {1,…,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 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
Mp00221: Set partitions —conjugate⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 80% ●values known / values provided: 87%●distinct values known / distinct values provided: 80%
St000502: Set partitions ⟶ ℤResult quality: 80% ●values known / values provided: 87%●distinct values known / distinct values provided: 80%
Values
{{1}}
=> {{1}}
=> ? = 0
{{1,2}}
=> {{1},{2}}
=> 0
{{1},{2}}
=> {{1,2}}
=> 1
{{1,2,3}}
=> {{1},{2},{3}}
=> 0
{{1,2},{3}}
=> {{1,2},{3}}
=> 1
{{1,3},{2}}
=> {{1},{2,3}}
=> 1
{{1},{2,3}}
=> {{1,3},{2}}
=> 0
{{1},{2},{3}}
=> {{1,2,3}}
=> 2
{{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
{{1,2,3},{4}}
=> {{1,2},{3},{4}}
=> 1
{{1,2,4},{3}}
=> {{1},{2,3},{4}}
=> 1
{{1,2},{3,4}}
=> {{1,3},{2},{4}}
=> 0
{{1,2},{3},{4}}
=> {{1,2,3},{4}}
=> 2
{{1,3,4},{2}}
=> {{1},{2},{3,4}}
=> 1
{{1,3},{2,4}}
=> {{1,3},{2,4}}
=> 0
{{1,3},{2},{4}}
=> {{1,2},{3,4}}
=> 2
{{1,4},{2,3}}
=> {{1},{2,4},{3}}
=> 0
{{1},{2,3,4}}
=> {{1,4},{2},{3}}
=> 0
{{1},{2,3},{4}}
=> {{1,2,4},{3}}
=> 1
{{1,4},{2},{3}}
=> {{1},{2,3,4}}
=> 2
{{1},{2,4},{3}}
=> {{1,4},{2,3}}
=> 1
{{1},{2},{3,4}}
=> {{1,3,4},{2}}
=> 1
{{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> 3
{{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> 0
{{1,2,3,4},{5}}
=> {{1,2},{3},{4},{5}}
=> 1
{{1,2,3,5},{4}}
=> {{1},{2,3},{4},{5}}
=> 1
{{1,2,3},{4,5}}
=> {{1,3},{2},{4},{5}}
=> 0
{{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 2
{{1,2,4,5},{3}}
=> {{1},{2},{3,4},{5}}
=> 1
{{1,2,4},{3,5}}
=> {{1,3},{2,4},{5}}
=> 0
{{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> 2
{{1,2,5},{3,4}}
=> {{1},{2,4},{3},{5}}
=> 0
{{1,2},{3,4,5}}
=> {{1,4},{2},{3},{5}}
=> 0
{{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> 1
{{1,2,5},{3},{4}}
=> {{1},{2,3,4},{5}}
=> 2
{{1,2},{3,5},{4}}
=> {{1,4},{2,3},{5}}
=> 1
{{1,2},{3},{4,5}}
=> {{1,3,4},{2},{5}}
=> 1
{{1,2},{3},{4},{5}}
=> {{1,2,3,4},{5}}
=> 3
{{1,3,4,5},{2}}
=> {{1},{2},{3},{4,5}}
=> 1
{{1,3,4},{2,5}}
=> {{1,4},{2,5},{3}}
=> 0
{{1,3,4},{2},{5}}
=> {{1,2},{3},{4,5}}
=> 2
{{1,3,5},{2,4}}
=> {{1},{2,4},{3,5}}
=> 0
{{1,3},{2,4,5}}
=> {{1,4},{2},{3,5}}
=> 0
{{1,3},{2,4},{5}}
=> {{1,2,4},{3,5}}
=> 1
{{1,3,5},{2},{4}}
=> {{1},{2,3},{4,5}}
=> 2
{{1,3},{2,5},{4}}
=> {{1,4},{2,3,5}}
=> 1
{{1,3},{2},{4,5}}
=> {{1,3},{2},{4,5}}
=> 1
{{1,3},{2},{4},{5}}
=> {{1,2,3},{4,5}}
=> 3
{{1,4,5},{2,3}}
=> {{1},{2},{3,5},{4}}
=> 0
{{1,4},{2,3,5}}
=> {{1,3},{2,5},{4}}
=> 0
{{1,4},{2,3},{5}}
=> {{1,2},{3,5},{4}}
=> 1
{{1},{2},{3},{4},{5,8},{6},{7}}
=> {{1,5,6,7,8},{2,3,4}}
=> ? = 5
{{1},{2},{3},{4},{5,7,8},{6}}
=> {{1,5,6,7,8},{2},{3,4}}
=> ? = 4
{{1},{2},{3},{4},{5,6,7},{8}}
=> {{1,2,5,6,7,8},{3},{4}}
=> ? = 4
{{1},{2},{3},{4},{5,6,8},{7}}
=> {{1,5,6,7,8},{2,3},{4}}
=> ? = 4
{{1},{2},{3},{4},{5,6,7,8}}
=> {{1,5,6,7,8},{2},{3},{4}}
=> ? = 3
{{1},{2},{3},{4,8},{5},{6},{7}}
=> {{1,6,7,8},{2,3,4,5}}
=> ? = 5
{{1},{2,3},{4,5},{6,7},{8}}
=> {{1,2,4,6,8},{3},{5},{7}}
=> ? = 1
{{1},{2,3},{4,5},{6,8},{7}}
=> {{1,4,6,8},{2,3},{5},{7}}
=> ? = 1
{{1},{2,3},{4,5},{6,7,8}}
=> {{1,4,6,8},{2},{3},{5},{7}}
=> ? = 0
{{1},{2,3},{4,7},{5,6},{8}}
=> {{1,2,6,8},{3,5},{4},{7}}
=> ? = 1
{{1},{2,3},{4,8},{5,6},{7}}
=> ?
