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Your data matches 12 different statistics following compositions of up to 3 maps.
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Matching statistic: St000477
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000477: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000477: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [3,2]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [4,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [3,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,2]
=> [2]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [3,3]
=> [3]
=> 3
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [4,2]
=> [2]
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [3,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [4,2]
=> [2]
=> 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [4,3]
=> [3]
=> 3
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => [5,2]
=> [2]
=> 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [4,3]
=> [3]
=> 3
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [5,2]
=> [2]
=> 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [4,3]
=> [3]
=> 3
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [5,2]
=> [2]
=> 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,7,1,2,6,3,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => [5,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => [5,2]
=> [2]
=> 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [6,7,1,5,2,3,4] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => [4,3]
=> [3]
=> 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => [4,3]
=> [3]
=> 3
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => [5,2]
=> [2]
=> 2
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => [5,2]
=> [2]
=> 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [5,2]
=> [2]
=> 2
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => [5,2]
=> [2]
=> 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => [4,3]
=> [3]
=> 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => [5,2]
=> [2]
=> 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => [5,2]
=> [2]
=> 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => [4,3]
=> [3]
=> 3
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => [5,2]
=> [2]
=> 2
Description
The weight of a partition according to Alladi.
Matching statistic: St001075
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St001075: Set partitions ⟶ ℤResult quality: 25% ●values known / values provided: 29%●distinct values known / distinct values provided: 25%
Mp00151: Permutations —to cycle type⟶ Set partitions
St001075: Set partitions ⟶ ℤResult quality: 25% ●values known / values provided: 29%●distinct values known / distinct values provided: 25%
Values
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => {{1,3,5},{2,4}}
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => {{1,2,4,6},{3,5}}
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => {{1,3,5,6},{2,4}}
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => {{1,3,5},{2,4,6}}
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => {{1,3,5,6},{2,4}}
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => {{1,3,6},{2,4,5}}
=> 3
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => {{1,2,4,6},{3,5}}
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => {{1,4,6},{2,3,5}}
=> 3
[1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => {{1,3,4,6},{2,5}}
=> 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => {{1,2,3,5,7},{4,6}}
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => {{1,2,4,6,7},{3,5}}
=> 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => {{1,2,4,6},{3,5,7}}
=> 3
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => {{1,2,4,6,7},{3,5}}
=> 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => {{1,2,4,7},{3,5,6}}
=> 3
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => {{1,2,3,5,7},{4,6}}
=> 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => {{1,2,5,7},{3,4,6}}
=> 3
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => {{1,2,4,5,7},{3,6}}
=> 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => {{1,2,3,5,7},{4,6}}
=> 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => {{1,3,5,6,7},{2,4}}
=> 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => {{1,3,5,6,7},{2,4}}
=> 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => {{1,3,5},{2,4,6,7}}
=> 3
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => {{1,3,5,7},{2,4,6}}
=> 3
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => {{1,3,5},{2,4,6,7}}
=> 3
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => {{1,3,5,6,7},{2,4}}
=> 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => {{1,3,5,6,7},{2,4}}
=> 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,7,1,2,6,3,4] => {{1,3,5,6},{2,4,7}}
=> 3
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => {{1,3,5,6,7},{2,4}}
=> 2
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => {{1,2,3,5,7},{4,6}}
=> 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => {{1,3,6,7},{2,4,5}}
=> 3
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [6,7,1,5,2,3,4] => {{1,3,6},{2,4,5,7}}
=> 3
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => {{1,3,6,7},{2,4,5}}
=> 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => {{1,3,7},{2,4,5,6}}
=> 3
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => {{1,2,4,6,7},{3,5}}
=> 2
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => {{1,2,4,6},{3,5,7}}
=> 3
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => {{1,2,4,6,7},{3,5}}
=> 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => {{1,2,4,7},{3,5,6}}
=> 3
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => {{1,4,6,7},{2,3,5}}
=> 3
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => {{1,4,6},{2,3,5,7}}
=> 3
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => {{1,4,6,7},{2,3,5}}
=> 3
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => {{1,3,4,6,7},{2,5}}
=> 2
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => {{1,3,4,6},{2,5,7}}
=> 3
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => {{1,3,4,6,7},{2,5}}
=> 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => {{1,4,7},{2,3,5,6}}
=> 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => {{1,3,4,7},{2,5,6}}
=> 3
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => {{1,2,3,5,7},{4,6}}
=> 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => {{1,2,5,7},{3,4,6}}
=> 3
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => {{1,2,4,5,7},{3,6}}
=> 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => {{1,5,7},{2,3,4,6}}
=> 3
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => {{1,4,5,7},{2,3,6}}
=> 3
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => {{1,3,4,5,7},{2,6}}
=> 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,1,2,8,3,4,5,6] => {{1,2,3,5,7},{4,6,8}}
=> ? = 3
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [8,1,2,6,3,7,4,5] => {{1,2,3,5,8},{4,6,7}}
=> ? = 3
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [8,1,2,5,7,3,4,6] => {{1,2,3,6,8},{4,5,7}}
=> ? = 3
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,1,8,2,3,4,5,7] => {{1,2,4,6},{3,5,7,8}}
=> ? = 4
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [8,1,7,2,3,4,5,6] => {{1,2,4,6,8},{3,5,7}}
=> ? = 3
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [6,1,7,2,3,4,8,5] => {{1,2,4,6},{3,5,7,8}}
=> ? = 4
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [6,1,8,2,3,7,4,5] => {{1,2,4,6,7},{3,5,8}}
=> ? = 3
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [8,1,5,2,6,3,4,7] => {{1,2,4,7,8},{3,5,6}}
=> ? = 3
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [7,1,8,2,6,3,4,5] => {{1,2,4,7},{3,5,6,8}}
=> ? = 4
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [7,1,5,2,6,3,8,4] => {{1,2,4,7,8},{3,5,6}}
=> ? = 3
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [8,1,5,2,6,7,3,4] => {{1,2,4,8},{3,5,6,7}}
=> ? = 4
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,7,8,2,4,5,6] => {{1,2,3,5,7},{4,6,8}}
=> ? = 3
