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Your data matches 18 different statistics following compositions of up to 3 maps.
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Matching statistic: St000458
St000458: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,2] => 2
[2,1] => 2
[1,2,3] => 3
[1,3,2] => 3
[2,1,3] => 3
[2,3,1] => 3
[3,1,2] => 3
[3,2,1] => 3
[1,2,3,4] => 5
[1,2,4,3] => 5
[1,3,2,4] => 5
[1,3,4,2] => 3
[1,4,2,3] => 3
[1,4,3,2] => 3
[2,1,3,4] => 5
[2,1,4,3] => 5
[2,3,1,4] => 3
[2,3,4,1] => 3
[2,4,1,3] => 1
[2,4,3,1] => 3
[3,1,2,4] => 3
[3,1,4,2] => 1
[3,2,1,4] => 3
[3,2,4,1] => 3
[3,4,1,2] => 5
[3,4,2,1] => 5
[4,1,2,3] => 3
[4,1,3,2] => 3
[4,2,1,3] => 3
[4,2,3,1] => 5
[4,3,1,2] => 5
[4,3,2,1] => 5
[1,2,3,4,5] => 8
[1,2,3,5,4] => 8
[1,2,4,3,5] => 8
[1,2,4,5,3] => 6
[1,2,5,3,4] => 6
[1,2,5,4,3] => 6
[1,3,2,4,5] => 8
[1,3,2,5,4] => 8
[1,3,4,2,5] => 3
[1,3,4,5,2] => 3
[1,3,5,2,4] => 1
[1,3,5,4,2] => 3
[1,4,2,3,5] => 3
[1,4,2,5,3] => 1
[1,4,3,2,5] => 3
[1,4,3,5,2] => 3
[1,4,5,2,3] => 5
Description
The number of permutations obtained by switching adjacencies or successions.
For a permutation π, this statistic is the size of its equivalence class of the equivalence relation generated by the interchange of any two adjacent elements πi and πi+1 such that |πi−πi+1|=1.
Matching statistic: St001605
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 30%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 30%
Values
[1] => [] => []
=> ?
=> ? = 1
[1,2] => [1] => [1]
=> []
=> ? ∊ {2,2}
[2,1] => [1] => [1]
=> []
=> ? ∊ {2,2}
[1,2,3] => [1,2] => [1,1]
=> [1]
=> ? ∊ {3,3,3,3,3,3}
[1,3,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {3,3,3,3,3,3}
[2,1,3] => [2,1] => [2]
=> []
=> ? ∊ {3,3,3,3,3,3}
[2,3,1] => [2,1] => [2]
=> []
=> ? ∊ {3,3,3,3,3,3}
[3,1,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {3,3,3,3,3,3}
[3,2,1] => [2,1] => [2]
=> []
=> ? ∊ {3,3,3,3,3,3}
[1,2,3,4] => [1,2,3] => [1,1,1]
=> [1,1]
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,2,4,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,3,2,4] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,3,4,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,4,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,4,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,1,3,4] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,1,4,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,3,1,4] => [2,3,1] => [3]
=> []
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,3,4,1] => [2,3,1] => [3]
=> []
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,4,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,4,3,1] => [2,3,1] => [3]
=> []
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,1,2,4] => [3,1,2] => [3]
=> []
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,1,4,2] => [3,1,2] => [3]
=> []
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,2,1,4] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,2,4,1] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,4,1,2] => [3,1,2] => [3]
=> []
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,4,2,1] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,2,3,1] => [2,3,1] => [3]
=> []
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,3,1,2] => [3,1,2] => [3]
=> []
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,3,2,1] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,2,3,4,5] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 2
[1,2,3,5,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 2
[1,2,4,3,5] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,2,4,5,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,2,5,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 2
[1,2,5,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,2,4,5] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,2,5,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,4,2,5] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,4,5,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,5,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,5,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,2,3,5] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,2,5,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,3,2,5] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,3,5,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,5,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,5,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,5,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 2
[1,5,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,5,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[5,1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 2
[1,2,3,4,5,6] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 6
[1,2,3,4,6,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 6
[1,2,3,5,4,6] => [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,2,3,5,6,4] => [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,2,3,6,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 6
[1,2,3,6,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,2,4,3,5,6] => [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,2,4,3,6,5] => [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,2,4,6,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,2,5,4,3,6] => [1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,2,5,4,6,3] => [1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,2,5,6,4,3] => [1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,2,6,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 6
[1,2,6,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,2,6,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,2,6,5,4,3] => [1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,3,2,4,5,6] => [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,3,2,4,6,5] => [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,3,2,5,4,6] => [1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 1
[1,3,2,5,6,4] => [1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 1
[1,3,2,6,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,3,2,6,5,4] => [1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 1
[1,3,6,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,3,6,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 1
[1,4,3,2,5,6] => [1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,4,3,2,6,5] => [1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,4,3,6,2,5] => [1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,4,5,2,3,6] => [1,4,5,2,3] => [2,2,1]
=> [2,1]
=> 1
[1,4,5,2,6,3] => [1,4,5,2,3] => [2,2,1]
=> [2,1]
=> 1
[1,4,5,6,2,3] => [1,4,5,2,3] => [2,2,1]
=> [2,1]
=> 1
[1,4,6,3,2,5] => [1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,4,6,5,2,3] => [1,4,5,2,3] => [2,2,1]
=> [2,1]
=> 1
[1,5,3,4,2,6] => [1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,5,3,4,6,2] => [1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,5,3,6,4,2] => [1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,5,4,3,2,6] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,5,4,3,6,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,5,4,6,3,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,5,6,3,4,2] => [1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,5,6,4,3,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,6,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 6
[1,6,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,6,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,6,2,5,4,3] => [1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,6,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 2
Description
The number of colourings of a cycle such that the multiplicities of colours are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the cyclic group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St000260
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 20%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 20%
Values
[1] => => [] => ?