=> ? = 1
{{1},{2,3},{4,8},{5,7},{6}}
=> {{1,6,8},{2,5},{3,4},{7}}
=> ? = 1
{{1},{2,3},{4,8},{5,6,7}}
=> {{1,6,8},{2,5},{3},{4},{7}}
=> ? = 0
{{1},{2,4,5,6,7,8},{3}}
=> {{1,8},{2},{3},{4},{5},{6,7}}
=> ? = 1
{{1},{2,5},{3,4},{6,7},{8}}
=> {{1,2,4,8},{3},{5,7},{6}}
=> ? = 1
{{1},{2,5},{3,4},{6,8},{7}}
=> {{1,4,8},{2,3},{5,7},{6}}
=> ? = 1
{{1},{2,5},{3,4},{6,7,8}}
=> {{1,4,8},{2},{3},{5,7},{6}}
=> ? = 0
{{1},{2,7},{3,4},{5,6},{8}}
=> {{1,2,8},{3,5,7},{4},{6}}
=> ? = 1
{{1},{2,8},{3,4},{5,6},{7}}
=> ?
=> ? = 1
{{1},{2,8},{3,4},{5,7},{6}}
=> {{1,8},{2,5,7},{3,4},{6}}
=> ? = 1
{{1},{2,8},{3,4},{5,6,7}}
=> {{1,8},{2,5,7},{3},{4},{6}}
=> ? = 0
{{1},{2,3,5,6,7,8},{4}}
=> {{1,8},{2},{3},{4},{5,6},{7}}
=> ? = 1
{{1},{2,7},{3,6},{4,5},{8}}
=> {{1,2,8},{3,7},{4,6},{5}}
=> ? = 1
{{1},{2,8},{3,6},{4,5},{7}}
=> {{1,8},{2,3,7},{4,6},{5}}
=> ? = 1
{{1},{2,8},{3,7},{4,5},{6}}
=> {{1,8},{2,7},{3,4,6},{5}}
=> ? = 1
{{1},{2,8},{3,7},{4,6},{5}}
=> {{1,8},{2,7},{3,6},{4,5}}
=> ? = 1
{{1},{2,3,4,6,7,8},{5}}
=> {{1,8},{2},{3},{4,5},{6},{7}}
=> ? = 1
{{1},{2,8},{3,7},{4,5,6}}
=> {{1,8},{2,7},{3,6},{4},{5}}
=> ? = 0
{{1},{2,3,4,5,7,8},{6}}
=> {{1,8},{2},{3,4},{5},{6},{7}}
=> ? = 1
{{1},{2,3,4,5,6,7},{8}}
=> {{1,2,8},{3},{4},{5},{6},{7}}
=> ? = 1
{{1},{2,3,4,5,6,8},{7}}
=> {{1,8},{2,3},{4},{5},{6},{7}}
=> ? = 1
{{1},{2,3,4,5,6,7,8}}
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? = 0
{{1,2},{3,5,6,7,8},{4}}
=> {{1,7},{2},{3},{4},{5,6},{8}}
=> ? = 1
{{1,2},{3,4,6,7,8},{5}}
=> {{1,7},{2},{3},{4,5},{6},{8}}
=> ? = 1
{{1,2},{3,4,5,7,8},{6}}
=> {{1,7},{2},{3,4},{5},{6},{8}}
=> ? = 1
{{1,2},{3,4,5,6,7},{8}}
=> {{1,2,7},{3},{4},{5},{6},{8}}
=> ? = 1
{{1,2},{3,4,5,6,8},{7}}
=> {{1,7},{2,3},{4},{5},{6},{8}}
=> ? = 1
{{1,4},{2},{3},{5},{6},{7},{8}}
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 6
{{1,5},{2},{3},{4},{6},{7},{8}}
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 6
{{1,6},{2},{3},{4},{5},{7},{8}}
=> {{1,2,3},{4,5,6,7,8}}
=> ? = 6
{{1,4,5,6,7,8},{2},{3}}
=> {{1},{2},{3},{4},{5},{6,7,8}}
=> ? = 2
{{1,3,5,6,7,8},{2},{4}}
=> {{1},{2},{3},{4},{5,6},{7,8}}
=> ? = 2
{{1,3,4,6,7,8},{2},{5}}
=> {{1},{2},{3},{4,5},{6},{7,8}}
=> ? = 2
{{1,3,4,5,7,8},{2},{6}}
=> {{1},{2},{3,4},{5},{6},{7,8}}
=> ? = 2
{{1,3,4,5,6,7},{2},{8}}
=> {{1,2},{3},{4},{5},{6},{7,8}}
=> ? = 2
{{1,3,4,5,6,8},{2},{7}}
=> {{1},{2,3},{4},{5},{6},{7,8}}
=> ? = 2
{{1,3,4,5,6,7,8},{2}}
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 1
{{1,4,5,6,7,8},{2,3}}
=> {{1},{2},{3},{4},{5},{6,8},{7}}
=> ? = 0
{{1,2,5,6,7,8},{3},{4}}
=> {{1},{2},{3},{4},{5,6,7},{8}}
=> ? = 2
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: St001067
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001067: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 86%●distinct values known / distinct values provided: 70%
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001067: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 86%●distinct values known / distinct values provided: 70%
Values
{{1}}
=> [1] => [1] => [1,0]
=> 0
{{1,2}}
=> [2] => [2] => [1,1,0,0]
=> 0
{{1},{2}}
=> [1,1] => [1,1] => [1,0,1,0]
=> 1
{{1,2,3}}
=> [3] => [3] => [1,1,1,0,0,0]
=> 0
{{1,2},{3}}
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
{{1,3},{2}}
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
{{1},{2,3}}
=> [1,2] => [2,1] => [1,1,0,0,1,0]
=> 0
{{1},{2},{3}}
=> [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2
{{1,2,3,4}}
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
{{1,2,3},{4}}
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1,2,4},{3}}
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1,2},{3,4}}
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
{{1,3,4},{2}}
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1,3},{2,4}}
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
{{1,4},{2,3}}
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
{{1},{2,3,4}}
=> [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
{{1},{2,3},{4}}
=> [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
{{1},{2,4},{3}}
=> [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
{{1},{2},{3,4}}
=> [1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3
{{1,2,3,4,5}}
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
{{1,2,3,4},{5}}
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,3,5},{4}}
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,3},{4,5}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
{{1,2,4,5},{3}}
