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Your data matches 35 different statistics following compositions of up to 3 maps.
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Matching statistic: St000489
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
St000489: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 2
[2,1] => 1
[1,2,3] => 3
[1,3,2] => 2
[2,1,3] => 2
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 2
[1,2,3,4] => 4
[1,2,4,3] => 3
[1,3,2,4] => 3
[1,3,4,2] => 2
[1,4,2,3] => 2
[1,4,3,2] => 3
[2,1,3,4] => 3
[2,1,4,3] => 2
[2,3,1,4] => 2
[2,3,4,1] => 0
[2,4,1,3] => 0
[2,4,3,1] => 2
[3,1,2,4] => 2
[3,1,4,2] => 0
[3,2,1,4] => 3
[3,2,4,1] => 2
[3,4,1,2] => 2
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,3,2] => 2
[4,2,1,3] => 2
[4,2,3,1] => 3
[4,3,1,2] => 0
[4,3,2,1] => 2
[1,2,3,4,5] => 5
[1,2,3,5,4] => 4
[1,2,4,3,5] => 4
[1,2,4,5,3] => 3
[1,2,5,3,4] => 3
[1,2,5,4,3] => 4
[1,3,2,4,5] => 4
[1,3,2,5,4] => 3
[1,3,4,2,5] => 3
[1,3,4,5,2] => 1
[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] => 4
[1,4,3,5,2] => 3
[1,4,5,2,3] => 3
[1,4,5,3,2] => 1
Description
The number of cycles of a permutation of length at most 3.
Matching statistic: St001604
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 35%●distinct values known / distinct values provided: 29%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 35%●distinct values known / distinct values provided: 29%
Values
[1,2] => [1,1]
=> [1]
=> [1]
=> ? ∊ {1,2}
[2,1] => [2]
=> []
=> []
=> ? ∊ {1,2}
[1,2,3] => [1,1,1]
=> [1,1]
=> [2]
=> ? ∊ {1,1,2,2,2,3}
[1,3,2] => [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,2,2,3}
[2,1,3] => [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,2,2,3}
[2,3,1] => [3]
=> []
=> []
=> ? ∊ {1,1,2,2,2,3}
[3,1,2] => [3]
=> []
=> []
=> ? ∊ {1,1,2,2,2,3}
[3,2,1] => [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,2,2,3}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [2]
=> ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [2]
=> ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,4,2] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,4,2,3] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [2]
=> ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [2]
=> ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,1,4,3] => [2,2]
=> [2]
=> [1,1]
=> ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,3,1,4] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,3,4,1] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,4,1,3] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,4,3,1] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[3,1,2,4] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[3,1,4,2] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [2]
=> ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[3,2,4,1] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[3,4,1,2] => [2,2]
=> [2]
=> [1,1]
=> ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[3,4,2,1] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[4,1,2,3] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[4,1,3,2] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[4,2,1,3] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [2]
=> ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[4,3,1,2] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[4,3,2,1] => [2,2]
=> [2]
=> [1,1]
=> ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [3]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,4,5,2] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,5,2,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,2,5,3] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [3]
=> 1
[1,4,5,3,2] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,2,3,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,5,4,2,3] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [3]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [3]
=> 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [3]
=> 1
[2,1,4,5,3] => [3,2]
=> [2]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,1,5,3,4] => [3,2]
=> [2]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [3]
=> 1
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,1,5,4] => [3,2]
=> [2]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,4,1,5] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [3]
=> 1
[3,4,1,2,5] => [2,2,1]
=> [2,1]
=> [3]
=> 1
[3,5,1,4,2] => [2,2,1]
=> [2,1]
=> [3]
=> 1
[4,2,3,1,5] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[4,2,5,1,3] => [2,2,1]
=> [2,1]
=> [3]
=> 1
[4,3,2,1,5] => [2,2,1]
=> [2,1]
=> [3]
=> 1
[4,5,3,1,2] => [2,2,1]
=> [2,1]
=> [3]
=> 1
[5,2,3,4,1] => [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[5,2,4,3,1] => [2,2,1]
=> [2,1]
=> [3]
=> 1
[5,3,2,4,1] => [2,2,1]
=> [2,1]
=> [3]
=> 1
[5,4,3,2,1] => [2,2,1]
=> [2,1]
=> [3]
=> 1
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [3,2]
=> 1
[1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 0
[1,2,3,5,4,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 0
[1,2,3,5,6,4] => [3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,2,3,6,4,5] => [3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,2,3,6,5,4] => [2,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 0
[1,2,4,3,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 0
[1,2,4,3,6,5] => [2,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1
[1,2,4,5,3,6] => [3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,2,4,6,5,3] => [3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,2,5,3,4,6] => [3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,2,5,4,3,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 0
[1,2,5,4,6,3] => [3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,2,5,6,3,4] => [2,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1
[1,2,6,3,5,4] => [3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,2,6,4,3,5] => [3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,2,6,4,5,3] => [2,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 0
[1,2,6,5,4,3] => [2,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1
[1,3,2,4,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 0
