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Your data matches 12 different statistics following compositions of up to 3 maps.
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Matching statistic: St000460
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(load all 2 compositions to match this statistic)
St000460: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 2
[1,1]
=> 2
[3]
=> 3
[2,1]
=> 3
[1,1,1]
=> 3
[4]
=> 4
[3,1]
=> 4
[2,2]
=> 1
[2,1,1]
=> 4
[1,1,1,1]
=> 4
[5]
=> 5
[4,1]
=> 5
[3,2]
=> 1
[3,1,1]
=> 5
[2,2,1]
=> 1
[2,1,1,1]
=> 5
[1,1,1,1,1]
=> 5
[6]
=> 6
[5,1]
=> 6
[4,2]
=> 1
[4,1,1]
=> 6
[3,3]
=> 2
[3,2,1]
=> 1
[3,1,1,1]
=> 6
[2,2,2]
=> 2
[2,2,1,1]
=> 1
[2,1,1,1,1]
=> 6
[1,1,1,1,1,1]
=> 6
[7]
=> 7
[6,1]
=> 7
[5,2]
=> 1
[5,1,1]
=> 7
[4,3]
=> 2
[4,2,1]
=> 1
[4,1,1,1]
=> 7
[3,3,1]
=> 2
[3,2,2]
=> 2
[3,2,1,1]
=> 1
[3,1,1,1,1]
=> 7
[2,2,2,1]
=> 2
[2,2,1,1,1]
=> 1
[2,1,1,1,1,1]
=> 7
[1,1,1,1,1,1,1]
=> 7
[8]
=> 8
[7,1]
=> 8
[6,2]
=> 1
[6,1,1]
=> 8
[5,3]
=> 2
[5,2,1]
=> 1
Description
The hook length of the last cell along the main diagonal of an integer partition.
Matching statistic: St000461
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000461: Permutations ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 58%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000461: Permutations ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 58%
Values
[1]
=> [1,0]
=> [1] => [1] => ? = 1
[2]
=> [1,0,1,0]
=> [1,2] => [1,2] => 2
[1,1]
=> [1,1,0,0]
=> [2,1] => [1,2] => 2
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 3
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,2,3] => 3
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => 3
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3,4] => 4
[2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => [1,3,2] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,2,3,4] => 4
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => 4
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4,5] => 5
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,4,3] => 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,3,4,5] => 5
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => 5
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 5
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [1,2,3,4,5,6] => 6
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,3,5,4] => 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [1,2,3,4,5,6] => 6
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,4,2,3] => 2
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,2,4,5,3] => 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => [1,2,3,4,5,6] => 6
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,4,2,3] => 2
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [1,3,4,5,2] => 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => [1,2,3,4,5,6] => 6
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => 6
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 7
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [1,2,3,4,5,6,7] => 7
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => [1,2,3,4,6,5] => 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [1,2,3,4,5,6,7] => 7
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,2,5,3,4] => 2
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => [1,2,3,5,6,4] => 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [1,2,3,4,5,6,7] => 7
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,4,5,2,3] => 2
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,2,5,3,4] => 2
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,3,5,6,2] => [1,2,4,5,6,3] => 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [1,2,3,4,5,6,7] => 7
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [1,4,5,2,3] => 2
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,1] => [1,3,4,5,6,2] => 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => [1,2,3,4,5,6,7] => 7
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => 7
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => ? ∊ {8,8,8,8,8,8,8,8}
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => [1,2,3,4,5,6,7,8] => ? ∊ {8,8,8,8,8,8,8,8}
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [1,2,3,4,5,7,6] => 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,7,8,6] => [1,2,3,4,5,6,7,8] => ? ∊ {8,8,8,8,8,8,8,8}
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,6,4,5,3] => [1,2,3,6,4,5] => 2
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => [1,2,3,4,6,7,5] => 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,5] => [1,2,3,4,5,6,7,8] => ? ∊ {8,8,8,8,8,8,8,8}
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => [1,5,2,3,4] => 3
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,5,3,4,6,2] => [1,2,5,6,3,4] => 2
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => [1,2,3,6,4,5] => 2
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => [1,2,3,5,6,7,4] => 1
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,4] => [1,2,3,4,5,6,7,8] => ? ∊ {8,8,8,8,8,8,8,8}
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,3] => [1,2,3,4,5,6,7,8] => ? ∊ {8,8,8,8,8,8,8,8}
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,2] => [1,2,3,4,5,6,7,8] => ? ∊ {8,8,8,8,8,8,8,8}
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,1] => [1,2,3,4,5,6,7,8] => ? ∊ {8,8,8,8,8,8,8,8}
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => ? ∊ {1,1,1,1,1,1,9,9,9,9,9,9,9,9,9}
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,9,8] => [1,2,3,4,5,6,7,8,9] => ? ∊ {1,1,1,1,1,1,9,9,9,9,9,9,9,9,9}
