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Your data matches 7 different statistics following compositions of up to 3 maps.
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Matching statistic: St001062
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St001062: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St001062: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> [1,2] => {{1},{2}}
=> 1
[1,1]
=> [1,1,0,0]
=> [2,1] => {{1,2}}
=> 2
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => {{1},{2},{3}}
=> 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> 2
[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}}
=> 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => {{1,3},{2}}
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => {{1},{2,3,4}}
=> 3
[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}}
=> 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => {{1},{2,4},{3}}
=> 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 3
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => {{1,3,4},{2}}
=> 3
[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
[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}}
=> 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}}
=> 2
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 2
[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}}
=> 3
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => {{1,2,3,4}}
=> 4
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => {{1},{2,4,5},{3}}
=> 3
[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}}
=> 4
[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}}
=> 4
[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
[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}}
=> 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}}
=> 2
[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}}
=> 2
[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}}
=> 3
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => {{1},{2,3,4,5}}
=> 4
[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}}
=> 3
[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}}
=> 4
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => {{1,2,3,4,5}}
=> 5
[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}}
=> 4
[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}}
=> 5
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => {{1,4,5},{2,3}}
=> 3
[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}}
=> 5
[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
[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
[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}}
=> 2
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,5,6,4,3] => {{1},{2},{3,4,5,6}}
=> 4
[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}}
=> 3
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => {{1,3,5},{2,4}}
=> 3
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,4,5,3,6,2] => {{1},{2,3,4,5,6}}
=> 5
[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}}
=> 4
Description
The maximal size of a block of a set partition.
Matching statistic: St000147
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> [1,2] => [1,1]
=> 1
[1,1]
=> [1,1,0,0]
=> [2,1] => [2]
=> 2
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => [3]
=> 3
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => [2,1]
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1]
=> 3
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4]
=> 4
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,1]
=> 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,1,1]
=> 3
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> 3
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1]
=> 4
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5]
=> 5
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,1,1,1,1,1]
=> 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [2,1,1,1,1]
=> 2
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [2,1,1,1]
=> 2
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [3,1,1,1]
=> 3
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [4]
=> 4
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> 3
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => [4,1,1]
=> 4
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [2,2]
=> 2
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [4,1]
=> 4
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => [5,1]
=> 5
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => [6]
=> 6
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [1,1,1,1,1,1,1]
=> 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [2,1,1,1,1,1]
=> 2
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => [2,1,1,1,1]
=> 2
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [3,1,1,1,1]
=> 3
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [4,1]
=> 4
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => [3,1,1,1]
=> 3
[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] => [4,1,1,1]
=> 4
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => [5]
=> 5
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,2,1]
=> 2
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,3,5,6,2] => [4,1,1]
=> 4
[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] => [5,1,1]
=> 5
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [3,2]
=> 3
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,1] => [5,1]
=> 5
[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] => [6,1]
=> 6
[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] => [7]
=> 7
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [2,1,1,1,1,1]
=> 2
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,5,6,4,3] => [4,1,1]
=> 4
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => [3,1,1,1,1]
=> 3
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => [3,2]
=> 3
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,4,5,3,6,2] => [5,1]
=> 5
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => [2,2,1,1]
=> 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] => [4,1,1,1]
=> 4
[4,4,2,2,1]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [3,4,6,7,5,2,8,1] => ?
=> ? = 7
Description
The largest part of an integer partition.