[1,3,2,4,6,5] => [2,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1
[1,3,2,5,4,6] => [2,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1
[1,3,2,5,6,4] => [3,2,1]
=> [2,1]
=> [3]
=> 1
[1,3,2,6,4,5] => [3,2,1]
=> [2,1]
=> [3]
=> 1
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St000771
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 57%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 57%
Values
[1,2] => [1,2] => [1,2] => ([],2)
=> ? ∊ {1,2}
[2,1] => [1,2] => [1,2] => ([],2)
=> ? ∊ {1,2}
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {2,2,2,3}
[1,3,2] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[2,1,3] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[2,3,1] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {2,2,2,3}
[3,1,2] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {2,2,2,3}
[3,2,1] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {2,2,2,3}
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,4,2] => [1,3,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,4}
[1,4,3,2] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,4}
[2,1,3,4] => [1,3,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,1,4,3] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,4}
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,4}
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,4}
[2,4,1,3] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,4}
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,1,2,4] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,1,4,2] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,4}
[3,2,1,4] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,4}
[3,2,4,1] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,4}
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,4}
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,4}
[4,1,3,2] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,4}
[4,2,1,3] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,4}
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,4}
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,4}
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,2,4,5,3] => [1,2,4,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,2,5,3,4] => [1,2,5,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,2,5,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,2,5,4] => [1,3,2,5,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[1,3,4,2,5] => [1,3,4,2,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,4,5,2] => [1,3,4,5,2] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,5,4,2] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,2,5,3] => [1,4,2,5,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,3,2,5] => [1,4,2,5,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,3,5,2] => [1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,5,2,3] => [1,4,5,2,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[1,4,5,3,2] => [1,4,5,2,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[1,5,2,3,4] => [1,5,2,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,2,4,3] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,5,3,2,4] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,5,3,4,2] => [1,5,2,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,4,2,3] => [1,5,2,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,4,3,2] => [1,5,2,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,1,3,4,5] => [1,3,4,5,2] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,1,3,5,4] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,1,4,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,1,4,5,3] => [1,4,5,2,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[2,1,5,3,4] => [1,5,2,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,1,5,4,3] => [1,5,2,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,1,4,5] => [1,4,5,2,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[2,3,1,5,4] => [1,5,2,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,4,1,5] => [1,5,2,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,5,1,4] => [1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,5,4,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[2,4,1,3,5] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,4,1,5,3] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,4,3,1,5] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,4,3,5,1] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,5,1,3] => [1,3,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,5,3,1] => [1,2,4,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,5,1,3,4] => [1,3,4,2,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,5,1,4,3] => [1,4,2,5,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,5,3,1,4] => [1,4,2,5,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,5,4,1,3] => [1,3,2,5,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[3,1,2,4,5] => [1,2,4,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[3,1,4,5,2] => [1,4,5,2,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[3,1,5,2,4] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,2,1,4,5] => [1,4,5,2,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[3,2,4,1,5] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,2,4,5,1] => [1,2,4,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[3,5,4,1,2] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[3,5,4,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[4,1,2,3,5] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[4,1,3,2,5] => [1,3,2,5,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[4,1,3,5,2] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[4,2,1,3,5] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[4,2,3,5,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[4,2,5,1,3] => [1,3,2,5,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[4,3,5,1,2] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[4,3,5,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[1,2,3,5,6,4] => [1,2,3,5,6,4] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [4,6,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[1,2,4,5,6,3] => [1,2,4,5,6,3] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 4