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,4},{3,5}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
{{1,2,5},{3,4}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
{{1,2},{3,4,5}}
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 3
{{1,3,4,5},{2}}
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,3,4},{2,5}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
{{1,3,5},{2,4}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
{{1,3},{2,4,5}}
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 3
{{1,4,5},{2,3}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
{{1,4},{2,3,5}}
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
{{1},{2},{3},{4},{5},{6},{7},{8}}
=> [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
{{1},{2},{3},{4},{5},{6},{7,8}}
=> [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5
{{1},{2},{3},{4},{5},{6,8},{7}}
=> [1,1,1,1,1,2,1] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5
{{1},{2},{3},{4},{5},{6,7,8}}
=> [1,1,1,1,1,3] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4
{{1},{2},{3},{4},{5,6},{7},{8}}
=> [1,1,1,1,2,1,1] => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
{{1},{2},{3},{4},{5,7},{6},{8}}
=> [1,1,1,1,2,1,1] => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
{{1},{2},{3},{4},{5,8},{6},{7}}
=> [1,1,1,1,2,1,1] => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
{{1},{2},{3},{4},{5,7,8},{6}}
=> [1,1,1,1,3,1] => [1,3,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4
{{1},{2},{3},{4},{5,6,7},{8}}
=> [1,1,1,1,3,1] => [1,3,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4
{{1},{2},{3},{4},{5,6,8},{7}}
=> [1,1,1,1,3,1] => [1,3,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4
{{1},{2},{3},{4},{5,6,7,8}}
=> [1,1,1,1,4] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
{{1},{2},{3},{4,8},{5},{6},{7}}
=> [1,1,1,2,1,1,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 5
{{1},{2,3},{4,5},{6,7},{8}}
=> [1,2,2,2,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1
{{1},{2,3},{4,5},{6,8},{7}}
=> [1,2,2,2,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1
{{1},{2,3},{4,5},{6,7,8}}
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
{{1},{2,3},{4,7},{5,6},{8}}
=> [1,2,2,2,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1
{{1},{2,3},{4,8},{5,6},{7}}
=> [1,2,2,2,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1
{{1},{2,3},{4,8},{5,7},{6}}
=> [1,2,2,2,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1
{{1},{2,3},{4,8},{5,6,7}}
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
{{1},{2,8},{3},{4},{5},{6},{7}}
=> [1,2,1,1,1,1,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 5
{{1},{2,4,5,6,7,8},{3}}
=> [1,6,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
{{1},{2,5},{3,4},{6,7},{8}}
=> [1,2,2,2,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1
{{1},{2,5},{3,4},{6,8},{7}}
=> [1,2,2,2,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1
{{1},{2,5},{3,4},{6,7,8}}
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
{{1},{2,7},{3,4},{5,6},{8}}
=> [1,2,2,2,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1
{{1},{2,8},{3,4},{5,6},{7}}
=> [1,2,2,2,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1
{{1},{2,8},{3,4},{5,7},{6}}
=> [1,2,2,2,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1
{{1},{2,8},{3,4},{5,6,7}}
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
{{1},{2,3,5,6,7,8},{4}}
=> [1,6,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
{{1},{2,7},{3,6},{4,5},{8}}
=> [1,2,2,2,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1
{{1},{2,8},{3,6},{4,5},{7}}
=> [1,2,2,2,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1
{{1},{2,8},{3,7},{4,5},{6}}
=> [1,2,2,2,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1
{{1},{2,8},{3,7},{4,6},{5}}
=> [1,2,2,2,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1
{{1},{2,3,4,6,7,8},{5}}
=> [1,6,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
{{1},{2,8},{3,7},{4,5,6}}
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
{{1},{2,3,4,5,7,8},{6}}
=> [1,6,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
{{1},{2,3,4,5,6,7},{8}}
=> [1,6,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
{{1},{2,3,4,5,6,8},{7}}
=> [1,6,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
{{1},{2,3,4,5,6,7,8}}
=> [1,7] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
{{1,2},{3},{4},{5},{6},{7},{8}}
=> [2,1,1,1,1,1,1] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 6