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,8,7,6] => [1,2,3,4,5,6,8,7] => ? ∊ {1,1,1,1,1,1,9,9,9,9,9,9,9,9,9}
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,8,9,7] => [1,2,3,4,5,6,7,8,9] => ? ∊ {1,1,1,1,1,1,9,9,9,9,9,9,9,9,9}
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,4,7,6,8,5] => [1,2,3,4,5,7,8,6] => ? ∊ {1,1,1,1,1,1,9,9,9,9,9,9,9,9,9}
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,5,7,8,9,6] => [1,2,3,4,5,6,7,8,9] => ? ∊ {1,1,1,1,1,1,9,9,9,9,9,9,9,9,9}
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,3,6,5,7,8,4] => [1,2,3,4,6,7,8,5] => ? ∊ {1,1,1,1,1,1,9,9,9,9,9,9,9,9,9}
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,9,5] => [1,2,3,4,5,6,7,8,9] => ? ∊ {1,1,1,1,1,1,9,9,9,9,9,9,9,9,9}
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,2,5,4,6,7,8,3] => [1,2,3,5,6,7,8,4] => ? ∊ {1,1,1,1,1,1,9,9,9,9,9,9,9,9,9}
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,9,4] => [1,2,3,4,5,6,7,8,9] => ? ∊ {1,1,1,1,1,1,9,9,9,9,9,9,9,9,9}
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,4,3,5,6,7,8,2] => [1,2,4,5,6,7,8,3] => ? ∊ {1,1,1,1,1,1,9,9,9,9,9,9,9,9,9}
[3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,9,3] => [1,2,3,4,5,6,7,8,9] => ? ∊ {1,1,1,1,1,1,9,9,9,9,9,9,9,9,9}
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,1] => [1,3,4,5,6,7,8,2] => ? ∊ {1,1,1,1,1,1,9,9,9,9,9,9,9,9,9}
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,9,2] => [1,2,3,4,5,6,7,8,9] => ? ∊ {1,1,1,1,1,1,9,9,9,9,9,9,9,9,9}
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,1] => [1,2,3,4,5,6,7,8,9] => ? ∊ {1,1,1,1,1,1,9,9,9,9,9,9,9,9,9}
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10}
[9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,10,9] => [1,2,3,4,5,6,7,8,9,10] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10}
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,6,9,8,7] => [1,2,3,4,5,6,7,9,8] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10}
[8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,7,9,10,8] => [1,2,3,4,5,6,7,8,9,10] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10}
[7,3]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,4,8,6,7,5] => [1,2,3,4,5,8,6,7] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10}
[7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,4,5,8,7,9,6] => [1,2,3,4,5,6,8,9,7] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10}
[7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,5,6,8,9,10,7] => [1,2,3,4,5,6,7,8,9,10] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10}
[6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,3,7,5,6,8,4] => [1,2,3,4,7,8,5,6] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10}
[6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,4,8,7,6,5] => [1,2,3,4,5,8,6,7] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10}
[6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,3,4,7,6,8,9,5] => [1,2,3,4,5,7,8,9,6] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10}
[6,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,4,5,7,8,9,10,6] => [1,2,3,4,5,6,7,8,9,10] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10}
[5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,2,6,4,5,7,8,3] => [1,2,3,6,7,8,4,5] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10}
[5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,3,7,6,5,8,4] => [1,2,3,4,7,8,5,6] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10}
[5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,2,3,6,5,7,8,9,4] => [1,2,3,4,6,7,8,9,5] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10}
[5,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,9,10,5] => [1,2,3,4,5,6,7,8,9,10] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10}
[4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [5,2,3,4,6,7,1] => [1,5,6,7,2,3,4] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10}
[4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,5,3,4,6,7,8,2] => [1,2,5,6,7,8,3,4] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10}
[4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,2,6,5,4,7,8,3] => [1,2,3,6,7,8,4,5] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10}
[4,2,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,2,5,4,6,7,8,9,3] => [1,2,3,5,6,7,8,9,4] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10}
[4,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,9,10,4] => [1,2,3,4,5,6,7,8,9,10] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10}
[3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [5,2,4,3,6,7,1] => [1,5,6,7,2,3,4] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10}
[3,3,1,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [4,2,3,5,6,7,8,1] => [1,4,5,6,7,8,2,3] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10}
[3,2,2,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,5,4,3,6,7,8,2] => [1,2,5,6,7,8,3,4] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10}
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,4,3,5,6,7,8,9,2] => [1,2,4,5,6,7,8,9,3] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10}
[3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,9,10,3] => [1,2,3,4,5,6,7,8,9,10] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10}
[2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [5,3,4,2,6,7,1] => [1,5,6,7,2,3,4] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10}
Description
The rix statistic of a permutation.
This statistic is defined recursively as follows: rix([])=0, and if wi=max, then
rix(w) := 0 if i = 1 < k,
rix(w) := 1 + rix(w_1,w_2,\dots,w_{k−1}) if i = k and
rix(w) := rix(w_{i+1},w_{i+2},\dots,w_k) if 1 < i < k.