Matching statistic: St000485
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000485: Permutations ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 75%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000485: Permutations ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 75%
Values
[2]
=> [1,0,1,0]
=> [1,2] => 1
[1,1]
=> [1,1,0,0]
=> [2,1] => 2
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 3
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 3
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 3
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 3
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 4
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 5
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 2
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 3
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 4
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 3
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => 4
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 2
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 4
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 5
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 6
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => 2
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => 2
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => 3
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 4
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => 3
[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] => 4
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => 5
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 2
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,3,5,6,2] => 4
[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] => 5
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 3
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,1] => 5
[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] => 6
[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] => ? = 7
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => 2
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,5,6,4,3] => 4
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => 3
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => 3
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,4,5,3,6,2] => 5
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => 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] => 4
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => 5
[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] => ? = 6
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,1] => ? = 6
[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] => ? = 7
[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] => ? = 8
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,4,2,5,6,7,1] => ? = 7
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,4,3,6,7,2] => ? = 4
[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] => ? = 6
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [4,3,2,5,6,7,1] => ? = 5
[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] => ? = 7
[5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,2,5,6,4,7,8,3] => ? = 6
[4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [3,4,5,2,6,7,1] => ? = 5
[4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,4,5,3,6,7,8,2] => ? = 7
[3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [3,5,4,2,6,7,1] => ? = 7
[3,3,1,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,4,2,5,6,7,8,1] => ? = 8
[3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,5,6,4,3,7,2] => ? = 5
[2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [4,5,3,2,6,7,1] => ? = 6
[2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,5,6,7,8,1] => ? = 6
[5,5,1]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,6,2,7,1] => ? = 7
[5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,2,5,7,6,4,8,3] => ? = 6
[4,4,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [3,4,6,5,2,7,1] => ? = 4
[4,4,1,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,4,5,2,6,7,8,1] => ? = 6
[4,3,3,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,5,4,3,7,2] => ? = 3
[4,3,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,6,7,5,3,2] => ? = 3
[3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [5,4,3,2,6,7,1] => ? = 4
[3,3,2,2,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [3,5,6,4,2,7,1] => ? = 4
[3,3,2,1,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> [3,5,4,2,6,7,8,1] => ? = 8
[3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,5,6,7,4,3,2] => ? = 4
[2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [4,5,6,3,2,7,1] => ? = 5
[2,2,2,2,1,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> [4,5,3,2,6,7,8,1] => ? = 7
[7,5]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,8,4,3] => ? = 6
[6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 4
[6,5,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,4,5,6,7,3,8,2] => ? = 7
[5,5,2]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,4,5,7,6,2,1] => ? = 7
[5,5,1,1]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [3,4,5,6,2,7,8,1] => ? = 8
[5,4,3]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 5
[4,4,3,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [3,6,5,4,2,7,1] => ? = 6
[4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [3,4,6,7,5,2,1] => ? = 6
[4,3,3,2]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,5,7,4,3,2] => ? = 6
[3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [5,4,6,3,2,7,1] => ? = 7
[3,3,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [3,5,6,7,4,2,1] => ? = 7
[2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 4
[2,2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> [4,5,6,3,2,7,8,1] => ? = 6
[5,5,3]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,4,7,6,5,2,1] => ? = 3
[5,5,2,1]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0]
=> [3,4,5,7,6,2,8,1] => ? = 8
[5,4,4]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => ? = 4
[5,3,3,2]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,2,7,6,8,5,4,3] => ? = 6
[4,4,4,1]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [5,6,4,3,2,7,1] => ? = 5
[4,4,3,2]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [3,6,5,7,4,2,1] => ? = 5
[4,4,3,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [3,6,5,4,2,7,8,1] => ? = 7
Description
The length of the longest cycle of a permutation.