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,1,8,6,2,7,4,5] => {{1,2,3,5,8},{4,6,7}}
=> ? = 3
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [8,1,4,6,2,3,5,7] => {{1,2,5,7,8},{3,4,6}}
=> ? = 3
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [7,1,4,8,2,3,5,6] => {{1,2,5,7},{3,4,6,8}}
=> ? = 4
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [7,1,4,6,2,3,8,5] => {{1,2,5,7,8},{3,4,6}}
=> ? = 3
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [7,1,8,5,2,3,4,6] => {{1,2,4,5,7},{3,6,8}}
=> ? = 3
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [8,1,4,6,2,7,3,5] => {{1,2,5,8},{3,4,6,7}}
=> ? = 4
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [8,1,6,5,2,7,3,4] => {{1,2,4,5,8},{3,6,7}}
=> ? = 3
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [3,1,8,5,7,2,4,6] => {{1,2,3,6,8},{4,5,7}}
=> ? = 3
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [8,1,4,5,7,2,3,6] => {{1,2,6,8},{3,4,5,7}}
=> ? = 4
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [8,1,4,7,6,2,3,5] => {{1,2,5,6,8},{3,4,7}}
=> ? = 3
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [2,7,1,8,3,4,5,6] => {{1,2,3,5,7},{4,6,8}}
=> ? = 3
[1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,8,1,5,7,3,4,6] => {{1,2,3,6,8},{4,5,7}}
=> ? = 3
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [5,8,1,2,3,4,6,7] => {{1,3,5},{2,4,6,7,8}}
=> ? = 3
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [5,7,1,2,3,4,8,6] => {{1,3,5},{2,4,6,7,8}}
=> ? = 3
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [8,6,1,2,3,4,5,7] => {{1,3,5,7,8},{2,4,6}}
=> ? = 3
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [7,6,1,2,3,4,8,5] => {{1,3,5,7,8},{2,4,6}}
=> ? = 3
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [5,6,1,2,3,8,4,7] => {{1,3,5},{2,4,6,7,8}}
=> ? = 3
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [5,8,1,2,3,7,4,6] => {{1,3,5},{2,4,6,7,8}}
=> ? = 3
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [8,6,1,2,3,7,4,5] => {{1,3,5,8},{2,4,6,7}}
=> ? = 4
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [5,6,1,2,3,7,8,4] => {{1,3,5},{2,4,6,7,8}}
=> ? = 3
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [8,4,1,2,7,3,5,6] => {{1,3,6,8},{2,4},{5,7}}
=> ? = -2
[1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [5,8,1,2,6,3,4,7] => {{1,3,5,6},{2,4,7,8}}
=> ? = 4
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [8,7,1,2,6,3,4,5] => {{1,3,5,6,8},{2,4,7}}
=> ? = 3
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [5,7,1,2,6,3,8,4] => {{1,3,5,6},{2,4,7,8}}
=> ? = 4
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [5,8,1,2,6,7,3,4] => {{1,3,5,6,7},{2,4,8}}
=> ? = 3
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [7,3,1,8,2,4,5,6] => {{1,2,3,5,7},{4,6,8}}
=> ? = 3
[1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [8,3,1,6,2,7,4,5] => {{1,2,3,5,8},{4,6,7}}
=> ? = 3
[1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [8,4,1,5,2,3,6,7] => {{1,3,6,7,8},{2,4,5}}
=> ? = 3
[1,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [7,4,1,5,2,3,8,6] => {{1,3,6,7,8},{2,4,5}}
=> ? = 3
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [6,8,1,5,2,3,4,7] => {{1,3,6},{2,4,5,7,8}}
=> ? = 3
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [8,7,1,5,2,3,4,6] => {{1,3,6,8},{2,4,5,7}}
=> ? = 4
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [6,7,1,5,2,3,8,4] => {{1,3,6},{2,4,5,7,8}}
=> ? = 3
[1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [6,4,1,5,2,8,3,7] => {{1,3,6,7,8},{2,4,5}}
=> ? = 3
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [8,4,1,5,2,7,3,6] => {{1,3,6,7,8},{2,4,5}}
=> ? = 3
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [6,8,1,5,2,7,3,4] => {{1,3,6,7},{2,4,5,8}}
=> ? = 4
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [6,4,1,5,2,7,8,3] => {{1,3,6,7,8},{2,4,5}}
=> ? = 3
[1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [8,3,1,5,7,2,4,6] => {{1,2,3,6,8},{4,5,7}}
=> ? = 3
[1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [8,4,1,5,6,2,3,7] => {{1,3,7,8},{2,4,5,6}}
=> ? = 4
Description
The minimal size of a block of a set partition.
Matching statistic: St000253
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00219: Set partitions —inverse Yip⟶ Set partitions
St000253: Set partitions ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 25%
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00219: Set partitions —inverse Yip⟶ Set partitions
St000253: Set partitions ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 25%
Values
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => {{1,3,5},{2,4}}
=> {{1,4,5},{2,3}}
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => {{1,2,4,6},{3,5}}
=> {{1,2,5,6},{3,4}}
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => {{1,3,5,6},{2,4}}
=> {{1,4,5,6},{2,3}}
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => {{1,3,5},{2,4,6}}
=> {{1,4,5},{2,3,6}}
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => {{1,3,5,6},{2,4}}
=> {{1,4,5,6},{2,3}}
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => {{1,3,6},{2,4,5}}
=> {{1,4,6},{2,3,5}}
=> 2 = 3 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => {{1,2,4,6},{3,5}}
=> {{1,2,5,6},{3,4}}
=> 1 = 2 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => {{1,4,6},{2,3,5}}
=> {{1,3,5},{2,4,6}}
=> 2 = 3 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => {{1,3,4,6},{2,5}}
=> {{1,5,6},{2,3,4}}
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => {{1,2,3,5,7},{4,6}}
=> {{1,2,3,6,7},{4,5}}
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => {{1,2,4,6,7},{3,5}}
=> {{1,2,5,6,7},{3,4}}
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => {{1,2,4,6},{3,5,7}}
=> {{1,2,5,6},{3,4,7}}
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => {{1,2,4,6,7},{3,5}}
=> {{1,2,5,6,7},{3,4}}
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => {{1,2,4,7},{3,5,6}}
=> {{1,2,5},{3,4,6,7}}
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => {{1,2,3,5,7},{4,6}}
=> {{1,2,3,6,7},{4,5}}
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => {{1,2,5,7},{3,4,6}}
=> {{1,2,4,5},{3,6,7}}
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => {{1,2,4,5,7},{3,6}}
=> {{1,2,6,7},{3,4,5}}
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => {{1,2,3,5,7},{4,6}}
=> {{1,2,3,6,7},{4,5}}
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => {{1,3,5,6,7},{2,4}}
=> {{1,4,5,6,7},{2,3}}
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => {{1,3,5,6,7},{2,4}}
=> {{1,4,5,6,7},{2,3}}
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => {{1,3,5},{2,4,6,7}}
=> {{1,4,5,7},{2,3,6}}
=> 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => {{1,3,5,7},{2,4,6}}
=> {{1,4,5},{2,3,6,7}}
=> 2 = 3 - 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => {{1,3,5},{2,4,6,7}}
=> {{1,4,5,7},{2,3,6}}
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => {{1,3,5,6,7},{2,4}}
=> {{1,4,5,6,7},{2,3}}
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => {{1,3,5,6,7},{2,4}}
=> {{1,4,5,6,7},{2,3}}
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,7,1,2,6,3,4] => {{1,3,5,6},{2,4,7}}
=> {{1,4,5,6},{2,3,7}}
=> 2 = 3 - 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => {{1,3,5,6,7},{2,4}}
=> {{1,4,5,6,7},{2,3}}
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => {{1,2,3,5,7},{4,6}}
=> {{1,2,3,6,7},{4,5}}
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => {{1,3,6,7},{2,4,5}}
=> {{1,4,6,7},{2,3,5}}
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [6,7,1,5,2,3,4] => {{1,3,6},{2,4,5,7}}
=> {{1,4,6},{2,3,5,7}}
=> 2 = 3 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => {{1,3,6,7},{2,4,5}}
=> {{1,4,6,7},{2,3,5}}
=> 2 = 3 - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => {{1,3,7},{2,4,5,6}}
=> {{1,4,7},{2,3,5,6}}
=> 2 = 3 - 1
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => {{1,2,4,6,7},{3,5}}
=> {{1,2,5,6,7},{3,4}}
=> 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => {{1,2,4,6},{3,5,7}}
=> {{1,2,5,6},{3,4,7}}
=> 2 = 3 - 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => {{1,2,4,6,7},{3,5}}
=> {{1,2,5,6,7},{3,4}}
=> 1 = 2 - 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => {{1,2,4,7},{3,5,6}}
=> {{1,2,5},{3,4,6,7}}