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,2,4,6,5,3] => [1,2,4,6,3,5] => [6,4,1,2,3,5] => ([(0,5),(1,4),(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
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $2$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St000772
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 57%
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 57%
Values
[1,2] => [[1,2]]
=> [2] => ([],2)
=> ? ∊ {1,2}
[2,1] => [[1],[2]]
=> [2] => ([],2)
=> ? ∊ {1,2}
[1,2,3] => [[1,2,3]]
=> [3] => ([],3)
=> ? ∊ {2,2,2,3}
[1,3,2] => [[1,2],[3]]
=> [3] => ([],3)
=> ? ∊ {2,2,2,3}
[2,1,3] => [[1,3],[2]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 1
[2,3,1] => [[1,2],[3]]
=> [3] => ([],3)
=> ? ∊ {2,2,2,3}
[3,1,2] => [[1,3],[2]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 1
[3,2,1] => [[1],[2],[3]]
=> [3] => ([],3)
=> ? ∊ {2,2,2,3}
[1,2,3,4] => [[1,2,3,4]]
=> [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,4}
[1,2,4,3] => [[1,2,3],[4]]
=> [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,4}
[1,3,2,4] => [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,3,4,2] => [[1,2,3],[4]]
=> [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,4}
[1,4,2,3] => [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [[1,2],[3],[4]]
=> [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,4}
[2,1,3,4] => [[1,3,4],[2]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,4}
[2,1,4,3] => [[1,3],[2,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,4}
[2,3,1,4] => [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,3,4,1] => [[1,2,3],[4]]
=> [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,4}
[2,4,1,3] => [[1,2],[3,4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,4,3,1] => [[1,2],[3],[4]]
=> [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,4}
[3,1,2,4] => [[1,3,4],[2]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,4}
[3,1,4,2] => [[1,3],[2,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,4}
[3,2,1,4] => [[1,4],[2],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => [[1,3],[2],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,4}
[3,4,1,2] => [[1,2],[3,4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,4,2,1] => [[1,2],[3],[4]]
=> [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,4}
[4,1,2,3] => [[1,3,4],[2]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,4}
[4,1,3,2] => [[1,3],[2],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,4}
[4,2,1,3] => [[1,4],[2],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[4,2,3,1] => [[1,3],[2],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,4}
[4,3,1,2] => [[1,4],[2],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[4,3,2,1] => [[1],[2],[3],[4]]
=> [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,4}
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,2,4,5,3] => [[1,2,3,4],[5]]
=> [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,2,5,3,4] => [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,2,5,4] => [[1,2,4],[3,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,4,2,5] => [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,3,4,5,2] => [[1,2,3,4],[5]]
=> [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,5,2,4] => [[1,2,3],[4,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,3,5,4,2] => [[1,2,3],[4],[5]]
=> [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,2,5,3] => [[1,2,4],[3,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,4,3,5,2] => [[1,2,4],[3],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,4,5,3,2] => [[1,2,3],[4],[5]]
=> [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,2,3,4] => [[1,2,4,5],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,2,4,3] => [[1,2,4],[3],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,3,2,4] => [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,5,3,4,2] => [[1,2,4],[3],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,4,2,3] => [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,1,3,4,5] => [[1,3,4,5],[2]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,1,3,5,4] => [[1,3,4],[2,5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,1,4,3,5] => [[1,3,5],[2,4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,1,4,5,3] => [[1,3,4],[2,5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,1,5,3,4] => [[1,3,5],[2,4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,1,5,4,3] => [[1,3],[2,4],[5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,1,4,5] => [[1,2,4,5],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,1,5,4] => [[1,2,4],[3,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,4,1,5] => [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[2,3,4,5,1] => [[1,2,3,4],[5]]
=> [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,5,1,4] => [[1,2,3],[4,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[2,3,5,4,1] => [[1,2,3],[4],[5]]
=> [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,1,3,5] => [[1,2,5],[3,4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,1,5,3] => [[1,2,4],[3,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,3,1,5] => [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[2,4,3,5,1] => [[1,2,4],[3],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,5,1,3] => [[1,2,3],[4,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[2,4,5,3,1] => [[1,2,3],[4],[5]]
=> [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,5,3,1,4] => [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[2,5,4,1,3] => [[1,2],[3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[3,1,4,2,5] => [[1,3,5],[2,4]]