Matching statistic: St001232
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 88%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 88%
Values
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 2 - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 2 - 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 3 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 4 - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 3 - 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 4 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? = 4 - 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[7]
=> [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]
=> 0 = 1 - 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 2 = 3 - 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? = 4 - 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> ? = 3 - 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3 = 4 - 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ? = 5 - 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 4 - 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 4 = 5 - 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 5 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 6 = 7 - 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 4 - 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> ? = 3 - 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? = 3 - 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 5 - 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> ? = 4 - 1
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 5 - 1
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 6 - 1
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[3,2,1,1,1]
=> [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]
=> ? = 5 - 1
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? = 6 - 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 4 - 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 4 - 1
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 6 - 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 7 - 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]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 8 - 1
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 4 - 1
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 3 - 1
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> ? = 5 - 1
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 4 - 1
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> ? = 5 - 1
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> ? = 6 - 1
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 6 - 1
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 7 - 1
[3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 4 - 1
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 4 - 1
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 6 - 1
[2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 5 - 1
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 5 - 1
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 7 - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000454
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 88%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 88%
Values
[2]
=> [[1,2]]
=> [2] => ([],2)
=> 0 = 1 - 1
[1,1]
=> [[1],[2]]
=> [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[3]
=> [[1,2,3]]
=> [3] => ([],3)
=> 0 = 1 - 1
[2,1]
=> [[1,3],[2]]
=> [1,2] => ([(1,2)],3)
=> 1 = 2 - 1
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[4]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> 0 = 1 - 1
[3,1]
=> [[1,3,4],[2]]
=> [1,3] => ([(2,3)],4)
=> 1 = 2 - 1
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 - 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[5]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> 0 = 1 - 1
[4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => ([(3,4)],5)
=> 1 = 2 - 1
[3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[6]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> 0 = 1 - 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => ([(4,5)],6)
=> 1 = 2 - 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? = 2 - 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [7] => ([],7)
=> 0 = 1 - 1
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [1,6] => ([(5,6)],7)
=> 1 = 2 - 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [2,5] => ([(4,6),(5,6)],7)
=> ? = 2 - 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 4 - 1
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 - 1
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 1
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 - 1
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 6 - 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 7 - 1
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [2,6] => ([(5,7),(6,7)],8)
=> ? = 2 - 1
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [3,5] => ([(4,7),(5,7),(6,7)],8)
=> ? = 4 - 1
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [1,2,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 - 1
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 3 - 1
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [1,3,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [1,1,2,4] => ([(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [1,1,3,3] => ([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 - 1
[3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [1,1,1,2,3] => ([(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [1,1,2,2,2] => ([(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [1,1,1,1,2,2] => ([(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 - 1
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 8 - 1
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [3,6] => ([(5,8),(6,8),(7,8)],9)
=> ? = 4 - 1
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [4,5] => ([(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 3 - 1
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [1,3,5] => ([(4,8),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 - 1
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [2,2,5] => ([(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2 - 1
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [1,4,4] => ([(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 4 - 1
[4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> [2,3,4] => ([(3,8),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 - 1
[4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> [1,1,3,4] => ([(3,8),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 6 - 1
[4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> [1,2,2,4] => ([(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [3,3,3] => ([(2,8),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2 - 1
[3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> [1,2,3,3] => ([(2,8),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 6 - 1
[3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> [1,1,1,3,3] => ([(2,8),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 7 - 1