{{1,2},{3,5,6,7,8},{4}}
=> [2,5,1] => [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 1
{{1,2},{3,4,6,7,8},{5}}
=> [2,5,1] => [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 1
{{1,2},{3,4,5,7,8},{6}}
=> [2,5,1] => [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 1
{{1,2},{3,4,5,6,7},{8}}
=> [2,5,1] => [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 1
{{1,2},{3,4,5,6,8},{7}}
=> [2,5,1] => [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 1
{{1,3},{2},{4},{5},{6},{7},{8}}
=> [2,1,1,1,1,1,1] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 6
{{1,4},{2},{3},{5},{6},{7},{8}}
=> [2,1,1,1,1,1,1] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 6
{{1,5},{2},{3},{4},{6},{7},{8}}
=> [2,1,1,1,1,1,1] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 6
{{1,6},{2},{3},{4},{5},{7},{8}}
=> [2,1,1,1,1,1,1] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 6
{{1,7},{2},{3},{4},{5},{6},{8}}
=> [2,1,1,1,1,1,1] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 6
Description
The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St001091
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St001091: Integer partitions ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 100%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St001091: Integer partitions ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [[1],[]]
=> [1]
=> 0
{{1,2}}
=> [2] => [[2],[]]
=> [2]
=> 0
{{1},{2}}
=> [1,1] => [[1,1],[]]
=> [1,1]
=> 1
{{1,2,3}}
=> [3] => [[3],[]]
=> [3]
=> 0
{{1,2},{3}}
=> [2,1] => [[2,2],[1]]
=> [2,2]
=> 1
{{1,3},{2}}
=> [2,1] => [[2,2],[1]]
=> [2,2]
=> 1
{{1},{2,3}}
=> [1,2] => [[2,1],[]]
=> [2,1]
=> 0
{{1},{2},{3}}
=> [1,1,1] => [[1,1,1],[]]
=> [1,1,1]
=> 2
{{1,2,3,4}}
=> [4] => [[4],[]]
=> [4]
=> 0
{{1,2,3},{4}}
=> [3,1] => [[3,3],[2]]
=> [3,3]
=> 1
{{1,2,4},{3}}
=> [3,1] => [[3,3],[2]]
=> [3,3]
=> 1
{{1,2},{3,4}}
=> [2,2] => [[3,2],[1]]
=> [3,2]
=> 0
{{1,2},{3},{4}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> [2,2,2]
=> 2
{{1,3,4},{2}}
=> [3,1] => [[3,3],[2]]
=> [3,3]
=> 1
{{1,3},{2,4}}
=> [2,2] => [[3,2],[1]]
=> [3,2]
=> 0
{{1,3},{2},{4}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> [2,2,2]
=> 2
{{1,4},{2,3}}
=> [2,2] => [[3,2],[1]]
=> [3,2]
=> 0
{{1},{2,3,4}}
=> [1,3] => [[3,1],[]]
=> [3,1]
=> 0
{{1},{2,3},{4}}
=> [1,2,1] => [[2,2,1],[1]]
=> [2,2,1]
=> 1
{{1,4},{2},{3}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> [2,2,2]
=> 2
{{1},{2,4},{3}}
=> [1,2,1] => [[2,2,1],[1]]
=> [2,2,1]
=> 1
{{1},{2},{3,4}}
=> [1,1,2] => [[2,1,1],[]]
=> [2,1,1]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1] => [[1,1,1,1],[]]
=> [1,1,1,1]
=> 3
{{1,2,3,4,5}}
=> [5] => [[5],[]]
=> [5]
=> 0
{{1,2,3,4},{5}}
=> [4,1] => [[4,4],[3]]
=> [4,4]
=> 1
{{1,2,3,5},{4}}
=> [4,1] => [[4,4],[3]]
=> [4,4]
=> 1
{{1,2,3},{4,5}}
=> [3,2] => [[4,3],[2]]
=> [4,3]
=> 0
{{1,2,3},{4},{5}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> [3,3,3]
=> 2
{{1,2,4,5},{3}}
=> [4,1] => [[4,4],[3]]
=> [4,4]
=> 1
{{1,2,4},{3,5}}
=> [3,2] => [[4,3],[2]]
=> [4,3]
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> [3,3,3]
=> 2
{{1,2,5},{3,4}}
=> [3,2] => [[4,3],[2]]
=> [4,3]
=> 0
{{1,2},{3,4,5}}
=> [2,3] => [[4,2],[1]]
=> [4,2]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> [3,3,3]
=> 2
{{1,2},{3,5},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [[3,2,2],[1,1]]
=> [3,2,2]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> 3
{{1,3,4,5},{2}}
=> [4,1] => [[4,4],[3]]
=> [4,4]
=> 1
{{1,3,4},{2,5}}
=> [3,2] => [[4,3],[2]]
=> [4,3]
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> [3,3,3]
=> 2
{{1,3,5},{2,4}}
=> [3,2] => [[4,3],[2]]
=> [4,3]
=> 0
{{1,3},{2,4,5}}
=> [2,3] => [[4,2],[1]]
=> [4,2]
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> [3,3,3]
=> 2
{{1,3},{2,5},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [[3,2,2],[1,1]]
=> [3,2,2]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> 3
{{1,4,5},{2,3}}
=> [3,2] => [[4,3],[2]]
=> [4,3]
=> 0
{{1,4},{2,3,5}}
=> [2,3] => [[4,2],[1]]
=> [4,2]
=> 0
{{1,2,3,4,5},{6},{7}}
=> [5,1,1] => [[5,5,5],[4,4]]
=> [5,5,5]
=> ? = 2
{{1,2,3,4,6},{5},{7}}
=> [5,1,1] => [[5,5,5],[4,4]]
=> [5,5,5]
=> ? = 2
{{1,2,3,4},{5,6},{7}}
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
{{1,2,3,4,7},{5},{6}}
=> [5,1,1] => [[5,5,5],[4,4]]
=> [5,5,5]
=> ? = 2
{{1,2,3,4},{5,7},{6}}
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
{{1,2,3,4},{5},{6,7}}
=> [4,1,2] => [[5,4,4],[3,3]]
=> [5,4,4]
=> ? = 1