Matching statistic: St000989
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000989: Permutations ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 58%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000989: Permutations ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 58%
Values
[1]
=> [1,0]
=> [1] => [1] => ? = 1 - 1
[2]
=> [1,0,1,0]
=> [1,2] => [1,2] => 1 = 2 - 1
[1,1]
=> [1,1,0,0]
=> [2,1] => [1,2] => 1 = 2 - 1
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 2 = 3 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,2,3] => 2 = 3 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => 2 = 3 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3,4] => 3 = 4 - 1
[2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => [1,3,2] => 0 = 1 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,2,3,4] => 3 = 4 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => 3 = 4 - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 4 = 5 - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4,5] => 4 = 5 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,4,3] => 0 = 1 - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,3,4,5] => 4 = 5 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => 0 = 1 - 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => 4 = 5 - 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 4 = 5 - 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 5 = 6 - 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [1,2,3,4,5,6] => 5 = 6 - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,3,5,4] => 0 = 1 - 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [1,2,3,4,5,6] => 5 = 6 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,4,2,3] => 1 = 2 - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,2,4,5,3] => 0 = 1 - 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => [1,2,3,4,5,6] => 5 = 6 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,4,2,3] => 1 = 2 - 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [1,3,4,5,2] => 0 = 1 - 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => [1,2,3,4,5,6] => 5 = 6 - 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => 5 = 6 - 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 6 = 7 - 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [1,2,3,4,5,6,7] => 6 = 7 - 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => [1,2,3,4,6,5] => 0 = 1 - 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [1,2,3,4,5,6,7] => 6 = 7 - 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,2,5,3,4] => 1 = 2 - 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => [1,2,3,5,6,4] => 0 = 1 - 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [1,2,3,4,5,6,7] => 6 = 7 - 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,4,5,2,3] => 1 = 2 - 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,2,5,3,4] => 1 = 2 - 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,3,5,6,2] => [1,2,4,5,6,3] => 0 = 1 - 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [1,2,3,4,5,6,7] => 6 = 7 - 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [1,4,5,2,3] => 1 = 2 - 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,1] => [1,3,4,5,6,2] => 0 = 1 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => [1,2,3,4,5,6,7] => 6 = 7 - 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => 6 = 7 - 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => ? ∊ {8,8,8,8,8,8,8,8} - 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => [1,2,3,4,5,6,7,8] => ? ∊ {8,8,8,8,8,8,8,8} - 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [1,2,3,4,5,7,6] => 0 = 1 - 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,7,8,6] => [1,2,3,4,5,6,7,8] => ? ∊ {8,8,8,8,8,8,8,8} - 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,6,4,5,3] => [1,2,3,6,4,5] => 1 = 2 - 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => [1,2,3,4,6,7,5] => 0 = 1 - 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,5] => [1,2,3,4,5,6,7,8] => ? ∊ {8,8,8,8,8,8,8,8} - 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => [1,5,2,3,4] => 2 = 3 - 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,5,3,4,6,2] => [1,2,5,6,3,4] => 1 = 2 - 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => [1,2,3,6,4,5] => 1 = 2 - 1
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => [1,2,3,5,6,7,4] => 0 = 1 - 1
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,4] => [1,2,3,4,5,6,7,8] => ? ∊ {8,8,8,8,8,8,8,8} - 1
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,3] => [1,2,3,4,5,6,7,8] => ? ∊ {8,8,8,8,8,8,8,8} - 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,2] => [1,2,3,4,5,6,7,8] => ? ∊ {8,8,8,8,8,8,8,8} - 1
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,1] => [1,2,3,4,5,6,7,8] => ? ∊ {8,8,8,8,8,8,8,8} - 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => ? ∊ {1,1,1,1,1,1,9,9,9,9,9,9,9,9,9} - 1
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,9,8] => [1,2,3,4,5,6,7,8,9] => ? ∊ {1,1,1,1,1,1,9,9,9,9,9,9,9,9,9} - 1
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,8,7,6] => [1,2,3,4,5,6,8,7] => ? ∊ {1,1,1,1,1,1,9,9,9,9,9,9,9,9,9} - 1
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,8,9,7] => [1,2,3,4,5,6,7,8,9] => ? ∊ {1,1,1,1,1,1,9,9,9,9,9,9,9,9,9} - 1
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,4,7,6,8,5] => [1,2,3,4,5,7,8,6] => ? ∊ {1,1,1,1,1,1,9,9,9,9,9,9,9,9,9} - 1