=> ? = 1 - 2
[1,2] => 1 => [1] => ([],1)
=> 0 = 2 - 2
[2,1] => 0 => [1] => ([],1)
=> 0 = 2 - 2
[1,2,3] => 11 => [2] => ([],2)
=> ? ∊ {3,3,3,3} - 2
[1,3,2] => 10 => [1,1] => ([(0,1)],2)
=> 1 = 3 - 2
[2,1,3] => 01 => [1,1] => ([(0,1)],2)
=> 1 = 3 - 2
[2,3,1] => 00 => [2] => ([],2)
=> ? ∊ {3,3,3,3} - 2
[3,1,2] => 00 => [2] => ([],2)
=> ? ∊ {3,3,3,3} - 2
[3,2,1] => 00 => [2] => ([],2)
=> ? ∊ {3,3,3,3} - 2
[1,2,3,4] => 111 => [3] => ([],3)
=> ? ∊ {1,1,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 2
[1,2,4,3] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 3 - 2
[1,3,2,4] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 3 - 2
[1,3,4,2] => 100 => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 2
[1,4,2,3] => 100 => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 2
[1,4,3,2] => 100 => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 2
[2,1,3,4] => 011 => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 2
[2,1,4,3] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 3 - 2
[2,3,1,4] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 3 - 2
[2,3,4,1] => 000 => [3] => ([],3)
=> ? ∊ {1,1,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 2
[2,4,1,3] => 000 => [3] => ([],3)
=> ? ∊ {1,1,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 2
[2,4,3,1] => 000 => [3] => ([],3)
=> ? ∊ {1,1,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 2
[3,1,2,4] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 3 - 2
[3,1,4,2] => 000 => [3] => ([],3)
=> ? ∊ {1,1,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 2
[3,2,1,4] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 3 - 2
[3,2,4,1] => 000 => [3] => ([],3)
=> ? ∊ {1,1,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 2
[3,4,1,2] => 000 => [3] => ([],3)
=> ? ∊ {1,1,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 2
[3,4,2,1] => 000 => [3] => ([],3)
=> ? ∊ {1,1,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 2
[4,1,2,3] => 000 => [3] => ([],3)
=> ? ∊ {1,1,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 2
[4,1,3,2] => 000 => [3] => ([],3)
=> ? ∊ {1,1,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 2
[4,2,1,3] => 000 => [3] => ([],3)
=> ? ∊ {1,1,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 2
[4,2,3,1] => 000 => [3] => ([],3)
=> ? ∊ {1,1,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 2
[4,3,1,2] => 000 => [3] => ([],3)
=> ? ∊ {1,1,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 2
[4,3,2,1] => 000 => [3] => ([],3)
=> ? ∊ {1,1,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 2
[1,2,3,4,5] => 1111 => [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,2,3,5,4] => 1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,2,4,3,5] => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,2,4,5,3] => 1100 => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,2,5,3,4] => 1100 => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,2,5,4,3] => 1100 => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,3,2,4,5] => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,3,2,5,4] => 1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,3,4,2,5] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,3,4,5,2] => 1000 => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,3,5,2,4] => 1000 => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,3,5,4,2] => 1000 => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,4,2,3,5] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,4,2,5,3] => 1000 => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,4,3,2,5] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,4,3,5,2] => 1000 => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,4,5,2,3] => 1000 => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,4,5,3,2] => 1000 => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,5,2,3,4] => 1000 => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,5,2,4,3] => 1000 => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,5,3,2,4] => 1000 => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,5,3,4,2] => 1000 => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,5,4,2,3] => 1000 => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[1,5,4,3,2] => 1000 => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[2,1,3,4,5] => 0111 => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[2,1,3,5,4] => 0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[2,1,4,3,5] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[2,1,4,5,3] => 0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[2,1,5,3,4] => 0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[2,1,5,4,3] => 0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[2,3,1,4,5] => 0011 => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[2,3,1,5,4] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[2,3,4,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 3 - 2
[2,3,4,5,1] => 0000 => [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[2,3,5,1,4] => 0000 => [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[2,3,5,4,1] => 0000 => [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[2,4,1,3,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 3 - 2
[2,4,1,5,3] => 0000 => [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 2
[2,4,3,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 3 - 2
[3,1,2,5,4] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[3,1,4,2,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 3 - 2
[3,2,1,5,4] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[3,2,4,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 3 - 2
[3,4,1,2,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 3 - 2
[3,4,2,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 3 - 2
[4,1,2,3,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 3 - 2
[4,1,3,2,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 3 - 2
[4,2,1,3,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 3 - 2
[4,2,3,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 3 - 2
[4,3,1,2,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 3 - 2
[4,3,2,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,2,3,4,6,5] => 11110 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,2,3,5,4,6] => 11101 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,2,4,3,6,5] => 11010 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,2,4,5,3,6] => 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,2,5,3,4,6] => 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,2,5,4,3,6] => 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,3,2,4,6,5] => 10110 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,3,2,5,4,6] => 10101 => [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)
=> 1 = 3 - 2
[1,3,4,2,6,5] => 10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,3,4,5,2,6] => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,3,5,2,4,6] => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,3,5,4,2,6] => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,4,2,3,6,5] => 10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,4,2,5,3,6] => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,4,3,2,6,5] => 10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,4,3,5,2,6] => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 3 - 2