{{1,2,3,4},{5},{6},{7}}
=> [4,1,1,1] => [[4,4,4,4],[3,3,3]]
=> [4,4,4,4]
=> ? = 3
{{1,2,3,5,6},{4},{7}}
=> [5,1,1] => [[5,5,5],[4,4]]
=> [5,5,5]
=> ? = 2
{{1,2,3,5},{4,6},{7}}
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
{{1,2,3,5,7},{4},{6}}
=> [5,1,1] => [[5,5,5],[4,4]]
=> [5,5,5]
=> ? = 2
{{1,2,3,5},{4,7},{6}}
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
{{1,2,3,5},{4},{6,7}}
=> [4,1,2] => [[5,4,4],[3,3]]
=> [5,4,4]
=> ? = 1
{{1,2,3,5},{4},{6},{7}}
=> [4,1,1,1] => [[4,4,4,4],[3,3,3]]
=> [4,4,4,4]
=> ? = 3
{{1,2,3,6},{4,5},{7}}
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
{{1,2,3},{4,5,6},{7}}
=> [3,3,1] => [[5,5,3],[4,2]]
=> [5,5,3]
=> ? = 1
{{1,2,3,7},{4,5},{6}}
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
{{1,2,3},{4,5,7},{6}}
=> [3,3,1] => [[5,5,3],[4,2]]
=> [5,5,3]
=> ? = 1
{{1,2,3},{4,5},{6},{7}}
=> [3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> [4,4,4,3]
=> ? = 2
{{1,2,3,6,7},{4},{5}}
=> [5,1,1] => [[5,5,5],[4,4]]
=> [5,5,5]
=> ? = 2
{{1,2,3,6},{4,7},{5}}
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
{{1,2,3,6},{4},{5,7}}
=> [4,1,2] => [[5,4,4],[3,3]]
=> [5,4,4]
=> ? = 1
{{1,2,3,6},{4},{5},{7}}
=> [4,1,1,1] => [[4,4,4,4],[3,3,3]]
=> [4,4,4,4]
=> ? = 3
{{1,2,3,7},{4,6},{5}}
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
{{1,2,3},{4,6,7},{5}}
=> [3,3,1] => [[5,5,3],[4,2]]
=> [5,5,3]
=> ? = 1
{{1,2,3},{4,6},{5},{7}}
=> [3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> [4,4,4,3]
=> ? = 2
{{1,2,3,7},{4},{5,6}}
=> [4,1,2] => [[5,4,4],[3,3]]
=> [5,4,4]
=> ? = 1
{{1,2,3},{4},{5,6},{7}}
=> [3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> [4,4,3,3]
=> ? = 2
{{1,2,3,7},{4},{5},{6}}
=> [4,1,1,1] => [[4,4,4,4],[3,3,3]]
=> [4,4,4,4]
=> ? = 3
{{1,2,3},{4,7},{5},{6}}
=> [3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> [4,4,4,3]
=> ? = 2
{{1,2,3},{4},{5,7},{6}}
=> [3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> [4,4,3,3]
=> ? = 2
{{1,2,3},{4},{5},{6,7}}
=> [3,1,1,2] => [[4,3,3,3],[2,2,2]]
=> [4,3,3,3]
=> ? = 2
{{1,2,3},{4},{5},{6},{7}}
=> [3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]]
=> [3,3,3,3,3]
=> ? = 4
{{1,2,4,5,6},{3},{7}}
=> [5,1,1] => [[5,5,5],[4,4]]
=> [5,5,5]
=> ? = 2
{{1,2,4,5},{3,6},{7}}
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
{{1,2,4,5,7},{3},{6}}
=> [5,1,1] => [[5,5,5],[4,4]]
=> [5,5,5]
=> ? = 2
{{1,2,4,5},{3,7},{6}}
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
{{1,2,4,5},{3},{6,7}}
=> [4,1,2] => [[5,4,4],[3,3]]
=> [5,4,4]
=> ? = 1
{{1,2,4,5},{3},{6},{7}}
=> [4,1,1,1] => [[4,4,4,4],[3,3,3]]
=> [4,4,4,4]
=> ? = 3
{{1,2,4,6},{3,5},{7}}
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
{{1,2,4},{3,5,6},{7}}
=> [3,3,1] => [[5,5,3],[4,2]]
=> [5,5,3]
=> ? = 1
{{1,2,4,7},{3,5},{6}}
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
{{1,2,4},{3,5,7},{6}}
=> [3,3,1] => [[5,5,3],[4,2]]
=> [5,5,3]
=> ? = 1
{{1,2,4},{3,5},{6},{7}}
=> [3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> [4,4,4,3]
=> ? = 2
{{1,2,4,6,7},{3},{5}}
=> [5,1,1] => [[5,5,5],[4,4]]
=> [5,5,5]
=> ? = 2
{{1,2,4,6},{3,7},{5}}
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
{{1,2,4,6},{3},{5,7}}
=> [4,1,2] => [[5,4,4],[3,3]]
=> [5,4,4]
=> ? = 1
{{1,2,4,6},{3},{5},{7}}
=> [4,1,1,1] => [[4,4,4,4],[3,3,3]]
=> [4,4,4,4]
=> ? = 3
{{1,2,4,7},{3,6},{5}}
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
{{1,2,4},{3,6,7},{5}}
=> [3,3,1] => [[5,5,3],[4,2]]
=> [5,5,3]
=> ? = 1
{{1,2,4},{3,6},{5},{7}}
=> [3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> [4,4,4,3]
=> ? = 2
Description
The number of parts in an integer partition whose next smaller part has the same size.
In other words, this is the number of distinct parts subtracted from the number of all parts.
Matching statistic: St001640
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 70%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 70%
Values
{{1}}
=> [1] => [1,0]
=> [1] => 0
{{1,2}}
=> [2] => [1,1,0,0]
=> [2,1] => 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [1,2] => 1
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [3,2,1] => 0
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 1
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 1
{{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] => 2
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0
{{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] => 2
{{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] => 2