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,5,7,8,9,6] => [1,2,3,4,5,6,7,8,9] => ? ∊ {1,1,1,1,1,1,9,9,9,9,9,9,9,9,9} - 1
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,3,6,5,7,8,4] => [1,2,3,4,6,7,8,5] => ? ∊ {1,1,1,1,1,1,9,9,9,9,9,9,9,9,9} - 1
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,9,5] => [1,2,3,4,5,6,7,8,9] => ? ∊ {1,1,1,1,1,1,9,9,9,9,9,9,9,9,9} - 1
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,2,5,4,6,7,8,3] => [1,2,3,5,6,7,8,4] => ? ∊ {1,1,1,1,1,1,9,9,9,9,9,9,9,9,9} - 1
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,9,4] => [1,2,3,4,5,6,7,8,9] => ? ∊ {1,1,1,1,1,1,9,9,9,9,9,9,9,9,9} - 1
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,4,3,5,6,7,8,2] => [1,2,4,5,6,7,8,3] => ? ∊ {1,1,1,1,1,1,9,9,9,9,9,9,9,9,9} - 1
[3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,9,3] => [1,2,3,4,5,6,7,8,9] => ? ∊ {1,1,1,1,1,1,9,9,9,9,9,9,9,9,9} - 1
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,1] => [1,3,4,5,6,7,8,2] => ? ∊ {1,1,1,1,1,1,9,9,9,9,9,9,9,9,9} - 1
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,9,2] => [1,2,3,4,5,6,7,8,9] => ? ∊ {1,1,1,1,1,1,9,9,9,9,9,9,9,9,9} - 1
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,1] => [1,2,3,4,5,6,7,8,9] => ? ∊ {1,1,1,1,1,1,9,9,9,9,9,9,9,9,9} - 1
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10} - 1
[9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,10,9] => [1,2,3,4,5,6,7,8,9,10] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10} - 1
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,6,9,8,7] => [1,2,3,4,5,6,7,9,8] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10} - 1
[8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,7,9,10,8] => [1,2,3,4,5,6,7,8,9,10] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10} - 1
[7,3]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,4,8,6,7,5] => [1,2,3,4,5,8,6,7] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10} - 1
[7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,4,5,8,7,9,6] => [1,2,3,4,5,6,8,9,7] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10} - 1
[7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,5,6,8,9,10,7] => [1,2,3,4,5,6,7,8,9,10] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10} - 1
[6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,3,7,5,6,8,4] => [1,2,3,4,7,8,5,6] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10} - 1
[6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,4,8,7,6,5] => [1,2,3,4,5,8,6,7] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10} - 1
[6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,3,4,7,6,8,9,5] => [1,2,3,4,5,7,8,9,6] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10} - 1
[6,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,4,5,7,8,9,10,6] => [1,2,3,4,5,6,7,8,9,10] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10} - 1
[5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,2,6,4,5,7,8,3] => [1,2,3,6,7,8,4,5] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10} - 1
[5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,3,7,6,5,8,4] => [1,2,3,4,7,8,5,6] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10} - 1
[5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,2,3,6,5,7,8,9,4] => [1,2,3,4,6,7,8,9,5] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10} - 1
[5,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,9,10,5] => [1,2,3,4,5,6,7,8,9,10] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10} - 1
[4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [5,2,3,4,6,7,1] => [1,5,6,7,2,3,4] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10} - 1
[4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,5,3,4,6,7,8,2] => [1,2,5,6,7,8,3,4] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10} - 1
[4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,2,6,5,4,7,8,3] => [1,2,3,6,7,8,4,5] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10} - 1
[4,2,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,2,5,4,6,7,8,9,3] => [1,2,3,5,6,7,8,9,4] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10} - 1
[4,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,9,10,4] => [1,2,3,4,5,6,7,8,9,10] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10} - 1
[3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [5,2,4,3,6,7,1] => [1,5,6,7,2,3,4] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10} - 1
[3,3,1,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [4,2,3,5,6,7,8,1] => [1,4,5,6,7,8,2,3] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10} - 1
[3,2,2,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,5,4,3,6,7,8,2] => [1,2,5,6,7,8,3,4] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10} - 1
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,4,3,5,6,7,8,9,2] => [1,2,4,5,6,7,8,9,3] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10} - 1
[3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,9,10,3] => [1,2,3,4,5,6,7,8,9,10] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10} - 1
[2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [5,3,4,2,6,7,1] => [1,5,6,7,2,3,4] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,10,10,10,10,10,10,10,10,10,10} - 1
Description
The number of final rises of a permutation.
For a permutation \pi of length n, this is the maximal k such that
\pi(n-k) \leq \pi(n-k+1) \leq \cdots \leq \pi(n-1) \leq \pi(n).
Equivalently, this is n-1 minus the position of the last descent [[St000653]].