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000771
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 30%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 30%
Values
[1] => [1] => [1] => ([],1)
=> 1
[1,2] => [1,2] => [2] => ([],2)
=> ? ∊ {2,2}
[2,1] => [1,2] => [2] => ([],2)
=> ? ∊ {2,2}
[1,2,3] => [1,2,3] => [3] => ([],3)
=> ? ∊ {3,3,3,3,3,3}
[1,3,2] => [1,2,3] => [3] => ([],3)
=> ? ∊ {3,3,3,3,3,3}
[2,1,3] => [1,2,3] => [3] => ([],3)
=> ? ∊ {3,3,3,3,3,3}
[2,3,1] => [1,2,3] => [3] => ([],3)
=> ? ∊ {3,3,3,3,3,3}
[3,1,2] => [1,3,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {3,3,3,3,3,3}
[3,2,1] => [1,3,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {3,3,3,3,3,3}
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,2,4,3] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,3,2,4] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,3,4,2] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,4,2,3] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,4,3,2] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,1,3,4] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,1,4,3] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,3,1,4] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,3,4,1] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,4,1,3] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,4,3,1] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,1,2,4] => [1,3,2,4] => [1,3] => ([(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,1,4,2] => [1,3,4,2] => [1,3] => ([(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,2,1,4] => [1,3,2,4] => [1,3] => ([(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,2,4,1] => [1,3,4,2] => [1,3] => ([(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,4,1,2] => [1,3,2,4] => [1,3] => ([(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,4,2,1] => [1,3,2,4] => [1,3] => ([(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,1,2,3] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,1,3,2] => [1,4,2,3] => [1,3] => ([(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,2,1,3] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,2,3,1] => [1,4,2,3] => [1,3] => ([(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,3,1,2] => [1,4,2,3] => [1,3] => ([(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,3,2,1] => [1,4,2,3] => [1,3] => ([(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,2,3,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,2,4,3,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,2,4,5,3] => [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,2,5,3,4] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,2,5,4,3] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,2,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,2,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,4,2,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,4,5,2] => [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,5,2,4] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,5,4,2] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,2,3,5] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,2,5,3] => [1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,3,2,5] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,3,5,2] => [1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,5,2,3] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,5,3,2] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,5,2,3,4] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,5,2,4,3] => [1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,5,3,2,4] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,5,3,4,2] => [1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[2,5,1,3,4] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,5,3,1,4] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,1,5,2,4] => [1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,2,5,1,4] => [1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[4,1,2,5,3] => [1,4,5,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[4,2,1,5,3] => [1,4,5,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,1,2,3,4] => [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,1,4,2,3] => [1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,2,1,3,4] => [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,2,4,1,3] => [1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,4,1,3,2] => [1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,4,2,3,1] => [1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,4,3,1,2] => [1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,4,3,2,1] => [1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,6,3,4,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,2,6,4,3,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,3,6,2,4,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,3,6,4,2,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,4,2,6,3,5] => [1,2,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,4,3,6,2,5] => [1,2,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,5,2,3,6,4] => [1,2,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,5,3,2,6,4] => [1,2,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,6,2,3,4,5] => [1,2,6,5,4,3] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,6,2,5,3,4] => [1,2,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,6,3,2,4,5] => [1,2,6,5,4,3] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,6,3,5,2,4] => [1,2,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,6,5,2,4,3] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,6,5,3,4,2] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,6,5,4,2,3] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,6,5,4,3,2] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,1,6,3,4,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,1,6,4,3,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,3,6,1,4,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,3,6,4,1,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,4,1,6,3,5] => [1,2,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,4,3,6,1,5] => [1,2,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,5,1,3,6,4] => [1,2,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,5,3,1,6,4] => [1,2,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,6,1,3,4,5] => [1,2,6,5,4,3] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,6,1,5,3,4] => [1,2,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,6,3,1,4,5] => [1,2,6,5,4,3] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,6,3,5,1,4] => [1,2,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,6,5,1,4,3] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,6,5,3,4,1] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,6,5,4,1,3] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums 0, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
(4−1−2−1−14−1−2−2−14−1−1−2−14).