{{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] => 0
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 3
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 0
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 0
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 2
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 0
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 2
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 0
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 0
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 2
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 3
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 0
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 2
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 0
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 0
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 2
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 3
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 0
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 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] => ? = 0
{{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] => ? = 1
{{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] => ? = 1
{{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] => ? = 0
{{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] => ? = 2
{{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] => ? = 1
{{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] => ? = 0
{{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] => ? = 2
{{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] => ? = 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] => ? = 0
{{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] => ? = 1
{{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] => ? = 2
{{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] => ? = 1
{{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] => ? = 1
{{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] => ? = 3
{{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] => ? = 1
{{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] => ? = 0
{{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] => ? = 2
{{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] => ? = 0
{{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] => ? = 0
{{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] => ? = 1
{{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] => ? = 2
{{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] => ? = 1
{{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] => ? = 1
{{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] => ? = 3
{{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] => ? = 0
{{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] => ? = 0
{{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] => ? = 1
{{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] => ? = 0
{{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] => ? = 0
{{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] => ? = 1
{{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] => ? = 1
{{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] => ? = 1
{{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] => ? = 0
{{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] => ? = 2
{{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] => ? = 2
{{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] => ? = 1
{{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] => ? = 1
{{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] => ? = 3
{{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] => ? = 1
{{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] => ? = 1
{{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] => ? = 0
{{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] => ? = 2
{{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] => ? = 1
{{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] => ? = 0
{{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] => ? = 1
{{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] => ? = 2
{{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] => ? = 3
{{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] => ? = 2
{{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] => ? = 2
Description
The number of ascent tops in the permutation such that all smaller elements appear before.