=> 2 = 3 - 1
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => {{1,4,6,7},{2,3,5}}
=> {{1,3,5},{2,4,6,7}}
=> 2 = 3 - 1
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => {{1,4,6},{2,3,5,7}}
=> {{1,3,5,7},{2,4,6}}
=> 2 = 3 - 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => {{1,4,6,7},{2,3,5}}
=> {{1,3,5},{2,4,6,7}}
=> 2 = 3 - 1
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => {{1,3,4,6,7},{2,5}}
=> {{1,5,6,7},{2,3,4}}
=> 1 = 2 - 1
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => {{1,3,4,6},{2,5,7}}
=> {{1,5,6},{2,3,4,7}}
=> 2 = 3 - 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => {{1,3,4,6,7},{2,5}}
=> {{1,5,6,7},{2,3,4}}
=> 1 = 2 - 1
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => {{1,4,7},{2,3,5,6}}
=> {{1,3,5,6},{2,4,7}}
=> 2 = 3 - 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => {{1,3,4,7},{2,5,6}}
=> {{1,5},{2,3,4,6,7}}
=> 2 = 3 - 1
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => {{1,2,3,5,7},{4,6}}
=> {{1,2,3,6,7},{4,5}}
=> 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => {{1,2,5,7},{3,4,6}}
=> {{1,2,4,5},{3,6,7}}
=> 2 = 3 - 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => {{1,2,4,5,7},{3,6}}
=> {{1,2,6,7},{3,4,5}}
=> 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => {{1,5,7},{2,3,4,6}}
=> {{1,3,4,7},{2,5,6}}
=> 2 = 3 - 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => {{1,4,5,7},{2,3,6}}
=> {{1,3,6,7},{2,4,5}}
=> 2 = 3 - 1
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => {{1,3,4,5,7},{2,6}}
=> {{1,6,7},{2,3,4,5}}
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,1,2,8,3,4,5,6] => {{1,2,3,5,7},{4,6,8}}
=> {{1,2,3,6,7},{4,5,8}}
=> ? = 3 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [8,1,2,6,3,7,4,5] => {{1,2,3,5,8},{4,6,7}}
=> {{1,2,3,6},{4,5,7,8}}
=> ? = 3 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [8,1,2,5,7,3,4,6] => {{1,2,3,6,8},{4,5,7}}
=> {{1,2,3,5,6},{4,7,8}}
=> ? = 3 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [8,1,2,7,6,3,4,5] => {{1,2,3,5,6,8},{4,7}}
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,1,8,2,3,4,5,7] => {{1,2,4,6},{3,5,7,8}}
=> {{1,2,5,6,8},{3,4,7}}
=> ? = 4 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [8,1,7,2,3,4,5,6] => {{1,2,4,6,8},{3,5,7}}
=> {{1,2,5,6},{3,4,7,8}}
=> ? = 3 - 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [6,1,7,2,3,4,8,5] => {{1,2,4,6},{3,5,7,8}}
=> {{1,2,5,6,8},{3,4,7}}
=> ? = 4 - 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [6,1,8,2,3,7,4,5] => {{1,2,4,6,7},{3,5,8}}
=> {{1,2,5,6,7},{3,4,8}}
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [8,1,5,2,6,3,4,7] => {{1,2,4,7,8},{3,5,6}}
=> {{1,2,5},{3,4,6,7,8}}
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [7,1,8,2,6,3,4,5] => {{1,2,4,7},{3,5,6,8}}
=> {{1,2,5,8},{3,4,6,7}}
=> ? = 4 - 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [7,1,5,2,6,3,8,4] => {{1,2,4,7,8},{3,5,6}}
=> {{1,2,5},{3,4,6,7,8}}
=> ? = 3 - 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [8,1,5,2,6,7,3,4] => {{1,2,4,8},{3,5,6,7}}
=> {{1,2,5,7},{3,4,6,8}}
=> ? = 4 - 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,7,8,2,4,5,6] => {{1,2,3,5,7},{4,6,8}}
=> {{1,2,3,6,7},{4,5,8}}
=> ? = 3 - 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,1,8,6,2,7,4,5] => {{1,2,3,5,8},{4,6,7}}
=> {{1,2,3,6},{4,5,7,8}}
=> ? = 3 - 1
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [8,1,4,6,2,3,5,7] => {{1,2,5,7,8},{3,4,6}}
=> {{1,2,4,5},{3,6,7,8}}
=> ? = 3 - 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [7,1,4,8,2,3,5,6] => {{1,2,5,7},{3,4,6,8}}
=> {{1,2,4,5,8},{3,6,7}}
=> ? = 4 - 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [7,1,4,6,2,3,8,5] => {{1,2,5,7,8},{3,4,6}}
=> {{1,2,4,5},{3,6,7,8}}
=> ? = 3 - 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [8,1,6,5,2,3,4,7] => {{1,2,4,5,7,8},{3,6}}
=> {{1,2,6,7,8},{3,4,5}}
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [7,1,8,5,2,3,4,6] => {{1,2,4,5,7},{3,6,8}}
=> {{1,2,6,7},{3,4,5,8}}
=> ? = 3 - 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [7,1,6,5,2,3,8,4] => {{1,2,4,5,7,8},{3,6}}
=> {{1,2,6,7,8},{3,4,5}}
=> ? = 2 - 1
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [8,1,4,6,2,7,3,5] => {{1,2,5,8},{3,4,6,7}}
=> {{1,2,4,5,7},{3,6,8}}
=> ? = 4 - 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [8,1,6,5,2,7,3,4] => {{1,2,4,5,8},{3,6,7}}
=> {{1,2,6},{3,4,5,7,8}}
=> ? = 3 - 1
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [3,1,8,5,7,2,4,6] => {{1,2,3,6,8},{4,5,7}}
=> {{1,2,3,5,6},{4,7,8}}
=> ? = 3 - 1
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,1,8,7,6,2,4,5] => {{1,2,3,5,6,8},{4,7}}
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [8,1,4,5,7,2,3,6] => {{1,2,6,8},{3,4,5,7}}
=> {{1,2,4,6,8},{3,5,7}}
=> ? = 4 - 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [8,1,4,7,6,2,3,5] => {{1,2,5,6,8},{3,4,7}}
=> {{1,2,4,5,6},{3,7,8}}
=> ? = 3 - 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [8,1,7,5,6,2,3,4] => {{1,2,4,5,6,8},{3,7}}
=> {{1,2,7,8},{3,4,5,6}}
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [2,7,1,8,3,4,5,6] => {{1,2,3,5,7},{4,6,8}}
=> {{1,2,3,6,7},{4,5,8}}
=> ? = 3 - 1
[1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,8,1,5,7,3,4,6] => {{1,2,3,6,8},{4,5,7}}
=> {{1,2,3,5,6},{4,7,8}}
=> ? = 3 - 1
[1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [2,8,1,7,6,3,4,5] => {{1,2,3,5,6,8},{4,7}}
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [5,8,1,2,3,4,6,7] => {{1,3,5},{2,4,6,7,8}}
=> {{1,4,5,7,8},{2,3,6}}
=> ? = 3 - 1
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [5,7,1,2,3,4,8,6] => {{1,3,5},{2,4,6,7,8}}
=> {{1,4,5,7,8},{2,3,6}}
=> ? = 3 - 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [8,6,1,2,3,4,5,7] => {{1,3,5,7,8},{2,4,6}}
=> {{1,4,5},{2,3,6,7,8}}
=> ? = 3 - 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [7,6,1,2,3,4,8,5] => {{1,3,5,7,8},{2,4,6}}
=> {{1,4,5},{2,3,6,7,8}}
=> ? = 3 - 1
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [5,6,1,2,3,8,4,7] => {{1,3,5},{2,4,6,7,8}}
=> {{1,4,5,7,8},{2,3,6}}
=> ? = 3 - 1
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [5,8,1,2,3,7,4,6] => {{1,3,5},{2,4,6,7,8}}
=> {{1,4,5,7,8},{2,3,6}}
=> ? = 3 - 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [8,6,1,2,3,7,4,5] => {{1,3,5,8},{2,4,6,7}}
=> {{1,4,5,7},{2,3,6,8}}
=> ? = 4 - 1
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [5,6,1,2,3,7,8,4] => {{1,3,5},{2,4,6,7,8}}
=> {{1,4,5,7,8},{2,3,6}}
=> ? = 3 - 1
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [8,4,1,2,7,3,5,6] => {{1,3,6,8},{2,4},{5,7}}
=> {{1,4},{2,3,7,8},{5,6}}
=> ? = -2 - 1
[1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [5,8,1,2,6,3,4,7] => {{1,3,5,6},{2,4,7,8}}
=> {{1,4,5,6,8},{2,3,7}}
=> ? = 4 - 1
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [8,7,1,2,6,3,4,5] => {{1,3,5,6,8},{2,4,7}}
=> {{1,4,5,6},{2,3,7,8}}
=> ? = 3 - 1
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [5,7,1,2,6,3,8,4] => {{1,3,5,6},{2,4,7,8}}
=> {{1,4,5,6,8},{2,3,7}}
=> ? = 4 - 1
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [5,8,1,2,6,7,3,4] => {{1,3,5,6,7},{2,4,8}}
=> {{1,4,5,6,7},{2,3,8}}
=> ? = 3 - 1
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [7,3,1,8,2,4,5,6] => {{1,2,3,5,7},{4,6,8}}
=> {{1,2,3,6,7},{4,5,8}}
=> ? = 3 - 1
[1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [8,3,1,6,2,7,4,5] => {{1,2,3,5,8},{4,6,7}}
=> {{1,2,3,6},{4,5,7,8}}
=> ? = 3 - 1
[1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [8,4,1,5,2,3,6,7] => {{1,3,6,7,8},{2,4,5}}
=> {{1,4,6,7,8},{2,3,5}}
=> ? = 3 - 1
[1,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [7,4,1,5,2,3,8,6] => {{1,3,6,7,8},{2,4,5}}
=> {{1,4,6,7,8},{2,3,5}}
=> ? = 3 - 1
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [6,8,1,5,2,3,4,7] => {{1,3,6},{2,4,5,7,8}}
=> {{1,4,6},{2,3,5,7,8}}
=> ? = 3 - 1
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [8,7,1,5,2,3,4,6] => {{1,3,6,8},{2,4,5,7}}
=> {{1,4,6,8},{2,3,5,7}}
=> ? = 4 - 1
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [6,7,1,5,2,3,8,4] => {{1,3,6},{2,4,5,7,8}}
=> {{1,4,6},{2,3,5,7,8}}
=> ? = 3 - 1
Description
The crossing number of a set partition.