Its eigenvalues are 0,4,4,6, so the statistic is 2.
The path on four vertices has eigenvalues 0,4.7…,6,9.2… and therefore statistic 1.
Matching statistic: St000772
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 30%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 30%
Values
[1] => [1] => [1] => ([],1)
=> 1
[1,2] => [1,2] => [2] => ([],2)
=> ? ∊ {2,2}
[2,1] => [1,2] => [2] => ([],2)
=> ? ∊ {2,2}
[1,2,3] => [1,2,3] => [3] => ([],3)
=> ? ∊ {3,3,3,3,3,3}
[1,3,2] => [1,2,3] => [3] => ([],3)
=> ? ∊ {3,3,3,3,3,3}
[2,1,3] => [1,2,3] => [3] => ([],3)
=> ? ∊ {3,3,3,3,3,3}
[2,3,1] => [1,2,3] => [3] => ([],3)
=> ? ∊ {3,3,3,3,3,3}
[3,1,2] => [1,3,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {3,3,3,3,3,3}
[3,2,1] => [1,3,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {3,3,3,3,3,3}
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,2,4,3] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,3,2,4] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,3,4,2] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,4,2,3] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,4,3,2] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,1,3,4] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,1,4,3] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,3,1,4] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,3,4,1] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,4,1,3] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,4,3,1] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,1,2,4] => [1,3,2,4] => [1,3] => ([(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,1,4,2] => [1,3,4,2] => [1,3] => ([(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,2,1,4] => [1,3,2,4] => [1,3] => ([(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,2,4,1] => [1,3,4,2] => [1,3] => ([(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,4,1,2] => [1,3,2,4] => [1,3] => ([(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,4,2,1] => [1,3,2,4] => [1,3] => ([(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,1,2,3] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,1,3,2] => [1,4,2,3] => [1,3] => ([(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,2,1,3] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,2,3,1] => [1,4,2,3] => [1,3] => ([(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,3,1,2] => [1,4,2,3] => [1,3] => ([(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,3,2,1] => [1,4,2,3] => [1,3] => ([(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,2,3,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,2,4,3,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,2,4,5,3] => [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,2,5,3,4] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,2,5,4,3] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,2,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,2,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,4,2,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,4,5,2] => [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,5,2,4] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,5,4,2] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,2,3,5] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,2,5,3] => [1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,3,2,5] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,3,5,2] => [1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,5,2,3] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,5,3,2] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,5,2,3,4] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,5,2,4,3] => [1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,5,3,2,4] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,5,3,4,2] => [1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[2,5,1,3,4] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,5,3,1,4] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,1,5,2,4] => [1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,2,5,1,4] => [1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[4,1,2,5,3] => [1,4,5,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[4,2,1,5,3] => [1,4,5,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,1,2,3,4] => [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,1,4,2,3] => [1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,2,1,3,4] => [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,2,4,1,3] => [1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,4,1,3,2] => [1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,4,2,3,1] => [1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,4,3,1,2] => [1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,4,3,2,1] => [1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,6,3,4,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,2,6,4,3,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,3,6,2,4,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,3,6,4,2,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,4,2,6,3,5] => [1,2,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,4,3,6,2,5] => [1,2,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,5,2,3,6,4] => [1,2,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,5,3,2,6,4] => [1,2,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,6,2,3,4,5] => [1,2,6,5,4,3] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,6,2,5,3,4] => [1,2,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,6,3,2,4,5] => [1,2,6,5,4,3] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,6,3,5,2,4] => [1,2,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,6,5,2,4,3] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,6,5,3,4,2] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,6,5,4,2,3] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,6,5,4,3,2] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,1,6,3,4,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[2,1,6,4,3,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[2,3,6,1,4,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[2,3,6,4,1,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[2,4,1,6,3,5] => [1,2,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,4,3,6,1,5] => [1,2,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,5,1,3,6,4] => [1,2,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,5,3,1,6,4] => [1,2,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,6,1,3,4,5] => [1,2,6,5,4,3] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[2,6,1,5,3,4] => [1,2,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,6,3,1,4,5] => [1,2,6,5,4,3] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[2,6,3,5,1,4] => [1,2,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,6,5,1,4,3] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,6,5,3,4,1] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,6,5,4,1,3] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums 0, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
(4−1−2−1−14−1−2−2−14−1−1−2−14).