Matching statistic: St001199
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 33%
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 33%
Values
[1]
=> [1,0]
=> [1,0]
=> [1,0]
=> ? = 1
[2]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {2,2}
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {2,2}
[3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {3,3,3}
[2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {3,3,3}
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {3,3,3}
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,4,4,4,4}
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,4,4,4,4}
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {1,4,4,4,4}
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,4,4,4,4}
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,4,4,4,4}
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,5,5,5,5,5}
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,5,5,5,5,5}
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,5,5,5,5,5}
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,5,5,5,5,5}
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,5,5,5,5,5}
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,5,5,5,5,5}
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,5,5,5,5,5}
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,2,2,6,6,6,6,6,6}
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,2,2,6,6,6,6,6,6}
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,6,6,6,6,6,6}
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,2,2,6,6,6,6,6,6}
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,6,6,6,6,6,6}
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,6,6,6,6,6,6}
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,2,2,6,6,6,6,6,6}
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,6,6,6,6,6,6}
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,2,2,6,6,6,6,6,6}
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,2,2,6,6,6,6,6,6}
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,7,7,7,7,7,7,7}
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,7,7,7,7,7,7,7}
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,7,7,7,7,7,7,7}
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,7,7,7,7,7,7,7}
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,7,7,7,7,7,7,7}
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,7,7,7,7,7,7,7}
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,7,7,7,7,7,7,7}
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,7,7,7,7,7,7,7}
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,7,7,7,7,7,7,7}
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,7,7,7,7,7,7,7}
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,7,7,7,7,7,7,7}
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,7,7,7,7,7,7,7}
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,7,7,7,7,7,7,7}
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 1
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> 1
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1
[3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 2
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> 1
[2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> 2
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 1
[4,4,2]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 1
[4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> 1
[4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> 2
[3,3,3,1]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 1
[3,3,2,2]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 2
[3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> 1
[3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> 2
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 3
[2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> 2
[5,4,2]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 1
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> 1
[4,4,3]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 1
[4,4,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> 1
[4,3,3,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> 1
[4,3,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> 2
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> 2
[3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> 1
[3,3,2,2,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> 2
[3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [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]
=> 3
[2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [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]
=> 3
[5,5,2]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 1
[5,4,3]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> 1
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 1
[4,4,3,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> 1
[4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> 2
[4,3,3,2]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> 2
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1
[3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> 2
[3,3,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [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]
=> 3
[2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [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]
=> 4
Description
The dominant dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA.
Matching statistic: St001772
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001772: Signed permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 83%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001772: Signed permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 83%
Values
[1]
=> [1,0,1,0]
=> [1,2] => [1,2] => 1
[2]
=> [1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 2
[1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => 3
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 3
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => 3
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1,4] => 4
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 4
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,4,2] => 4
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => 4
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => [5,4,3,2,1,6] => ? ∊ {1,1,5}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => [3,4,2,1,5] => ? ∊ {1,1,5}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 5
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 5
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 5
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,5,3,2] => 5
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => [1,6,5,4,3,2] => ? ∊ {1,1,5}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => ? ∊ {1,1,1,2,2,6,6,6}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,3,2,1,6] => [4,5,3,2,1,6] => ? ∊ {1,1,1,2,2,6,6,6}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [3,2,4,1,5] => ? ∊ {1,1,1,2,2,6,6,6}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,4,3,1,5] => ? ∊ {1,1,1,2,2,6,6,6}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [3,2,1,5,4] => ? ∊ {1,1,1,2,2,6,6,6}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 6
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,5,2] => 6
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => ? ∊ {1,1,1,2,2,6,6,6}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,5,4,2] => 6
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => [1,5,6,4,3,2] => ? ∊ {1,1,1,2,2,6,6,6}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? ∊ {1,1,1,2,2,6,6,6}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1,8] => ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,4,3,2,1,7] => [5,6,4,3,2,1,7] => ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,3,5,2,1,6] => [4,3,5,2,1,6] => ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,5,4,2,1,6] => [3,5,4,2,1,6] => ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4,5] => ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7}
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2,5] => 7
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,5,4] => ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,5,3] => ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7}
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,5,2] => 7
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,4,6,3,2] => [1,5,4,6,3,2] => ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,4,3] => 7
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,5,3,2] => [1,4,6,5,3,2] => ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => [1,6,7,5,4,3,2] => ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,7,6,5,4,3,2,1,9] => [8,7,6,5,4,3,2,1,9] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,5,4,3,2,1,8] => [6,7,5,4,3,2,1,8] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,4,6,3,2,1,7] => [5,4,6,3,2,1,7] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,5,3,2,1,7] => [4,6,5,3,2,1,7] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,3,2,5,1,6] => [4,3,2,5,1,6] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,2,1,6] => [3,4,5,2,1,6] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8}
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,4,3,1,6] => [2,5,4,3,1,6] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8}
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => [4,3,2,1,6,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8}
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8}
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8}
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2,5] => 8
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,4,3,6,2] => [1,5,4,3,6,2] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8}
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8}
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 8
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,5,3] => 8
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,4,5,6,3,2] => [1,4,5,6,3,2] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8}
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,5,7,4,3,2] => [1,6,5,7,4,3,2] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8}
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,5,4,3] => [2,1,6,5,4,3] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8}
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,6,5,4,2] => [1,3,6,5,4,2] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8}
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => [1,5,7,6,4,3,2] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8}
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,7,8,6,5,4,3,2] => [1,7,8,6,5,4,3,2] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8}
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,8,7,6,5,4,3,2] => [1,9,8,7,6,5,4,3,2] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8}
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,8,7,6,5,4,3,2,1,10] => [9,8,7,6,5,4,3,2,1,10] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,9,9,9,9,9,9}
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [7,8,6,5,4,3,2,1,9] => [7,8,6,5,4,3,2,1,9] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,9,9,9,9,9,9}
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [6,5,7,4,3,2,1,8] => [6,5,7,4,3,2,1,8] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,9,9,9,9,9,9}
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [5,7,6,4,3,2,1,8] => [5,7,6,4,3,2,1,8] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,9,9,9,9,9,9}
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [5,4,3,6,2,1,7] => [5,4,3,6,2,1,7] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,9,9,9,9,9,9}
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [4,5,6,3,2,1,7] => [4,5,6,3,2,1,7] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,9,9,9,9,9,9}
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [3,6,5,4,2,1,7] => [3,6,5,4,2,1,7] => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,9,9,9,9,9,9}
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 9
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 9
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 9
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 10
Description
The number of occurrences of the signed pattern 12 in a signed permutation.