This is the maximal number of chords in the standard representation of a set partition, that mutually cross.
Matching statistic: St000614
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00218: Set partitions —inverse Wachs-White-rho⟶ Set partitions
St000614: Set partitions ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 25%
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00218: Set partitions —inverse Wachs-White-rho⟶ Set partitions
St000614: Set partitions ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 25%
Values
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => {{1,3,5},{2,4}}
=> {{1,4},{2,3,5}}
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => {{1,2,4,6},{3,5}}
=> {{1,2,5},{3,4,6}}
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => {{1,3,5,6},{2,4}}
=> {{1,4},{2,3,5,6}}
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => {{1,3,5},{2,4,6}}
=> {{1,4,5},{2,3,6}}
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => {{1,3,5,6},{2,4}}
=> {{1,4},{2,3,5,6}}
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => {{1,3,6},{2,4,5}}
=> {{1,4,6},{2,3,5}}
=> 2 = 3 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => {{1,2,4,6},{3,5}}
=> {{1,2,5},{3,4,6}}
=> 1 = 2 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => {{1,4,6},{2,3,5}}
=> {{1,3,4,6},{2,5}}
=> 2 = 3 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => {{1,3,4,6},{2,5}}
=> {{1,5},{2,3,4,6}}
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => {{1,2,3,5,7},{4,6}}
=> {{1,2,3,6},{4,5,7}}
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => {{1,2,4,6,7},{3,5}}
=> {{1,2,5},{3,4,6,7}}
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => {{1,2,4,6},{3,5,7}}
=> {{1,2,5,6},{3,4,7}}
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => {{1,2,4,6,7},{3,5}}
=> {{1,2,5},{3,4,6,7}}
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => {{1,2,4,7},{3,5,6}}
=> {{1,2,5,7},{3,4,6}}
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => {{1,2,3,5,7},{4,6}}
=> {{1,2,3,6},{4,5,7}}
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => {{1,2,5,7},{3,4,6}}
=> {{1,2,4,5,7},{3,6}}
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => {{1,2,4,5,7},{3,6}}
=> {{1,2,6},{3,4,5,7}}
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => {{1,2,3,5,7},{4,6}}
=> {{1,2,3,6},{4,5,7}}
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => {{1,3,5,6,7},{2,4}}
=> {{1,4},{2,3,5,6,7}}
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => {{1,3,5,6,7},{2,4}}
=> {{1,4},{2,3,5,6,7}}
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => {{1,3,5},{2,4,6,7}}
=> {{1,4,5},{2,3,6,7}}
=> 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => {{1,3,5,7},{2,4,6}}
=> {{1,4,5,7},{2,3,6}}
=> 2 = 3 - 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => {{1,3,5},{2,4,6,7}}
=> {{1,4,5},{2,3,6,7}}
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => {{1,3,5,6,7},{2,4}}
=> {{1,4},{2,3,5,6,7}}
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => {{1,3,5,6,7},{2,4}}
=> {{1,4},{2,3,5,6,7}}
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,7,1,2,6,3,4] => {{1,3,5,6},{2,4,7}}
=> {{1,4,5,6},{2,3,7}}
=> 2 = 3 - 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => {{1,3,5,6,7},{2,4}}
=> {{1,4},{2,3,5,6,7}}
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => {{1,2,3,5,7},{4,6}}
=> {{1,2,3,6},{4,5,7}}
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => {{1,3,6,7},{2,4,5}}
=> {{1,4,6,7},{2,3,5}}
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [6,7,1,5,2,3,4] => {{1,3,6},{2,4,5,7}}
=> {{1,4,7},{2,3,5,6}}
=> 2 = 3 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => {{1,3,6,7},{2,4,5}}
=> {{1,4,6,7},{2,3,5}}
=> 2 = 3 - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => {{1,3,7},{2,4,5,6}}
=> {{1,4,6},{2,3,5,7}}
=> 2 = 3 - 1
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => {{1,2,4,6,7},{3,5}}
=> {{1,2,5},{3,4,6,7}}
=> 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => {{1,2,4,6},{3,5,7}}
=> {{1,2,5,6},{3,4,7}}
=> 2 = 3 - 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => {{1,2,4,6,7},{3,5}}
=> {{1,2,5},{3,4,6,7}}
=> 1 = 2 - 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => {{1,2,4,7},{3,5,6}}
=> {{1,2,5,7},{3,4,6}}
=> 2 = 3 - 1
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => {{1,4,6,7},{2,3,5}}
=> {{1,3,4,6,7},{2,5}}
=> 2 = 3 - 1
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => {{1,4,6},{2,3,5,7}}
=> {{1,3,4,7},{2,5,6}}
=> 2 = 3 - 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => {{1,4,6,7},{2,3,5}}
=> {{1,3,4,6,7},{2,5}}
=> 2 = 3 - 1
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => {{1,3,4,6,7},{2,5}}
=> {{1,5},{2,3,4,6,7}}
=> 1 = 2 - 1
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => {{1,3,4,6},{2,5,7}}
=> {{1,5,6},{2,3,4,7}}
=> 2 = 3 - 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => {{1,3,4,6,7},{2,5}}
=> {{1,5},{2,3,4,6,7}}
=> 1 = 2 - 1
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => {{1,4,7},{2,3,5,6}}
=> {{1,3,4,6},{2,5,7}}
=> 2 = 3 - 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => {{1,3,4,7},{2,5,6}}
=> {{1,5,7},{2,3,4,6}}
=> 2 = 3 - 1
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => {{1,2,3,5,7},{4,6}}
=> {{1,2,3,6},{4,5,7}}
=> 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => {{1,2,5,7},{3,4,6}}
=> {{1,2,4,5,7},{3,6}}
=> 2 = 3 - 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => {{1,2,4,5,7},{3,6}}
=> {{1,2,6},{3,4,5,7}}
=> 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => {{1,5,7},{2,3,4,6}}
=> {{1,3,6},{2,4,5,7}}
=> 2 = 3 - 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => {{1,4,5,7},{2,3,6}}
=> {{1,3,4,5,7},{2,6}}
=> 2 = 3 - 1
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => {{1,3,4,5,7},{2,6}}
=> {{1,6},{2,3,4,5,7}}
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [8,1,2,3,7,4,5,6] => {{1,2,3,4,6,8},{5,7}}
=> {{1,2,3,4,7},{5,6,8}}
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [8,1,2,6,3,4,5,7] => {{1,2,3,5,7,8},{4,6}}
=> {{1,2,3,6},{4,5,7,8}}
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,1,2,8,3,4,5,6] => {{1,2,3,5,7},{4,6,8}}