Matching statistic: St000237
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 100%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => 0
{{1,2}}
=> [2,1] => [1,2] => 0
{{1},{2}}
=> [1,2] => [2,1] => 1
{{1,2,3}}
=> [2,3,1] => [1,2,3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,3,2] => 1
{{1,3},{2}}
=> [3,2,1] => [2,1,3] => 1
{{1},{2,3}}
=> [1,3,2] => [3,2,1] => 0
{{1},{2},{3}}
=> [1,2,3] => [2,3,1] => 2
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,4,3] => 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,3,2,4] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,4,3,2] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,3,4,2] => 2
{{1,3,4},{2}}
=> [3,2,4,1] => [2,1,3,4] => 1
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,2,3] => 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,1,4,3] => 2
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,1,4] => 0
{{1},{2,3,4}}
=> [1,3,4,2] => [4,2,3,1] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [3,2,4,1] => 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,1,4] => 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,3,2,1] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [2,4,3,1] => 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [2,3,4,1] => 3
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,5,4] => 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,4,3,5] => 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,5,4,3] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,4,5,3] => 2
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,3,2,4,5] => 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,5,2,3,4] => 0
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,3,2,5,4] => 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,4,3,2,5] => 0
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,5,3,4,2] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,4,3,5,2] => 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,3,4,2,5] => 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,5,4,3,2] => 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,3,5,4,2] => 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,3,4,5,2] => 3
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,1,3,4,5] => 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,1,3,2,4] => 0
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,1,3,5,4] => 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,1,2,3,5] => 0
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [5,1,2,4,3] => 0
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [4,1,2,5,3] => 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,1,4,3,5] => 2
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [5,1,4,2,3] => 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,1,5,4,3] => 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,1,4,5,3] => 3
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,1,4,5] => 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,2,1,3,4] => 0
{{1,2,3,4,5},{6,7}}
=> [2,3,4,5,1,7,6] => [1,2,3,4,7,6,5] => ? = 0
{{1,2,3,4,5},{6},{7}}
=> [2,3,4,5,1,6,7] => [1,2,3,4,6,7,5] => ? = 2
{{1,2,3,4,6,7},{5}}
=> [2,3,4,6,5,7,1] => [1,2,3,5,4,6,7] => ? = 1
{{1,2,3,4,6},{5,7}}
=> [2,3,4,6,7,1,5] => [1,2,3,7,4,5,6] => ? = 0
{{1,2,3,4,6},{5},{7}}
=> [2,3,4,6,5,1,7] => [1,2,3,5,4,7,6] => ? = 2
{{1,2,3,4,7},{5,6}}
=> [2,3,4,7,6,5,1] => [1,2,3,6,5,4,7] => ? = 0
{{1,2,3,4},{5,6,7}}