This is the number of pairs 1\leq i < j\leq n such that 0 < \pi(i) < \pi(j).
Matching statistic: St001630
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 17%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 17%
Values
[1]
=> [1,0,1,0]
=> [[1,1],[]]
=> ([],1)
=> ? = 1
[2]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> ([],1)
=> ? ∊ {2,2}
[1,1]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> ([],1)
=> ? ∊ {2,2}
[3]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> ([],1)
=> ? ∊ {3,3,3}
[2,1]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> ([],1)
=> ? ∊ {3,3,3}
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> ([],1)
=> ? ∊ {3,3,3}
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ([],1)
=> ? ∊ {1,4,4,4,4}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> ([],1)
=> ? ∊ {1,4,4,4,4}
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {1,4,4,4,4}
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> ([],1)
=> ? ∊ {1,4,4,4,4}
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ([],1)
=> ? ∊ {1,4,4,4,4}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3],[2]]
=> ([],1)
=> ? ∊ {1,1,5,5,5,5,5}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ([],1)
=> ? ∊ {1,1,5,5,5,5,5}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {1,1,5,5,5,5,5}
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {1,1,5,5,5,5,5}
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,5,5,5,5,5}
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ([],1)
=> ? ∊ {1,1,5,5,5,5,5}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> ([],1)
=> ? ∊ {1,1,5,5,5,5,5}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3]]
=> ([],1)
=> ? ∊ {1,1,1,2,2,6,6,6,6,6,6}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[4,4,4],[3]]
=> ([],1)
=> ? ∊ {1,1,1,2,2,6,6,6,6,6,6}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,2,2,6,6,6,6,6,6}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ([],1)
=> ? ∊ {1,1,1,2,2,6,6,6,6,6,6}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,2,2,6,6,6,6,6,6}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,2,2,6,6,6,6,6,6}
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,2,2,6,6,6,6,6,6}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,2,2,6,6,6,6,6,6}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,2,2,6,6,6,6,6,6}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,2,2,6,6,6,6,6,6}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,2,2,6,6,6,6,6,6}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,4],[3]]
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[3,3,3,3,3],[2]]
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1]]
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [[3,3,3,3,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[4,4,4,4,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [[5,5,5,5,5],[4]]
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [[5,5,5,5],[4]]
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [[4,4,4,3],[3]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3,1]]
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [[4,4,3],[3]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,2],[1]]
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [[4,4,4,3],[3,1]]
=> ([(0,2),(2,1)],3)
=> 1
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [[4,4,3],[3,1]]
=> ([(0,2),(2,1)],3)
=> 1
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [[4,3,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [[4,3,3],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [[4,3,2],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [[4,4,2],[2,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[4,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[4,3,3,1]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[5,3,3,1]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
[5,3,2,2]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[4,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[4,3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
Matching statistic: St001875
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 17%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 17%
Values
[1]
=> [1,0,1,0]
=> [[1,1],[]]
=> ([],1)
=> ? = 1
[2]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> ([],1)
=> ? ∊ {2,2}
[1,1]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> ([],1)
=> ? ∊ {2,2}
[3]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> ([],1)
=> ? ∊ {3,3,3}
[2,1]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> ([],1)
=> ? ∊ {3,3,3}
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> ([],1)
=> ? ∊ {3,3,3}
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ([],1)
=> ? ∊ {1,4,4,4,4}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> ([],1)
=> ? ∊ {1,4,4,4,4}
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {1,4,4,4,4}
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> ([],1)
=> ? ∊ {1,4,4,4,4}
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ([],1)
=> ? ∊ {1,4,4,4,4}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3],[2]]
=> ([],1)
=> ? ∊ {1,1,5,5,5,5,5}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ([],1)
=> ? ∊ {1,1,5,5,5,5,5}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {1,1,5,5,5,5,5}
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {1,1,5,5,5,5,5}
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,5,5,5,5,5}
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ([],1)
=> ? ∊ {1,1,5,5,5,5,5}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> ([],1)
=> ? ∊ {1,1,5,5,5,5,5}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3]]
=> ([],1)
=> ? ∊ {1,1,1,2,2,6,6,6,6,6,6}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[4,4,4],[3]]
=> ([],1)
=> ? ∊ {1,1,1,2,2,6,6,6,6,6,6}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,2,2,6,6,6,6,6,6}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ([],1)
=> ? ∊ {1,1,1,2,2,6,6,6,6,6,6}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,2,2,6,6,6,6,6,6}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,2,2,6,6,6,6,6,6}
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,2,2,6,6,6,6,6,6}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,2,2,6,6,6,6,6,6}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,2,2,6,6,6,6,6,6}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,2,2,6,6,6,6,6,6}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,2,2,6,6,6,6,6,6}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,4],[3]]
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[3,3,3,3,3],[2]]