=> {{1,2,3,6,7},{4,5,8}}
=> ? = 3 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [7,1,2,6,3,4,8,5] => {{1,2,3,5,7,8},{4,6}}
=> {{1,2,3,6},{4,5,7,8}}
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [8,1,2,6,3,7,4,5] => {{1,2,3,5,8},{4,6,7}}
=> {{1,2,3,6,8},{4,5,7}}
=> ? = 3 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,1,2,8,7,3,5,6] => {{1,2,3,4,6,8},{5,7}}
=> {{1,2,3,4,7},{5,6,8}}
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [8,1,2,7,6,3,4,5] => {{1,2,3,5,6,8},{4,7}}
=> {{1,2,3,7},{4,5,6,8}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [3,1,8,2,7,4,5,6] => {{1,2,3,4,6,8},{5,7}}
=> {{1,2,3,4,7},{5,6,8}}
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [8,1,5,2,3,4,6,7] => {{1,2,4,6,7,8},{3,5}}
=> {{1,2,5},{3,4,6,7,8}}
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [7,1,5,2,3,4,8,6] => {{1,2,4,6,7,8},{3,5}}
=> {{1,2,5},{3,4,6,7,8}}
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,1,8,2,3,4,5,7] => {{1,2,4,6},{3,5,7,8}}
=> {{1,2,5,6},{3,4,7,8}}
=> ? = 4 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [8,1,7,2,3,4,5,6] => {{1,2,4,6,8},{3,5,7}}
=> {{1,2,5,6,8},{3,4,7}}
=> ? = 3 - 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [6,1,7,2,3,4,8,5] => {{1,2,4,6},{3,5,7,8}}
=> {{1,2,5,6},{3,4,7,8}}
=> ? = 4 - 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [6,1,5,2,3,8,4,7] => {{1,2,4,6,7,8},{3,5}}
=> {{1,2,5},{3,4,6,7,8}}
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [8,1,5,2,3,7,4,6] => {{1,2,4,6,7,8},{3,5}}
=> {{1,2,5},{3,4,6,7,8}}
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [6,1,8,2,3,7,4,5] => {{1,2,4,6,7},{3,5,8}}
=> {{1,2,5,6,7},{3,4,8}}
=> ? = 3 - 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [6,1,5,2,3,7,8,4] => {{1,2,4,6,7,8},{3,5}}
=> {{1,2,5},{3,4,6,7,8}}
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [8,1,4,2,7,3,5,6] => {{1,2,3,4,6,8},{5,7}}
=> {{1,2,3,4,7},{5,6,8}}
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [8,1,5,2,6,3,4,7] => {{1,2,4,7,8},{3,5,6}}
=> {{1,2,5,7,8},{3,4,6}}
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [7,1,8,2,6,3,4,5] => {{1,2,4,7},{3,5,6,8}}
=> {{1,2,5,8},{3,4,6,7}}
=> ? = 4 - 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [7,1,5,2,6,3,8,4] => {{1,2,4,7,8},{3,5,6}}
=> {{1,2,5,7,8},{3,4,6}}
=> ? = 3 - 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [8,1,5,2,6,7,3,4] => {{1,2,4,8},{3,5,6,7}}
=> {{1,2,5,7},{3,4,6,8}}
=> ? = 4 - 1
[1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [3,1,8,6,2,4,5,7] => {{1,2,3,5,7,8},{4,6}}
=> {{1,2,3,6},{4,5,7,8}}
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,7,8,2,4,5,6] => {{1,2,3,5,7},{4,6,8}}
=> {{1,2,3,6,7},{4,5,8}}
=> ? = 3 - 1
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,1,7,6,2,4,8,5] => {{1,2,3,5,7,8},{4,6}}
=> {{1,2,3,6},{4,5,7,8}}
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,1,8,6,2,7,4,5] => {{1,2,3,5,8},{4,6,7}}
=> {{1,2,3,6,8},{4,5,7}}
=> ? = 3 - 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [7,1,4,8,2,3,5,6] => {{1,2,5,7},{3,4,6,8}}
=> {{1,2,4,5,8},{3,6,7}}
=> ? = 4 - 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [8,1,6,5,2,3,4,7] => {{1,2,4,5,7,8},{3,6}}
=> {{1,2,6},{3,4,5,7,8}}
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [7,1,8,5,2,3,4,6] => {{1,2,4,5,7},{3,6,8}}
=> {{1,2,6,7},{3,4,5,8}}
=> ? = 3 - 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [7,1,6,5,2,3,8,4] => {{1,2,4,5,7,8},{3,6}}
=> {{1,2,6},{3,4,5,7,8}}
=> ? = 2 - 1
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [8,1,4,6,2,7,3,5] => {{1,2,5,8},{3,4,6,7}}
=> {{1,2,4,5,7},{3,6,8}}
=> ? = 4 - 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [8,1,6,5,2,7,3,4] => {{1,2,4,5,8},{3,6,7}}
=> {{1,2,6,8},{3,4,5,7}}
=> ? = 3 - 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [3,1,4,8,7,2,5,6] => {{1,2,3,4,6,8},{5,7}}
=> {{1,2,3,4,7},{5,6,8}}
=> ? = 2 - 1
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,1,8,7,6,2,4,5] => {{1,2,3,5,6,8},{4,7}}
=> {{1,2,3,7},{4,5,6,8}}
=> ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [8,1,4,5,7,2,3,6] => {{1,2,6,8},{3,4,5,7}}
=> {{1,2,4,7},{3,5,6,8}}
=> ? = 4 - 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [8,1,7,5,6,2,3,4] => {{1,2,4,5,6,8},{3,7}}
=> {{1,2,7},{3,4,5,6,8}}
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [2,8,1,3,7,4,5,6] => {{1,2,3,4,6,8},{5,7}}
=> {{1,2,3,4,7},{5,6,8}}
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [2,8,1,6,3,4,5,7] => {{1,2,3,5,7,8},{4,6}}
=> {{1,2,3,6},{4,5,7,8}}
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [2,7,1,8,3,4,5,6] => {{1,2,3,5,7},{4,6,8}}
=> {{1,2,3,6,7},{4,5,8}}
=> ? = 3 - 1
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [2,7,1,6,3,4,8,5] => {{1,2,3,5,7,8},{4,6}}
=> {{1,2,3,6},{4,5,7,8}}
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,8,7,3,5,6] => {{1,2,3,4,6,8},{5,7}}
=> {{1,2,3,4,7},{5,6,8}}
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [2,8,1,7,6,3,4,5] => {{1,2,3,5,6,8},{4,7}}
=> {{1,2,3,7},{4,5,6,8}}
=> ? = 2 - 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [8,3,1,2,7,4,5,6] => {{1,2,3,4,6,8},{5,7}}
=> {{1,2,3,4,7},{5,6,8}}
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [5,8,1,2,3,4,6,7] => {{1,3,5},{2,4,6,7,8}}
=> {{1,4,5},{2,3,6,7,8}}
=> ? = 3 - 1
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [5,7,1,2,3,4,8,6] => {{1,3,5},{2,4,6,7,8}}
=> {{1,4,5},{2,3,6,7,8}}
=> ? = 3 - 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [8,6,1,2,3,4,5,7] => {{1,3,5,7,8},{2,4,6}}
=> {{1,4,5,7,8},{2,3,6}}
=> ? = 3 - 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [7,6,1,2,3,4,8,5] => {{1,3,5,7,8},{2,4,6}}
=> {{1,4,5,7,8},{2,3,6}}
=> ? = 3 - 1
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [5,6,1,2,3,8,4,7] => {{1,3,5},{2,4,6,7,8}}
=> {{1,4,5},{2,3,6,7,8}}
=> ? = 3 - 1
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [5,8,1,2,3,7,4,6] => {{1,3,5},{2,4,6,7,8}}
=> {{1,4,5},{2,3,6,7,8}}
=> ? = 3 - 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [8,6,1,2,3,7,4,5] => {{1,3,5,8},{2,4,6,7}}
=> {{1,4,5,7},{2,3,6,8}}
=> ? = 4 - 1
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block.