=> [2,3,4,1,6,7,5] => [1,2,3,7,5,6,4] => ? = 0
{{1,2,3,4},{5,6},{7}}
=> [2,3,4,1,6,5,7] => [1,2,3,6,5,7,4] => ? = 1
{{1,2,3,4,7},{5},{6}}
=> [2,3,4,7,5,6,1] => [1,2,3,5,6,4,7] => ? = 2
{{1,2,3,4},{5,7},{6}}
=> [2,3,4,1,7,6,5] => [1,2,3,7,6,5,4] => ? = 1
{{1,2,3,4},{5},{6,7}}
=> [2,3,4,1,5,7,6] => [1,2,3,5,7,6,4] => ? = 1
{{1,2,3,4},{5},{6},{7}}
=> [2,3,4,1,5,6,7] => [1,2,3,5,6,7,4] => ? = 3
{{1,2,3,5,6,7},{4}}
=> [2,3,5,4,6,7,1] => [1,2,4,3,5,6,7] => ? = 1
{{1,2,3,5,6},{4,7}}
=> [2,3,5,7,6,1,4] => [1,2,7,3,5,4,6] => ? = 0
{{1,2,3,5,6},{4},{7}}
=> [2,3,5,4,6,1,7] => [1,2,4,3,5,7,6] => ? = 2
{{1,2,3,5,7},{4,6}}
=> [2,3,5,6,7,4,1] => [1,2,6,3,4,5,7] => ? = 0
{{1,2,3,5},{4,6,7}}
=> [2,3,5,6,1,7,4] => [1,2,7,3,4,6,5] => ? = 0
{{1,2,3,5,7},{4},{6}}
=> [2,3,5,4,7,6,1] => [1,2,4,3,6,5,7] => ? = 2
{{1,2,3,5},{4,7},{6}}
=> [2,3,5,7,1,6,4] => [1,2,7,3,6,4,5] => ? = 1
{{1,2,3,5},{4},{6,7}}
=> [2,3,5,4,1,7,6] => [1,2,4,3,7,6,5] => ? = 1
{{1,2,3,5},{4},{6},{7}}
=> [2,3,5,4,1,6,7] => [1,2,4,3,6,7,5] => ? = 3
{{1,2,3,6,7},{4,5}}
=> [2,3,6,5,4,7,1] => [1,2,5,4,3,6,7] => ? = 0
{{1,2,3,6},{4,5,7}}
=> [2,3,6,5,7,1,4] => [1,2,7,4,3,5,6] => ? = 0
{{1,2,3,6},{4,5},{7}}
=> [2,3,6,5,4,1,7] => [1,2,5,4,3,7,6] => ? = 1
{{1,2,3,7},{4,5,6}}
=> [2,3,7,5,6,4,1] => [1,2,6,4,5,3,7] => ? = 0
{{1,2,3},{4,5,6,7}}
=> [2,3,1,5,6,7,4] => [1,2,7,4,5,6,3] => ? = 0
{{1,2,3},{4,5,6},{7}}
=> [2,3,1,5,6,4,7] => [1,2,6,4,5,7,3] => ? = 1
{{1,2,3,7},{4,5},{6}}
=> [2,3,7,5,4,6,1] => [1,2,5,4,6,3,7] => ? = 1
{{1,2,3},{4,5,7},{6}}
=> [2,3,1,5,7,6,4] => [1,2,7,4,6,5,3] => ? = 1
{{1,2,3},{4,5},{6,7}}
=> [2,3,1,5,4,7,6] => [1,2,5,4,7,6,3] => ? = 0
{{1,2,3},{4,5},{6},{7}}
=> [2,3,1,5,4,6,7] => [1,2,5,4,6,7,3] => ? = 2
{{1,2,3,6,7},{4},{5}}
=> [2,3,6,4,5,7,1] => [1,2,4,5,3,6,7] => ? = 2
{{1,2,3,6},{4,7},{5}}
=> [2,3,6,7,5,1,4] => [1,2,7,5,3,4,6] => ? = 1
{{1,2,3,6},{4},{5,7}}
=> [2,3,6,4,7,1,5] => [1,2,4,7,3,5,6] => ? = 1
{{1,2,3,6},{4},{5},{7}}
=> [2,3,6,4,5,1,7] => [1,2,4,5,3,7,6] => ? = 3
{{1,2,3,7},{4,6},{5}}
=> [2,3,7,6,5,4,1] => [1,2,6,5,4,3,7] => ? = 1
{{1,2,3},{4,6,7},{5}}
=> [2,3,1,6,5,7,4] => [1,2,7,5,4,6,3] => ? = 1
{{1,2,3},{4,6},{5,7}}
=> [2,3,1,6,7,4,5] => [1,2,6,7,4,5,3] => ? = 0
{{1,2,3},{4,6},{5},{7}}
=> [2,3,1,6,5,4,7] => [1,2,6,5,4,7,3] => ? = 2
{{1,2,3,7},{4},{5,6}}
=> [2,3,7,4,6,5,1] => [1,2,4,6,5,3,7] => ? = 1
{{1,2,3},{4,7},{5,6}}
=> [2,3,1,7,6,5,4] => [1,2,7,6,5,4,3] => ? = 0
{{1,2,3},{4},{5,6,7}}
=> [2,3,1,4,6,7,5] => [1,2,4,7,5,6,3] => ? = 1
{{1,2,3},{4},{5,6},{7}}
=> [2,3,1,4,6,5,7] => [1,2,4,6,5,7,3] => ? = 2
{{1,2,3,7},{4},{5},{6}}
=> [2,3,7,4,5,6,1] => [1,2,4,5,6,3,7] => ? = 3
{{1,2,3},{4,7},{5},{6}}
=> [2,3,1,7,5,6,4] => [1,2,7,5,6,4,3] => ? = 2
{{1,2,3},{4},{5,7},{6}}
=> [2,3,1,4,7,6,5] => [1,2,4,7,6,5,3] => ? = 2
{{1,2,3},{4},{5},{6,7}}
=> [2,3,1,4,5,7,6] => [1,2,4,5,7,6,3] => ? = 2
{{1,2,3},{4},{5},{6},{7}}
=> [2,3,1,4,5,6,7] => [1,2,4,5,6,7,3] => ? = 4
{{1,2,4,5,6},{3,7}}
=> [2,4,7,5,6,1,3] => [1,7,2,4,5,3,6] => ? = 0
{{1,2,4,5,6},{3},{7}}
=> [2,4,3,5,6,1,7] => [1,3,2,4,5,7,6] => ? = 2
Description
The number of small exceedances.
This is the number of indices i such that \pi_i=i+1.
The following 5 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001172The number of 1-rises at odd height of a Dyck path. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001948The number of augmented double ascents of a permutation.
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