[3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> [2,2,2,3] => ([(2,8),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 4 - 1
[3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> [1,1,2,2,3] => ([(2,8),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 4 - 1
[3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> [1,1,1,1,2,3] => ([(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 6 - 1
[2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> [1,2,2,2,2] => ([(1,8),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 - 1
[2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> [1,1,1,2,2,2] => ([(1,8),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 - 1
[2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> [1,1,1,1,1,2,2] => ([(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 7 - 1
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001812
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001812: Graphs ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 62%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001812: Graphs ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 62%
Values
[2]
=> [[1,2]]
=> [1,2] => ([],2)
=> 0 = 1 - 1
[1,1]
=> [[1],[2]]
=> [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[3]
=> [[1,2,3]]
=> [1,2,3] => ([],3)
=> 0 = 1 - 1
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => ([(1,2)],3)
=> 1 = 2 - 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => ([(2,3)],4)
=> 1 = 2 - 1
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1 = 2 - 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => ([],5)
=> 0 = 1 - 1
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 1 = 2 - 1
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 1 = 2 - 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ([],6)
=> 0 = 1 - 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => ([(4,5)],6)
=> 1 = 2 - 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 1 = 2 - 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 4 - 1
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2 - 1
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => ([],7)
=> ? = 1 - 1
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => ([(5,6)],7)
=> ? = 2 - 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ([(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 - 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ([(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 4 - 1
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 1
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(4,5),(5,6)],7)
=> ? = 5 - 1
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 - 1
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ([(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 1
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 - 1
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => ([(0,1),(0,6),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 - 1
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 - 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 7 - 1
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ([(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2 - 1
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 4 - 1
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [4,2,5,1,3,6,7,8] => ([(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 3 - 1
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ([(0,4),(0,5),(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)],8)
=> ? = 3 - 1
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => ([(1,6),(1,7),(2,4),(2,5),(2,7),(3,4),(3,5),(3,7),(4,6),(5,6),(6,7)],8)
=> ? = 5 - 1
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8] => ([(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)
=> ? = 2 - 1
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8] => ([(2,3),(2,7),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5] => ([(0,1),(0,6),(0,7),(1,4),(1,5),(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)
=> ? = 5 - 1
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5] => ([(0,2),(0,3),(0,7),(1,2),(1,3),(1,7),(2,6),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8] => ([(1,2),(1,4),(1,5),(1,7),(2,3),(2,5),(2,6),(3,4),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 - 1
[3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8] => ([(1,2),(1,7),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(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)
=> ? = 4 - 1
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4] => ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6] => ([(0,1),(0,7),(1,6),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 - 1
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 8 - 1
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9] => ([(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ? = 4 - 1
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9] => ([(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8)],9)
=> ? = 3 - 1
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9] => ([(2,7),(2,8),(3,5),(3,6),(3,8),(4,5),(4,6),(4,8),(5,7),(6,7),(7,8)],9)
=> ? = 5 - 1
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9] => ([(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8)],9)
=> ? = 2 - 1
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5] => ([(0,7),(0,8),(1,4),(1,5),(1,6),(1,8),(2,4),(2,5),(2,6),(2,8),(3,4),(3,5),(3,6),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
=> ? = 4 - 1
[4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5,9] => ([(1,2),(1,7),(1,8),(2,5),(2,6),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8)],9)
=> ? = 5 - 1
[4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5,9] => ([(1,3),(1,4),(1,8),(2,3),(2,4),(2,8),(3,7),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 6 - 1
[4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8,9] => ([(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ? = 2 - 1
[3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6] => ([(0,1),(0,6),(0,8),(1,5),(1,7),(2,3),(2,5),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> ? = 6 - 1
[3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6] => ([(0,2),(0,3),(0,8),(1,2),(1,3),(1,8),(2,7),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 7 - 1
[3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8)],9)
=> ? = 4 - 1
[3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4,9] => ([(1,2),(1,3),(1,7),(1,8),(2,3),(2,6),(2,8),(3,5),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 4 - 1
[3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6,9] => ([(1,2),(1,8),(2,7),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 6 - 1
[2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> [8,6,9,4,7,2,5,1,3] => ([(0,1),(0,3),(0,4),(0,5),(0,6),(0,8),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 - 1
Description
The biclique partition number of a graph.
The biclique partition number of a graph is the minimum number of pairwise edge disjoint complete bipartite subgraphs so that each edge belongs to exactly one of them. A theorem of Graham and Pollak [1] asserts that the complete graph $K_n$ has biclique partition number $n - 1$.