=> [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],[2,4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,2,4,1,5] => [[1,3,5],[2],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,2,5,1,4] => [[1,3],[2,5],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,4,2,1,5] => [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[3,4,5,1,2] => [[1,2,3],[4,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[3,5,2,1,4] => [[1,2],[3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[3,5,4,1,2] => [[1,2],[3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[4,1,3,2,5] => [[1,3,5],[2],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[4,1,5,2,3] => [[1,3,5],[2,4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[4,2,3,1,5] => [[1,3,5],[2],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[4,2,5,1,3] => [[1,3],[2,5],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[4,3,2,1,5] => [[1,5],[2],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[4,3,5,1,2] => [[1,3],[2,5],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[4,5,2,1,3] => [[1,2],[3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[4,5,3,1,2] => [[1,2],[3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[5,1,3,2,4] => [[1,3,5],[2],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,1,4,2,3] => [[1,3,5],[2],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,2,3,1,4] => [[1,3,5],[2],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,2,4,1,3] => [[1,3],[2,5],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,3,2,1,4] => [[1,5],[2],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[5,3,4,1,2] => [[1,3],[2,5],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,4,2,1,3] => [[1,5],[2],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[5,4,3,1,2] => [[1,5],[2],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
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
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $1$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$.
The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Matching statistic: St000264
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Values
[1,2] => ([],2)
=> ([],2)
=> ([(0,1)],2)
=> ? ∊ {1,2}
[2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {1,2}
[1,2,3] => ([],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1,2,2,2}
[2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1,2,2,2}
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {1,1,2,2,2}
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {1,1,2,2,2}
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {1,1,2,2,2}
[1,2,3,4] => ([],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,2,4,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,3,2,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3}
[2,1,3,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3}
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3}
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3}
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3}
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3}
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3}
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3}
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3}
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3}
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3}
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3}
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3}
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3}
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3}
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3}
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,2,3,4,5] => ([],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,2,3,5,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,2,4,3,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,2,4,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,1,3,4,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[1,2,3,4,5,6] => ([],6)
=> ([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,2,3,4,6,5] => ([(4,5)],6)
=> ([(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,2,3,5,4,6] => ([(4,5)],6)
=> ([(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,2,4,3,5,6] => ([(4,5)],6)
=> ([(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,2,4,6,5,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,2,5,4,6,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 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: St000259
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 71%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 71%
Values
[1,2] => [1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 1
[2,1] => [1,1,0,0]
=> [1,2] => ([],2)
=> ? = 2
[1,2,3] => [1,0,1,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> ? ∊ {1,1,2,3}
[1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[2,3,1] => [1,1,0,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> ? ∊ {1,1,2,3}
[3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,2,3}
[3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,2,3}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,3,3,3,3,4}
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 3
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,3,3,3,3,4}
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,3,3,3,3,4}
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 3
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,3,3,3,3,4}
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,3,3,3,3,4}
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,3,3,3,3,4}
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,3,3,3,3,4}
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,3,3,3,3,4}
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,3,3,3,3,4}
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,3,3,3,3,4}
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,3,3,3,3,4}
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,3,3,3,3,4}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,3,3,3,3,4}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,3,3,3,3,4}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,3,3,3,3,4}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,3,3,3,3,4}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,3,3,3,3,4}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5}
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5}
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5}
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5}