Matching statistic: St001596
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001596: Skew partitions ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 25%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001596: Skew partitions ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 25%
Values
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [3,2]
=> [[3,2],[]]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [4,2]
=> [[4,2],[]]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,2]
=> [[4,2],[]]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [3,3]
=> [[3,3],[]]
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,2]
=> [[4,2],[]]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [3,3]
=> [[3,3],[]]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [4,2]
=> [[4,2],[]]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [3,3]
=> [[3,3],[]]
=> 2 = 3 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [4,2]
=> [[4,2],[]]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [5,2]
=> [[5,2],[]]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => [5,2]
=> [[5,2],[]]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [4,3]
=> [[4,3],[]]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => [5,2]
=> [[5,2],[]]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [4,3]
=> [[4,3],[]]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [5,2]
=> [[5,2],[]]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [4,3]
=> [[4,3],[]]
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [5,2]
=> [[5,2],[]]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => [5,2]
=> [[5,2],[]]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => [5,2]
=> [[5,2],[]]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => [5,2]
=> [[5,2],[]]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => [4,3]
=> [[4,3],[]]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => [4,3]
=> [[4,3],[]]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => [4,3]
=> [[4,3],[]]
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => [5,2]
=> [[5,2],[]]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => [5,2]
=> [[5,2],[]]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,7,1,2,6,3,4] => [4,3]
=> [[4,3],[]]
=> 2 = 3 - 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => [5,2]
=> [[5,2],[]]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => [5,2]
=> [[5,2],[]]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => [4,3]
=> [[4,3],[]]
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [6,7,1,5,2,3,4] => [4,3]
=> [[4,3],[]]
=> 2 = 3 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => [4,3]
=> [[4,3],[]]
=> 2 = 3 - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => [4,3]
=> [[4,3],[]]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => [5,2]
=> [[5,2],[]]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => [4,3]
=> [[4,3],[]]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => [5,2]
=> [[5,2],[]]
=> 1 = 2 - 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => [4,3]
=> [[4,3],[]]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => [4,3]
=> [[4,3],[]]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => [4,3]
=> [[4,3],[]]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => [4,3]
=> [[4,3],[]]
=> 2 = 3 - 1
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [5,2]
=> [[5,2],[]]
=> 1 = 2 - 1
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => [4,3]
=> [[4,3],[]]
=> 2 = 3 - 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => [5,2]
=> [[5,2],[]]
=> 1 = 2 - 1
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => [4,3]
=> [[4,3],[]]
=> 2 = 3 - 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => [4,3]
=> [[4,3],[]]
=> 2 = 3 - 1
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => [5,2]
=> [[5,2],[]]
=> 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => [4,3]
=> [[4,3],[]]
=> 2 = 3 - 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => [5,2]
=> [[5,2],[]]
=> 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => [4,3]
=> [[4,3],[]]
=> 2 = 3 - 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => [4,3]
=> [[4,3],[]]
=> 2 = 3 - 1
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => [5,2]
=> [[5,2],[]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [8,1,2,3,7,4,5,6] => [6,2]
=> [[6,2],[]]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [8,1,2,6,3,4,5,7] => [6,2]
=> [[6,2],[]]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,1,2,8,3,4,5,6] => [5,3]
=> [[5,3],[]]
=> ? = 3 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [7,1,2,6,3,4,8,5] => [6,2]
=> [[6,2],[]]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [8,1,2,6,3,7,4,5] => [5,3]
=> [[5,3],[]]
=> ? = 3 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,1,2,8,7,3,5,6] => [6,2]
=> [[6,2],[]]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [8,1,2,5,7,3,4,6] => [5,3]
=> [[5,3],[]]
=> ? = 3 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [8,1,2,7,6,3,4,5] => [6,2]
=> [[6,2],[]]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [3,1,8,2,7,4,5,6] => [6,2]
=> [[6,2],[]]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [8,1,5,2,3,4,6,7] => [6,2]
=> [[6,2],[]]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [7,1,5,2,3,4,8,6] => [6,2]
=> [[6,2],[]]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,1,8,2,3,4,5,7] => [4,4]
=> [[4,4],[]]
=> ? = 4 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [8,1,7,2,3,4,5,6] => [5,3]
=> [[5,3],[]]
=> ? = 3 - 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [6,1,7,2,3,4,8,5] => [4,4]
=> [[4,4],[]]
=> ? = 4 - 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [6,1,5,2,3,8,4,7] => [6,2]
=> [[6,2],[]]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [8,1,5,2,3,7,4,6] => [6,2]
=> [[6,2],[]]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [6,1,8,2,3,7,4,5] => [5,3]
=> [[5,3],[]]
=> ? = 3 - 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [6,1,5,2,3,7,8,4] => [6,2]
=> [[6,2],[]]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [8,1,4,2,7,3,5,6] => [6,2]
=> [[6,2],[]]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [8,1,5,2,6,3,4,7] => [5,3]
=> [[5,3],[]]
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [7,1,8,2,6,3,4,5] => [4,4]
=> [[4,4],[]]
=> ? = 4 - 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [7,1,5,2,6,3,8,4] => [5,3]
=> [[5,3],[]]
=> ? = 3 - 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [8,1,5,2,6,7,3,4] => [4,4]
=> [[4,4],[]]
=> ? = 4 - 1
[1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [3,1,8,6,2,4,5,7] => [6,2]
=> [[6,2],[]]
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,7,8,2,4,5,6] => [5,3]
=> [[5,3],[]]
=> ? = 3 - 1
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,1,7,6,2,4,8,5] => [6,2]
=> [[6,2],[]]
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,1,8,6,2,7,4,5] => [5,3]
=> [[5,3],[]]
=> ? = 3 - 1
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [8,1,4,6,2,3,5,7] => [5,3]
=> [[5,3],[]]
=> ? = 3 - 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [7,1,4,8,2,3,5,6] => [4,4]
=> [[4,4],[]]
=> ? = 4 - 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [7,1,4,6,2,3,8,5] => [5,3]
=> [[5,3],[]]
=> ? = 3 - 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [8,1,6,5,2,3,4,7] => [6,2]
=> [[6,2],[]]
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [7,1,8,5,2,3,4,6] => [5,3]
=> [[5,3],[]]
=> ? = 3 - 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [7,1,6,5,2,3,8,4] => [6,2]
=> [[6,2],[]]
=> ? = 2 - 1
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [8,1,4,6,2,7,3,5] => [4,4]
=> [[4,4],[]]
=> ? = 4 - 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [8,1,6,5,2,7,3,4] => [5,3]
=> [[5,3],[]]
=> ? = 3 - 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [3,1,4,8,7,2,5,6] => [6,2]
=> [[6,2],[]]
=> ? = 2 - 1
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [3,1,8,5,7,2,4,6] => [5,3]
=> [[5,3],[]]
=> ? = 3 - 1
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,1,8,7,6,2,4,5] => [6,2]
=> [[6,2],[]]
=> ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [8,1,4,5,7,2,3,6] => [4,4]
=> [[4,4],[]]
=> ? = 4 - 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [8,1,4,7,6,2,3,5] => [5,3]
=> [[5,3],[]]
=> ? = 3 - 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [8,1,7,5,6,2,3,4] => [6,2]
=> [[6,2],[]]
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [2,8,1,3,7,4,5,6] => [6,2]
=> [[6,2],[]]
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [2,8,1,6,3,4,5,7] => [6,2]
=> [[6,2],[]]
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [2,7,1,8,3,4,5,6] => [5,3]
=> [[5,3],[]]
=> ? = 3 - 1
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [2,7,1,6,3,4,8,5] => [6,2]
=> [[6,2],[]]
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,8,7,3,5,6] => [6,2]
=> [[6,2],[]]
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,8,1,5,7,3,4,6] => [5,3]
=> [[5,3],[]]
=> ? = 3 - 1
[1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [2,8,1,7,6,3,4,5] => [6,2]
=> [[6,2],[]]
=> ? = 2 - 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [8,3,1,2,7,4,5,6] => [6,2]
=> [[6,2],[]]
=> ? = 2 - 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [8,4,1,2,3,5,6,7] => [6,2]
=> [[6,2],[]]
=> ? = 2 - 1
Description
The number of two-by-two squares inside a skew partition.