Its eigenvalues are 0,4,4,6, so the statistic is 1.
The path on four vertices has eigenvalues 0,4.7…,6,9.2… and therefore also statistic 1.
The graphs with statistic n−1, n−2 and n−3 have been characterised, see [1].
Matching statistic: St000259
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 20%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 20%
Values
[1] => [1] => [1] => ([],1)
=> 0 = 1 - 1
[1,2] => [1,2] => [2] => ([],2)
=> ? ∊ {2,2} - 1
[2,1] => [1,2] => [2] => ([],2)
=> ? ∊ {2,2} - 1
[1,2,3] => [1,2,3] => [3] => ([],3)
=> ? ∊ {3,3,3,3,3,3} - 1
[1,3,2] => [1,2,3] => [3] => ([],3)
=> ? ∊ {3,3,3,3,3,3} - 1
[2,1,3] => [1,2,3] => [3] => ([],3)
=> ? ∊ {3,3,3,3,3,3} - 1
[2,3,1] => [1,2,3] => [3] => ([],3)
=> ? ∊ {3,3,3,3,3,3} - 1
[3,1,2] => [1,3,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {3,3,3,3,3,3} - 1
[3,2,1] => [1,3,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {3,3,3,3,3,3} - 1
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 1
[1,2,4,3] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 1
[1,3,2,4] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 1
[1,3,4,2] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 1
[1,4,2,3] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 1
[1,4,3,2] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 1
[2,1,3,4] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 1
[2,1,4,3] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 1
[2,3,1,4] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 1
[2,3,4,1] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 1
[2,4,1,3] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 1
[2,4,3,1] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 1
[3,1,2,4] => [1,3,2,4] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 1
[3,1,4,2] => [1,3,4,2] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 1
[3,2,1,4] => [1,3,2,4] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 1
[3,2,4,1] => [1,3,4,2] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 1
[3,4,1,2] => [1,3,2,4] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 1
[3,4,2,1] => [1,3,2,4] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 1
[4,1,2,3] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[4,1,3,2] => [1,4,2,3] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 1
[4,2,1,3] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[4,2,3,1] => [1,4,2,3] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 1
[4,3,1,2] => [1,4,2,3] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 1
[4,3,2,1] => [1,4,2,3] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5} - 1
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,2,3,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,2,4,3,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,2,4,5,3] => [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,2,5,3,4] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,2,5,4,3] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,3,2,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,3,2,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,3,4,2,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,3,4,5,2] => [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,3,5,2,4] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,3,5,4,2] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,4,2,3,5] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,4,2,5,3] => [1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,4,3,2,5] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,4,3,5,2] => [1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,4,5,2,3] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,4,5,3,2] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,5,2,3,4] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,5,2,4,3] => [1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[1,5,3,2,4] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,5,3,4,2] => [1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8} - 1
[2,5,1,3,4] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,5,3,1,4] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,1,5,2,4] => [1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,2,5,1,4] => [1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[4,1,2,5,3] => [1,4,5,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[4,2,1,5,3] => [1,4,5,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[5,1,2,3,4] => [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[5,1,4,2,3] => [1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[5,2,1,3,4] => [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[5,2,4,1,3] => [1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[5,4,1,3,2] => [1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[5,4,2,3,1] => [1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[5,4,3,1,2] => [1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[5,4,3,2,1] => [1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,2,6,3,4,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,2,6,4,3,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,3,6,2,4,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,3,6,4,2,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,4,2,6,3,5] => [1,2,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,4,3,6,2,5] => [1,2,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,5,2,3,6,4] => [1,2,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,5,3,2,6,4] => [1,2,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,6,2,3,4,5] => [1,2,6,5,4,3] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,6,2,5,3,4] => [1,2,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,6,3,2,4,5] => [1,2,6,5,4,3] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,6,3,5,2,4] => [1,2,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,6,5,2,4,3] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,6,5,3,4,2] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,6,5,4,2,3] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,6,5,4,3,2] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,1,6,3,4,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,1,6,4,3,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,3,6,1,4,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,3,6,4,1,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,4,1,6,3,5] => [1,2,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,4,3,6,1,5] => [1,2,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,5,1,3,6,4] => [1,2,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,5,3,1,6,4] => [1,2,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,6,1,3,4,5] => [1,2,6,5,4,3] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,6,1,5,3,4] => [1,2,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,6,3,1,4,5] => [1,2,6,5,4,3] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,6,3,5,1,4] => [1,2,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,6,5,1,4,3] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,6,5,3,4,1] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,6,5,4,1,3] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000937
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000937: Integer partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 30%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000937: Integer partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 30%
Values
[1] => [1]
=> []
=> ?