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5}
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5}
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5}
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5}
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5}
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5}
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5}
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5}
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5}
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5}
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5}
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5}
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5}
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5}
[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5}
[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5}
[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5}
[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5}
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5}
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5}
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5}
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5}
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5}
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5}
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 3
[1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 3
[1,2,5,3,6,4] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 4
[1,2,5,4,6,3] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 4
[1,2,5,6,3,4] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,2,5,6,4,3] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[1,2,6,4,3,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St001060
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 43%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 43%
Values
[1,2] => 1 => [1] => ([],1)
=> ? ∊ {1,2}
[2,1] => 0 => [1] => ([],1)
=> ? ∊ {1,2}
[1,2,3] => 11 => [2] => ([],2)
=> ? ∊ {1,1,2,2,2,3}
[1,3,2] => 10 => [1,1] => ([(0,1)],2)
=> ? ∊ {1,1,2,2,2,3}
[2,1,3] => 01 => [1,1] => ([(0,1)],2)
=> ? ∊ {1,1,2,2,2,3}
[2,3,1] => 00 => [2] => ([],2)
=> ? ∊ {1,1,2,2,2,3}
[3,1,2] => 00 => [2] => ([],2)
=> ? ∊ {1,1,2,2,2,3}
[3,2,1] => 00 => [2] => ([],2)
=> ? ∊ {1,1,2,2,2,3}
[1,2,3,4] => 111 => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,4}
[1,2,4,3] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,3,2,4] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,3,4,2] => 100 => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,4}
[1,4,2,3] => 100 => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,4}
[1,4,3,2] => 100 => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,4}
[2,1,3,4] => 011 => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,4}
[2,1,4,3] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[2,3,1,4] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[2,3,4,1] => 000 => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,4}
[2,4,1,3] => 000 => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,4}
[2,4,3,1] => 000 => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,4}
[3,1,2,4] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[3,1,4,2] => 000 => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,4}
[3,2,1,4] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[3,2,4,1] => 000 => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,4}
[3,4,1,2] => 000 => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,4}
[3,4,2,1] => 000 => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,4}
[4,1,2,3] => 000 => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,4}
[4,1,3,2] => 000 => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,4}
[4,2,1,3] => 000 => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,4}
[4,2,3,1] => 000 => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,4}
[4,3,1,2] => 000 => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,4}
[4,3,2,1] => 000 => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,4}
[1,2,3,4,5] => 1111 => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,2,3,5,4] => 1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,2,4,3,5] => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,2,4,5,3] => 1100 => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,2,5,3,4] => 1100 => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,2,5,4,3] => 1100 => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,3,2,4,5] => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,3,2,5,4] => 1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,3,4,2,5] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,4,5,2] => 1000 => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,5,2,4] => 1000 => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,5,4,2] => 1000 => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,2,3,5] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,4,2,5,3] => 1000 => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,3,2,5] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,4,3,5,2] => 1000 => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,5,2,3] => 1000 => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,5,3,2] => 1000 => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,2,3,4] => 1000 => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,2,4,3] => 1000 => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,3,2,4] => 1000 => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,3,4,2] => 1000 => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,4,2,3] => 1000 => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,4,3,2] => 1000 => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,1,3,4,5] => 0111 => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,1,3,5,4] => 0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3,5] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,1,4,5,3] => 0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[2,1,5,3,4] => 0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[2,1,5,4,3] => 0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[2,3,1,4,5] => 