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1]]
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [[3,3,3,3,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[4,4,4,4,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [[5,5,5,5,5],[4]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,8,8,8,8,8,8,8,8}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [[5,5,5,5],[4]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,8,8,8,8,8,8,8,8}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [[4,4,4,3],[3]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,8,8,8,8,8,8,8,8}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3,1]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,8,8,8,8,8,8,8,8}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [[4,4,3],[3]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,8,8,8,8,8,8,8,8}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,2],[1]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,8,8,8,8,8,8,8,8}
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1]]
=> ([(0,2),(2,1)],3)
=> 3
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [[4,4,4,3],[3,1]]
=> ([(0,2),(2,1)],3)
=> 3
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [[4,4,3],[3,1]]
=> ([(0,2),(2,1)],3)
=> 3
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [[4,3,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [[4,3,3],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3
[4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [[4,3,2],[1,1]]
=> ([(0,2),(2,1)],3)
=> 3
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3
[3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [[4,4,2],[2,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3
[5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> 3
[5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1,1]]
=> ([(0,2),(2,1)],3)
=> 3
[5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3
[4,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3
[4,3,3,1]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 3
[3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[5,3,3,1]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 3
[5,3,2,2]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4
[4,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[4,3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3
Description
The number of simple modules with projective dimension at most 1.
Matching statistic: St001878
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 17%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 17%
Values
[1]
=> [1,0,1,0]
=> [[1,1],[]]
=> ([],1)
=> ? = 1
[2]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> ([],1)
=> ? ∊ {2,2}
[1,1]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> ([],1)
=> ? ∊ {2,2}
[3]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> ([],1)
=> ? ∊ {3,3,3}
[2,1]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> ([],1)
=> ? ∊ {3,3,3}
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> ([],1)
=> ? ∊ {3,3,3}
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ([],1)
=> ? ∊ {1,4,4,4,4}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> ([],1)
=> ? ∊ {1,4,4,4,4}
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {1,4,4,4,4}
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> ([],1)
=> ? ∊ {1,4,4,4,4}
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ([],1)
=> ? ∊ {1,4,4,4,4}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3],[2]]
=> ([],1)
=> ? ∊ {1,1,5,5,5,5,5}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ([],1)
=> ? ∊ {1,1,5,5,5,5,5}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {1,1,5,5,5,5,5}
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {1,1,5,5,5,5,5}
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,5,5,5,5,5}
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ([],1)
=> ? ∊ {1,1,5,5,5,5,5}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> ([],1)
=> ? ∊ {1,1,5,5,5,5,5}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3]]
=> ([],1)
=> ? ∊ {1,1,1,2,2,6,6,6,6,6,6}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[4,4,4],[3]]
=> ([],1)
=> ? ∊ {1,1,1,2,2,6,6,6,6,6,6}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,2,2,6,6,6,6,6,6}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ([],1)
=> ? ∊ {1,1,1,2,2,6,6,6,6,6,6}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,2,2,6,6,6,6,6,6}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,2,2,6,6,6,6,6,6}
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,2,2,6,6,6,6,6,6}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,2,2,6,6,6,6,6,6}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,2,2,6,6,6,6,6,6}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,2,2,6,6,6,6,6,6}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,2,2,6,6,6,6,6,6}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,4],[3]]
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[3,3,3,3,3],[2]]
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1]]
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [[3,3,3,3,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[4,4,4,4,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7,7,7,7}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [[5,5,5,5,5],[4]]
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [[5,5,5,5],[4]]
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [[4,4,4,3],[3]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3,1]]
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [[4,4,3],[3]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,2],[1]]
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [[4,4,4,3],[3,1]]
=> ([(0,2),(2,1)],3)
=> 1
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [[4,4,3],[3,1]]
=> ([(0,2),(2,1)],3)
=> 1
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [[4,3,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [[4,3,3],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [[4,3,2],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [[4,4,2],[2,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[4,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[4,3,3,1]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[5,3,3,1]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
[5,3,2,2]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[4,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[4,3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St001880
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 42%
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 42%
Values
[1]
=> [1,0]
=> [.,.]