=> ? = 1
[1,2] => [1,1]
=> [1]
=> []
=> ? ∊ {2,2}
[2,1] => [2]
=> []
=> ?
=> ? ∊ {2,2}
[1,2,3] => [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {3,3,3,3,3,3}
[1,3,2] => [2,1]
=> [1]
=> []
=> ? ∊ {3,3,3,3,3,3}
[2,1,3] => [2,1]
=> [1]
=> []
=> ? ∊ {3,3,3,3,3,3}
[2,3,1] => [3]
=> []
=> ?
=> ? ∊ {3,3,3,3,3,3}
[3,1,2] => [3]
=> []
=> ?
=> ? ∊ {3,3,3,3,3,3}
[3,2,1] => [2,1]
=> [1]
=> []
=> ? ∊ {3,3,3,3,3,3}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? ∊ {1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? ∊ {1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,3,4,1] => [4]
=> []
=> ?
=> ? ∊ {1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,4,1,3] => [4]
=> []
=> ?
=> ? ∊ {1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,4,3,1] => [3,1]
=> [1]
=> []
=> ? ∊ {1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,1,4,2] => [4]
=> []
=> ?
=> ? ∊ {1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,2,4,1] => [3,1]
=> [1]
=> []
=> ? ∊ {1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? ∊ {1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,4,2,1] => [4]
=> []
=> ?
=> ? ∊ {1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,1,2,3] => [4]
=> []
=> ?
=> ? ∊ {1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,1,3,2] => [3,1]
=> [1]
=> []
=> ? ∊ {1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,2,1,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,3,1,2] => [4]
=> []
=> ?
=> ? ∊ {1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,3,2,1] => [2,2]
=> [2]
=> []
=> ? ∊ {1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,5,2,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,2,5,3] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,5,3,2] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,5,4,2,3] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,2,3,1,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[5,2,3,4,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3
[1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,2,3,5,4,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,2,3,5,6,4] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,3,6,4,5] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,3,6,5,4] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,2,4,3,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,2,4,3,6,5] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,2,4,5,3,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,4,6,5,3] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,5,3,4,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,5,4,3,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,2,5,4,6,3] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,5,6,3,4] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,2,6,3,5,4] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,6,4,3,5] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,6,4,5,3] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,2,6,5,4,3] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,3,2,4,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,3,2,4,6,5] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,3,2,5,4,6] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,3,2,6,5,4] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,3,4,2,5,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,3,5,4,2,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,3,6,4,5,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,4,2,3,5,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,4,3,2,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,4,3,2,6,5] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,4,3,5,2,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,4,3,6,5,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,4,5,2,3,6] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,4,6,2,5,3] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,5,2,4,3,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,5,3,2,4,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,5,3,4,2,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,5,3,4,6,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,5,3,6,2,4] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,5,4,3,2,6] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
Description
The number of positive values of the symmetric group character corresponding to the partition.
For example, the character values of the irreducible representation S(2,2) are 2 on the conjugacy classes (4) and (2,2), 0 on the conjugacy classes (3,1) and (1,1,1,1), and −1 on the conjugacy class (2,1,1). Therefore, the statistic on the partition (2,2) is 2.
Matching statistic: St000264
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 14%●distinct values known / distinct values provided: 10%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 14%●distinct values known / distinct values provided: 10%
Values
[1] => [1] => ([],1)
=> ([],1)
=> ? = 1
[1,2] => [1,2] => ([],2)
=> ([],1)
=> ? ∊ {2,2}
[2,1] => [1,2] => ([],2)
=> ([],1)
=> ? ∊ {2,2}
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> ? ∊ {3,3,3,3,3,3}
[1,3,2] => [1,3,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {3,3,3,3,3,3}
[2,1,3] => [1,3,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {3,3,3,3,3,3}
[2,3,1] => [1,2,3] => ([],3)
=> ([],1)
=> ? ∊ {3,3,3,3,3,3}
[3,1,2] => [1,2,3] => ([],3)
=> ([],1)
=> ? ∊ {3,3,3,3,3,3}
[3,2,1] => [1,2,3] => ([],3)
=> ([],1)
=> ? ∊ {3,3,3,3,3,3}
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,3,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,1,4,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,3,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,3,4,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,4,1,3] => [1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,1,2,4] => [1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,1,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,2,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,2,4,1] => [1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,4,1,2] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,4,2,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,1,2,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,1,3,2] => [1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,2,1,3] => [1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,2,3,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,3,1,2] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,3,2,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,2,4,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,2,5,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,2,5,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,4,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,4,5,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,5,2,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,5,4,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,2,5,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,3,2,5] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,3,5,2] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,5,2,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,5,2,4,3] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,5,3,2,4] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,1,5,3] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,3,1,5] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[3,1,5,2,4] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[3,2,4,1,5] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,6,3,5,4] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,6,4,3,5] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,3,6,2,5,4] => [1,3,6,2,5,4] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,3,6,4,2,5] => [1,3,6,2,5,4] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,4,6,2,5,3] => [1,4,6,2,5,3] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,4,6,3,2,5] => [1,4,6,2,5,3] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,5,2,4,3,6] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,5,2,4,6,3] => [1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,5,3,2,4,6] => [1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,5,3,6,2,4] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,5,6,2,4,3] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,5,6,3,2,4] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,2,3,5,4] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,2,4,3,5] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,2,4,5,3] => [1,6,2,4,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,2,5,3,4] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,2,5,4,3] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,3,2,4,5] => [1,6,2,4,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,3,2,5,4] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,3,4,2,5] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,3,5,2,4] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,3,5,4,2] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,4,2,3,5] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,4,2,5,3] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,4,3,2,5] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,4,3,5,2] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,5,2,4,3] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,5,3,2,4] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,1,6,3,5,4] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,1,6,4,3,5] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,3,5,1,6,4] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,3,5,4,1,6] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,1,5,3,6] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,1,5,6,3] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,1,6,3,5] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,1,6,5,3] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,3,1,5,6] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,3,1,6,5] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,3,5,1,6] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,3,6,1,5] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,5,1,6,3] => [1,6,2,4,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,5,3,1,6] => [1,6,2,4,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,6,1,5,3] => [1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,6,3,1,5] => [1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St001060
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 14%●distinct values known / distinct values provided: 10%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 14%●distinct values known / distinct values provided: 10%
Values
[1] => [1] => ([],1)
=> ([],1)
=> ? = 1
[1,2] => [1,2] => ([],2)
=> ([],1)
=> ? ∊ {2,2}
[2,1] => [1,2] => ([],2)
=> ([],1)
=> ? ∊ {2,2}
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> ? ∊ {3,3,3,3,3,3}
[1,3,2] => [1,3,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {3,3,3,3,3,3}
[2,1,3] => [1,3,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {3,3,3,3,3,3}
[2,3,1] => [1,2,3] => ([],3)
=> ([],1)
=> ? ∊ {3,3,3,3,3,3}
[3,1,2] => [1,2,3] => ([],3)
=> ([],1)
=> ? ∊ {3,3,3,3,3,3}
[3,2,1] => [1,2,3] => ([],3)
=> ([],1)
=> ? ∊ {3,3,3,3,3,3}
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,3,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,1,4,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,3,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,3,4,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,4,1,3] => [1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,1,2,4] => [1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,1,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,2,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,2,4,1] => [1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,4,1,2] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,4,2,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,1,2,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,1,3,2] => [1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,2,1,3] => [1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,2,3,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,3,1,2] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,3,2,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {1,1,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,2,4,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,2,5,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,2,5,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,4,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,4,5,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,5,2,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,5,4,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,2,5,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,3,2,5] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,3,5,2] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,5,2,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,5,2,4,3] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,5,3,2,4] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,1,5,3] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,3,1,5] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[3,1,5,2,4] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[3,2,4,1,5] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,6,3,5,4] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,6,4,3,5] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,3,6,2,5,4] => [1,3,6,2,5,4] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,3,6,4,2,5] => [1,3,6,2,5,4] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,4,6,2,5,3] => [1,4,6,2,5,3] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,4,6,3,2,5] => [1,4,6,2,5,3] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,5,2,4,3,6] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,5,2,4,6,3] => [1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,5,3,2,4,6] => [1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,5,3,6,2,4] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,5,6,2,4,3] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,5,6,3,2,4] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,2,3,5,4] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,2,4,3,5] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,2,4,5,3] => [1,6,2,4,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,2,5,3,4] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,2,5,4,3] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,3,2,4,5] => [1,6,2,4,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,3,2,5,4] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,3,4,2,5] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,3,5,2,4] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,3,5,4,2] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,4,2,3,5] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,4,2,5,3] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,4,3,2,5] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,4,3,5,2] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,5,2,4,3] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,6,5,3,2,4] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,1,6,3,5,4] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,1,6,4,3,5] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,3,5,1,6,4] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,3,5,4,1,6] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,1,5,3,6] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,1,5,6,3] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,1,6,3,5] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,1,6,5,3] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,3,1,5,6] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,3,1,6,5] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,3,5,1,6] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,3,6,1,5] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,5,1,6,3] => [1,6,2,4,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,5,3,1,6] => [1,6,2,4,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,6,1,5,3] => [1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,6,3,1,5] => [1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St001875
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 20%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 20%
Values