This is, the number of cells $(i,j)$ in a skew partition for which the box $(i+1,j+1)$ is also a cell inside the skew partition.
Matching statistic: St001816
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001816: Standard tableaux ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 25%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001816: Standard tableaux ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 25%
Values
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [3,2]
=> [[1,2,5],[3,4]]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [4,2]
=> [[1,2,5,6],[3,4]]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,2]
=> [[1,2,5,6],[3,4]]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [3,3]
=> [[1,2,3],[4,5,6]]
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,2]
=> [[1,2,5,6],[3,4]]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [3,3]
=> [[1,2,3],[4,5,6]]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [4,2]
=> [[1,2,5,6],[3,4]]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [3,3]
=> [[1,2,3],[4,5,6]]
=> 2 = 3 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [4,2]
=> [[1,2,5,6],[3,4]]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,7,1,2,6,3,4] => [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 2 = 3 - 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [6,7,1,5,2,3,4] => [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 2 = 3 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 2 = 3 - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 1 = 2 - 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 2 = 3 - 1
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 1 = 2 - 1
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 2 = 3 - 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 1 = 2 - 1
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 2 = 3 - 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 2 = 3 - 1
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 2 = 3 - 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 2 = 3 - 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 2 = 3 - 1
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [8,1,2,3,7,4,5,6] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [8,1,2,6,3,4,5,7] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,1,2,8,3,4,5,6] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> ? = 3 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [7,1,2,6,3,4,8,5] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [8,1,2,6,3,7,4,5] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> ? = 3 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,1,2,8,7,3,5,6] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [8,1,2,5,7,3,4,6] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> ? = 3 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [8,1,2,7,6,3,4,5] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [3,1,8,2,7,4,5,6] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [8,1,5,2,3,4,6,7] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [7,1,5,2,3,4,8,6] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,1,8,2,3,4,5,7] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> ? = 4 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [8,1,7,2,3,4,5,6] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> ? = 3 - 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [6,1,7,2,3,4,8,5] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> ? = 4 - 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [6,1,5,2,3,8,4,7] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [8,1,5,2,3,7,4,6] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [6,1,8,2,3,7,4,5] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> ? = 3 - 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [6,1,5,2,3,7,8,4] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [8,1,4,2,7,3,5,6] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [8,1,5,2,6,3,4,7] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [7,1,8,2,6,3,4,5] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> ? = 4 - 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [7,1,5,2,6,3,8,4] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> ? = 3 - 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [8,1,5,2,6,7,3,4] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> ? = 4 - 1
[1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [3,1,8,6,2,4,5,7] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,7,8,2,4,5,6] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> ? = 3 - 1
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,1,7,6,2,4,8,5] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,1,8,6,2,7,4,5] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> ? = 3 - 1
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [8,1,4,6,2,3,5,7] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> ? = 3 - 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [7,1,4,8,2,3,5,6] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> ? = 4 - 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [7,1,4,6,2,3,8,5] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> ? = 3 - 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [8,1,6,5,2,3,4,7] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [7,1,8,5,2,3,4,6] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> ? = 3 - 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [7,1,6,5,2,3,8,4] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> ? = 2 - 1
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [8,1,4,6,2,7,3,5] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> ? = 4 - 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [8,1,6,5,2,7,3,4] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> ? = 3 - 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [3,1,4,8,7,2,5,6] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> ? = 2 - 1
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [3,1,8,5,7,2,4,6] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> ? = 3 - 1
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,1,8,7,6,2,4,5] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [8,1,4,5,7,2,3,6] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> ? = 4 - 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [8,1,4,7,6,2,3,5] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> ? = 3 - 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [8,1,7,5,6,2,3,4] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [2,8,1,3,7,4,5,6] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [2,8,1,6,3,4,5,7] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [2,7,1,8,3,4,5,6] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> ? = 3 - 1
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [2,7,1,6,3,4,8,5] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,8,7,3,5,6] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,8,1,5,7,3,4,6] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> ? = 3 - 1
[1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [2,8,1,7,6,3,4,5] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> ? = 2 - 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [8,3,1,2,7,4,5,6] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> ? = 2 - 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [8,4,1,2,3,5,6,7] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> ? = 2 - 1
Description
Eigenvalues of the top-to-random operator acting on a simple module.
These eigenvalues are given in [1] and [3].
The simple module of the symmetric group indexed by a partition $\lambda$ has dimension equal to the number of standard tableaux of shape $\lambda$. Hence, the eigenvalues of any linear operator defined on this module can be indexed by standard tableaux of shape $\lambda$; this statistic gives all the eigenvalues of the operator acting on the module.
This statistic bears different names, such as the type in [2] or eig in [3].
Similarly, the eigenvalues of the random-to-random operator acting on a simple module is [[St000508]].
Matching statistic: St001258
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001258: Dyck paths ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 25%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001258: Dyck paths ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 25%
Values
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 4 = 2 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 4 = 2 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 4 = 2 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 3 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 4 = 2 + 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 5 = 3 + 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 4 = 2 + 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 5 = 3 + 2
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 4 = 2 + 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2 + 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 2 + 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 2 + 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 3 + 2
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [5,4,2]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 2 + 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 3 + 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 4 = 2 + 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 2 + 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 2 + 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 2 + 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 3 + 2
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 3 + 2
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 3 + 2
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 2 + 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 2 + 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 3 + 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 2 + 2
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 2 + 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 3 + 2
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 3 + 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 3 + 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 2
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 2 + 2
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 3 + 2
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 2 + 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 + 2
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 3 + 2
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 3 + 2
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 3 + 2
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 4 = 2 + 2
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 3 + 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 4 = 2 + 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 3 + 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 5 = 3 + 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [5,4,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 2 + 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 3 + 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 4 = 2 + 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 3 + 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 5 = 3 + 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 4 = 2 + 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 2 + 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3]
=> [1,0,1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 3 + 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 2 + 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [6,5,3]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 2 + 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [6,4,3]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 3 + 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [5,4,3]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2 + 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> ? = 2 + 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2]
=> [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> ? = 2 + 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2]
=> [1,0,1,0,1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> ? = 2 + 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> ? = 4 + 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> ? = 3 + 2
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> ? = 4 + 2
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2]
=> [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 2 + 2
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 4 = 2 + 2
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 4 = 2 + 2
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 4 = 2 + 2
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 5 = 3 + 2
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 4 = 2 + 2
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 5 = 3 + 2
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 4 = 2 + 2
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 4 = 2 + 2
[1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 4 = 2 + 2
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 4 = 2 + 2
[1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 4 = 2 + 2
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 4 = 2 + 2
[1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 4 = 2 + 2
[1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 4 = 2 + 2
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 4 = 2 + 2
[1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 4 = 2 + 2
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 4 = 2 + 2
[1,1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 4 = 2 + 2
[1,1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0,0,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 4 = 2 + 2
Description
Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra.