=> ([],1)
=> ? = 1
[2]
=> [1,0,1,0]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ? ∊ {2,2}
[1,1]
=> [1,1,0,0]
=> [[.,.],.]
=> ([(0,1)],2)
=> ? ∊ {2,2}
[3]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 3
[2,1]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,1]
=> [1,1,0,1,0,0]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 3
[4]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,2]
=> [1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ? ∊ {1,1}
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,1}
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? ∊ {1,1,1,2,2}
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,1,1,2,2}
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {1,1,1,2,2}
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,1,1,2,2}
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {1,1,1,2,2}
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[[[[[.,.],.],.],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ? ∊ {1,1,1,1,2,2,2,2}
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {1,1,1,1,2,2,2,2}
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? ∊ {1,1,1,1,2,2,2,2}
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[.,[[[[.,.],.],.],.]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,2,2,2,2}
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {1,1,1,1,2,2,2,2}
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? ∊ {1,1,1,1,2,2,2,2}
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[.,[[[[[.,.],.],.],.],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,2,2,2,2}
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[.,.],[[[[.,.],.],.],.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? ∊ {1,1,1,1,2,2,2,2}
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[[[[[[.,.],.],.],.],.],.]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [.,[.,[[.,.],[[[.,.],.],.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [[.,[.,.]],[[[.,.],.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [.,[[.,.],[[[[.,.],.],.],.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[.,[[[[[[.,.],.],.],.],.],.]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [[[.,.],.],[[[.,.],.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [[.,.],[[[[[.,.],.],.],.],.]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8}
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,9,9,9,9,9,9,9,9,9}
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,9,9,9,9,9,9,9,9,9}
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,9,9,9,9,9,9,9,9,9}
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,9,9,9,9,9,9,9,9,9}
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,9,9,9,9,9,9,9,9,9}
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,9,9,9,9,9,9,9,9,9}
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,9,9,9,9,9,9,9,9,9}
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,9,9,9,9,9,9,9,9,9}
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,9,9,9,9,9,9,9,9,9}
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001879
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 42%
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 42%
Values
[1]
=> [1,0]
=> [.,.]
=> ([],1)
=> ? = 1 - 1
[2]
=> [1,0,1,0]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ? ∊ {2,2} - 1
[1,1]
=> [1,1,0,0]
=> [[.,.],.]
=> ([(0,1)],2)
=> ? ∊ {2,2} - 1
[3]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[2,2]
=> [1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 1 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ? ∊ {1,1} - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,1} - 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? ∊ {1,1,1,2,2} - 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,1,1,2,2} - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {1,1,1,2,2} - 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,1,1,2,2} - 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {1,1,1,2,2} - 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[[[[[.,.],.],.],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ? ∊ {1,1,1,1,2,2,2,2} - 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {1,1,1,1,2,2,2,2} - 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? ∊ {1,1,1,1,2,2,2,2} - 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[.,[[[[.,.],.],.],.]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,2,2,2,2} - 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {1,1,1,1,2,2,2,2} - 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? ∊ {1,1,1,1,2,2,2,2} - 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[.,[[[[[.,.],.],.],.],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,2,2,2,2} - 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[.,.],[[[[.,.],.],.],.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? ∊ {1,1,1,1,2,2,2,2} - 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[[[[[[.,.],.],.],.],.],.]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8} - 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8} - 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8} - 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8} - 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8} - 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8} - 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8} - 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8} - 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8} - 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8} - 1
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [.,[.,[[.,.],[[[.,.],.],.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8} - 1
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8} - 1
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8} - 1
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [[.,[.,.]],[[[.,.],.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8} - 1
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8} - 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [.,[[.,.],[[[[.,.],.],.],.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8} - 1
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[.,[[[[[[.,.],.],.],.],.],.]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8} - 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8} - 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [[[.,.],.],[[[.,.],.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8} - 1
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [[.,.],[[[[[.,.],.],.],.],.]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8} - 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8} - 1
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,8,8,8,8,8,8,8,8} - 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,9,9,9,9,9,9,9,9,9} - 1
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,9,9,9,9,9,9,9,9,9} - 1
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,9,9,9,9,9,9,9,9,9} - 1
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,9,9,9,9,9,9,9,9,9} - 1
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,9,9,9,9,9,9,9,9,9} - 1
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,9,9,9,9,9,9,9,9,9} - 1
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,9,9,9,9,9,9,9,9,9} - 1
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,9,9,9,9,9,9,9,9,9} - 1
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,9,9,9,9,9,9,9,9,9} - 1
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
The following 2 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001198The number of simple modules in the algebra eAe with projective dimension at most 1 in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001206The maximal dimension of an indecomposable projective eAe-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module eA.
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