[1] => [1] => ([],1)
=> ([],1)
=> ? = 1
[1,2] => [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {2,2}
[2,1] => [2,1] => ([],2)
=> ([],1)
=> ? ∊ {2,2}
[1,2,3] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([],1)
=> ? ∊ {3,3,3,3,3,3}
[1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([],1)
=> ? ∊ {3,3,3,3,3,3}
[2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> ([],1)
=> ? ∊ {3,3,3,3,3,3}
[2,3,1] => [2,3,1] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {3,3,3,3,3,3}
[3,1,2] => [3,1,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {3,3,3,3,3,3}
[3,2,1] => [3,2,1] => ([],3)
=> ([],1)
=> ? ∊ {3,3,3,3,3,3}
[1,2,3,4] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> ? ∊ {1,1,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,2,4,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> ? ∊ {1,1,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,3,2,4] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> ? ∊ {1,1,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,3,4,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> ? ∊ {1,1,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,4,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> ? ∊ {1,1,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> ? ∊ {1,1,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,1,3,4] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,3,1,4] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[2,3,4,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> ([],1)
=> ? ∊ {1,1,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[2,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[2,4,3,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> ([],1)
=> ? ∊ {1,1,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,1,2,4] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[3,1,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ? ∊ {1,1,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,2,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> ([],1)
=> ? ∊ {1,1,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[3,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,1,2,3] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> ([],1)
=> ? ∊ {1,1,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,1,3,2] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> ([],1)
=> ? ∊ {1,1,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,2,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([],1)
=> ? ∊ {1,1,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,2,3,1] => [4,2,3,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,3,1,2] => [4,3,1,2] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[4,3,2,1] => [4,3,2,1] => ([],4)
=> ([],1)
=> ? ∊ {1,1,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5}
[1,2,3,4,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,2,3,5,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,2,4,3,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,2,4,5,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,2,5,3,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,2,5,4,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,2,4,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,2,5,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,4,2,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,4,5,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,5,2,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,3,5,4,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,2,3,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,2,5,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,3,2,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,3,5,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,5,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,4,5,3,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,5,2,3,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,5,2,4,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,5,3,2,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[1,5,3,4,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
[3,4,2,5,1] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[3,5,2,4,1] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[4,2,3,5,1] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[4,2,5,3,1] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[4,5,2,3,1] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[4,5,3,1,2] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[5,2,3,1,4] => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[5,2,4,1,3] => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[5,3,1,2,4] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[5,3,1,4,2] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[5,3,4,1,2] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[3,2,4,5,1,6] => [3,2,6,5,1,4] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[3,2,4,6,1,5] => [3,2,6,5,1,4] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[3,2,5,4,1,6] => [3,2,6,5,1,4] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[3,2,5,6,1,4] => [3,2,6,5,1,4] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[3,2,6,4,1,5] => [3,2,6,5,1,4] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[3,2,6,5,1,4] => [3,2,6,5,1,4] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[3,4,2,1,5,6] => [3,6,2,1,5,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[3,4,2,1,6,5] => [3,6,2,1,5,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[3,4,2,5,1,6] => [3,6,2,5,1,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,4,2,6,1,5] => [3,6,2,5,1,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,5,2,1,4,6] => [3,6,2,1,5,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[3,5,2,1,6,4] => [3,6,2,1,5,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[3,5,2,4,1,6] => [3,6,2,5,1,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,5,2,6,1,4] => [3,6,2,5,1,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,6,2,1,4,5] => [3,6,2,1,5,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[3,6,2,1,5,4] => [3,6,2,1,5,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[3,6,2,4,1,5] => [3,6,2,5,1,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,6,2,5,1,4] => [3,6,2,5,1,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,2,3,5,1,6] => [4,2,6,5,1,3] => ([(0,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[4,2,3,6,1,5] => [4,2,6,5,1,3] => ([(0,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[4,2,5,3,1,6] => [4,2,6,5,1,3] => ([(0,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[4,2,5,6,1,3] => [4,2,6,5,1,3] => ([(0,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[4,2,6,3,1,5] => [4,2,6,5,1,3] => ([(0,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[4,2,6,5,1,3] => [4,2,6,5,1,3] => ([(0,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[4,3,1,2,5,6] => [4,3,1,6,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[4,3,1,2,6,5] => [4,3,1,6,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[4,3,1,5,2,6] => [4,3,1,6,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[4,3,1,5,6,2] => [4,3,1,6,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[4,3,1,6,2,5] => [4,3,1,6,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[4,3,1,6,5,2] => [4,3,1,6,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[4,3,5,1,2,6] => [4,3,6,1,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[4,3,5,1,6,2] => [4,3,6,1,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[4,3,5,6,1,2] => [4,3,6,5,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[4,3,6,1,2,5] => [4,3,6,1,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
Description
The number of simple modules with projective dimension at most 1.
The following 8 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000454The largest eigenvalue of a graph if it is integral. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000455The second largest eigenvalue of a graph if it is integral. St001570The minimal number of edges to add to make a graph Hamiltonian. St001621The number of atoms of a lattice. St001632The number of indecomposable injective modules I with dimExt1(I,A)=1 for the incidence algebra A of a poset.
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