For at most 6 simple modules this statistic coincides with the injective dimension of the regular module as a bimodule.
Matching statistic: St001644
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001644: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 25%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001644: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 25%
Values
[1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,3,1,7,6,2,5] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => ([(0,6),(1,2),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [7,3,1,5,6,2,4] => ([(0,4),(0,6),(1,2),(1,3),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => ([(0,5),(1,4),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => ([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => ([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 3
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => ([(0,6),(1,6),(2,3),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 2
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => ([(0,6),(1,6),(2,3),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => ([(0,5),(1,4),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,4,5,1,7,2] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> ? = 3
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ([(0,6),(1,5),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => ([(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ([(0,6),(1,2),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 3
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ([(0,1),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,3,4,1,6,2,5] => ([(0,4),(0,6),(1,2),(1,3),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ([(0,1),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,4,7,6,1,5] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,4,5,6,1,3] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> ? = 3
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => ([(0,3),(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,7,5,6,1,4] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ([(0,6),(1,5),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => ([(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [5,3,4,1,6,7,2] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [7,3,4,5,6,1,2] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [5,6,1,2,3,7,8,4] => ([(0,7),(1,7),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [5,4,1,2,8,7,3,6] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,4),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [5,8,1,2,6,7,3,4] => ([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 3
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [5,4,1,2,6,8,3,7] => ([(0,7),(1,4),(2,5),(2,6),(3,5),(3,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [8,4,1,2,6,7,3,5] => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [5,3,1,2,6,7,8,4] => ([(0,6),(1,6),(2,6),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [5,7,1,2,6,3,8,4] => ([(0,6),(1,5),(1,7),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 3
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [5,4,1,2,6,7,8,3] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(4,5),(4,6),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [2,6,1,5,3,7,8,4] => ([(0,6),(1,6),(2,3),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3,1,8,7,2,5,6] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,4),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [4,3,1,8,6,2,5,7] => ([(0,7),(1,3),(1,4),(2,5),(2,7),(3,4),(3,6),(4,6),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [8,4,1,5,7,2,3,6] => ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [7,8,1,5,6,2,3,4] => ([(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 3
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [8,4,1,5,6,2,3,7] => ([(0,7),(1,6),(1,7),(2,4),(2,5),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,1,5,7,2,8,6] => ([(0,4),(1,7),(2,4),(2,7),(3,5),(3,6),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [4,3,1,7,6,2,8,5] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(2,7),(3,6),(3,7),(4,5),(4,7),(5,7)],8)
=> ? = 2
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [8,4,1,7,6,2,3,5] => ([(0,4),(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,1,5,8,2,6,7] => ([(0,4),(1,4),(2,7),(3,5),(3,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [6,3,1,5,2,7,8,4] => ([(0,6),(1,6),(2,5),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [8,3,1,5,7,2,4,6] => ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,6),(2,7),(3,5),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [7,4,1,5,6,2,8,3] => ([(0,6),(1,5),(1,7),(2,4),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [8,3,1,5,6,2,4,7] => ([(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [8,3,1,7,6,2,4,5] => ([(0,3),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [2,4,1,5,8,7,3,6] => ([(0,4),(1,7),(2,4),(2,7),(3,5),(3,6),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [2,8,1,5,6,7,3,4] => ([(0,1),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 3
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,4,1,5,6,8,3,7] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 2
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [2,4,1,8,6,7,3,5] => ([(0,2),(1,2),(1,7),(3,4),(3,5),(3,6),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [8,4,1,5,2,7,3,6] => ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,6),(2,7),(3,5),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [6,8,1,5,2,7,3,4] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 4
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [6,4,1,5,2,8,3,7] => ([(0,2),(1,5),(1,7),(2,6),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
Description
The dimension of a graph.
The dimension of a graph is the least integer $n$ such that there exists a representation of the graph in the Euclidean space of dimension $n$ with all vertices distinct and all edges having unit length. Edges are allowed to intersect, however.
Matching statistic: St001330
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 12%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 12%
Values
[1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,3,1,7,6,2,5] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => ([(0,6),(1,2),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [7,3,1,5,6,2,4] => ([(0,4),(0,6),(1,2),(1,3),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => ([(0,5),(1,4),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => ([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => ([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 3
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => ([(0,6),(1,6),(2,3),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 2
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => ([(0,6),(1,6),(2,3),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => ([(0,5),(1,4),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,4,5,1,7,2] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> ? = 3
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ([(0,6),(1,5),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => ([(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ([(0,6),(1,2),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 3
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ([(0,1),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,3,4,1,6,2,5] => ([(0,4),(0,6),(1,2),(1,3),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ([(0,1),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,4,7,6,1,5] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,4,5,6,1,3] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> ? = 3
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => ([(0,3),(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,7,5,6,1,4] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ([(0,6),(1,5),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => ([(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [5,3,4,1,6,7,2] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [7,3,4,5,6,1,2] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [5,6,1,2,3,7,8,4] => ([(0,7),(1,7),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [5,4,1,2,8,7,3,6] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,4),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [5,8,1,2,6,7,3,4] => ([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 3
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [5,4,1,2,6,8,3,7] => ([(0,7),(1,4),(2,5),(2,6),(3,5),(3,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [8,4,1,2,6,7,3,5] => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [5,3,1,2,6,7,8,4] => ([(0,6),(1,6),(2,6),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [5,7,1,2,6,3,8,4] => ([(0,6),(1,5),(1,7),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 3
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [5,4,1,2,6,7,8,3] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(4,5),(4,6),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [2,6,1,5,3,7,8,4] => ([(0,6),(1,6),(2,3),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3,1,8,7,2,5,6] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,4),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [4,3,1,8,6,2,5,7] => ([(0,7),(1,3),(1,4),(2,5),(2,7),(3,4),(3,6),(4,6),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,4,1,5,6,8,3,7] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> 2
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [2,5,1,3,6,7,8,4] => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> 2
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,4,1,5,6,7,8,3] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> 2
[1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,1,5,2,6,7,8,4] => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> 2
[1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [2,3,4,6,1,8,5,7] => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> 2
[1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [2,3,4,7,1,5,8,6] => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> 2
[1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [2,3,4,6,1,7,8,5] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> 2
[1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [2,3,4,8,1,5,6,7] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> 2
[1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [2,3,5,1,6,7,8,4] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> 2
[1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [3,1,4,5,7,2,8,6] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> 2
[1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [3,1,4,5,8,2,6,7] => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> 2
[1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [2,3,4,5,7,1,8,6] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> 2
[1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [2,3,4,5,8,1,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2
[1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,5,6,8,3,7] => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> 2
[1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [3,1,4,5,6,8,2,7] => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> 2
[1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [5,1,2,3,6,7,8,4] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> 2
[1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [4,1,2,5,6,7,8,3] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St001200
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 25%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 25%
Values
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 3
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 3
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 3
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 3
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 3
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 3
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 3
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 2
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 3
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 3
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 3
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 3
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 3
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 3
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 3
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 3
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 2
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 3
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 3
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 3
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ? = 3
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,3,2,2,1]
=> [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ? = 3
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [5,4,2,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [5,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ? = 3
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ?
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ?
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,1,1]
=> [1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ?
=> ? = 2
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
Description
The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
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