Matching statistic: St000181
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000181: Posets ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 62%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000181: Posets ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 62%
Values
[2]
=> [[1,2]]
=> [1,2] => ([(0,1)],2)
=> 1
[1,1]
=> [[1],[2]]
=> [2,1] => ([],2)
=> 2
[3]
=> [[1,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> 2
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => ([],3)
=> 3
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => ([(2,3)],4)
=> 3
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => ([],4)
=> 4
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> 2
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> 2
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> 3
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> 3
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => ([(3,4)],5)
=> 4
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ([],5)
=> 5
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> ? = 2
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
=> ? = 2
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ([(2,3),(3,5),(5,4)],6)
=> ? = 3
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
=> ? = 4
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ([(1,3),(2,4),(4,5)],6)
=> ? = 3
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ([(3,4),(4,5)],6)
=> ? = 4
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ([(0,5),(1,4),(2,3)],6)
=> ? = 2
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => ([(2,5),(3,4)],6)
=> ? = 4
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => ([(4,5)],6)
=> ? = 5
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => ([],6)
=> ? = 6
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 1
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 2
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 2
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ([(2,6),(4,5),(5,3),(6,4)],7)
=> ? = 3
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? = 4
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? = 3
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ([(3,4),(4,6),(6,5)],7)
=> ? = 4
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ([(1,6),(2,5),(5,3),(6,4)],7)
=> ? = 5
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ([(0,5),(1,4),(2,6),(6,3)],7)
=> ? = 2
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ([(2,4),(3,5),(5,6)],7)
=> ? = 4
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ([(4,5),(5,6)],7)
=> ? = 5
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ([(1,6),(2,5),(3,4)],7)
=> ? = 3
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ([(3,6),(4,5)],7)
=> ? = 5
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ([(5,6)],7)
=> ? = 6
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ([],7)
=> ? = 7
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ([(0,7),(1,3),(4,6),(5,4),(6,2),(7,5)],8)
=> ? = 2
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => ([(0,7),(1,6),(4,5),(5,3),(6,4),(7,2)],8)
=> ? = 4
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [8,6,7,1,2,3,4,5] => ([(1,7),(2,4),(5,6),(6,3),(7,5)],8)
=> ? = 3
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ([(0,7),(1,6),(4,2),(5,3),(6,4),(7,5)],8)
=> ? = 3
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [8,5,6,7,1,2,3,4] => ([(1,6),(2,7),(5,4),(6,5),(7,3)],8)
=> ? = 5
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [7,8,5,6,1,2,3,4] => ([(0,5),(1,4),(2,7),(6,3),(7,6)],8)
=> ? = 2
[4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [8,7,5,6,1,2,3,4] => ([(2,4),(3,5),(5,6),(6,7)],8)
=> ? = 4
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => ([(0,5),(1,7),(2,6),(6,3),(7,4)],8)
=> ? = 5
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [8,7,4,5,6,1,2,3] => ([(2,5),(3,4),(4,6),(5,7)],8)
=> ? = 6
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => ([(1,5),(2,4),(3,6),(6,7)],8)
=> ? = 3
[3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [8,7,6,4,5,1,2,3] => ([(3,5),(4,6),(6,7)],8)
=> ? = 5
[3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,1,2,3] => ([(5,6),(6,7)],8)
=> ? = 6
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ([(0,7),(1,6),(2,5),(3,4)],8)
=> ? = 4
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => ([(2,7),(3,6),(4,5)],8)
=> ? = 4
[2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => ([(4,7),(5,6)],8)
=> ? = 6
[2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,1,2] => ([(6,7)],8)
=> ? = 7
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => ([],8)
=> ? = 8
[6,3]
=> [[1,2,3,4,5,6],[7,8,9]]
=> [7,8,9,1,2,3,4,5,6] => ([(0,8),(1,7),(4,6),(5,4),(6,3),(7,5),(8,2)],9)
=> ? = 4
[5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> [6,7,8,9,1,2,3,4,5] => ([(0,7),(1,8),(4,5),(5,2),(6,3),(7,6),(8,4)],9)
=> ? = 3
[5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> [9,6,7,8,1,2,3,4,5] => ([(1,8),(2,7),(5,6),(6,4),(7,5),(8,3)],9)
=> ? = 5
[5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> [8,9,6,7,1,2,3,4,5] => ([(0,5),(1,4),(2,8),(6,7),(7,3),(8,6)],9)
=> ? = 2
[4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [9,5,6,7,8,1,2,3,4] => ([(1,8),(2,7),(5,3),(6,4),(7,5),(8,6)],9)
=> ? = 4
[4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [8,9,5,6,7,1,2,3,4] => ([(0,5),(1,7),(2,8),(6,3),(7,4),(8,6)],9)
=> ? = 5
[4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [9,8,5,6,7,1,2,3,4] => ([(2,7),(3,8),(6,5),(7,6),(8,4)],9)
=> ? = 6
Description
The number of connected components of the Hasse diagram for the poset.
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