0011 => [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,3,1,5,4] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3,4,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,3,4,5,1] => 0000 => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,5,1,4] => 0000 => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,5,4,1] => 0000 => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,1,3,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,4,1,5,3] => 0000 => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,3,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,4,3,5,1] => 0000 => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,5,1,3] => 0000 => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,5,3,1] => 0000 => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,5,1,3,4] => 0000 => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,5,1,4,3] => 0000 => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[3,1,2,4,5] => 0011 => [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,1,2,5,4] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,4,2,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[3,2,1,4,5] => 0011 => [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,2,1,5,4] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[3,4,1,2,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[3,4,2,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[4,1,2,3,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[4,1,3,2,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[4,2,1,3,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[4,2,3,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[4,3,1,2,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[4,3,2,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,2,3,4,6,5] => 11110 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[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)
=> 2
[1,2,3,5,6,4] => 11100 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,2,3,6,4,5] => 11100 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,2,3,6,5,4] => 11100 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,2,4,3,5,6] => 11011 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 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)
=> 2
[1,2,4,5,3,6] => 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,5,3,4,6] => 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,5,4,3,6] => 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
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: St001632
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 43%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 43%
Values
[1,2] => [1,2] => [1,2] => ([(0,1)],2)
=> 1
[2,1] => [2,1] => [2,1] => ([],2)
=> ? = 2
[1,2,3] => [1,3,2] => [2,3,1] => ([(1,2)],3)
=> ? ∊ {1,2,2,3}
[1,3,2] => [1,3,2] => [2,3,1] => ([(1,2)],3)
=> ? ∊ {1,2,2,3}
[2,1,3] => [2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[2,3,1] => [2,3,1] => [3,1,2] => ([(1,2)],3)
=> ? ∊ {1,2,2,3}
[3,1,2] => [3,1,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[3,2,1] => [3,2,1] => [3,2,1] => ([],3)
=> ? ∊ {1,2,2,3}
[1,2,3,4] => [1,4,3,2] => [3,4,2,1] => ([(2,3)],4)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,4,3] => [1,4,3,2] => [3,4,2,1] => ([(2,3)],4)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,2,4] => [1,4,3,2] => [3,4,2,1] => ([(2,3)],4)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,4,2] => [1,4,3,2] => [3,4,2,1] => ([(2,3)],4)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,4,2,3] => [1,4,3,2] => [3,4,2,1] => ([(2,3)],4)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,4,3,2] => [1,4,3,2] => [3,4,2,1] => ([(2,3)],4)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,1,3,4] => [2,1,4,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,1,4,3] => [2,1,4,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,3,1,4] => [2,4,1,3] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
[2,3,4,1] => [2,4,3,1] => [4,2,3,1] => ([(2,3)],4)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,4,1,3] => [2,4,1,3] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
[2,4,3,1] => [2,4,3,1] => [4,2,3,1] => ([(2,3)],4)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[3,1,2,4] => [3,1,4,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[3,1,4,2] => [3,1,4,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 0
[3,2,4,1] => [3,2,4,1] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[3,4,1,2] => [3,4,1,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[3,4,2,1] => [3,4,2,1] => [4,3,1,2] => ([(2,3)],4)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[4,1,2,3] => [4,1,3,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[4,1,3,2] => [4,1,3,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[4,2,1,3] => [4,2,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 0
[4,2,3,1] => [4,2,3,1] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[4,3,1,2] => [4,3,1,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 0
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => ([],4)
=> ? ∊ {0,0,0,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,3,4,5] => [1,5,4,3,2] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,2,3,5,4] => [1,5,4,3,2] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,2,4,3,5] => [1,5,4,3,2] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,2,4,5,3] => [1,5,4,3,2] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,2,5,3,4] => [1,5,4,3,2] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,2,5,4,3] => [1,5,4,3,2] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,2,4,5] => [1,5,4,3,2] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,2,5,4] => [1,5,4,3,2] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,4,2,5] => [1,5,4,3,2] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,4,5,2] => [1,5,4,3,2] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,5,2,4] => [1,5,4,3,2] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,5,4,2] => [1,5,4,3,2] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,2,3,5] => [1,5,4,3,2] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,2,5,3] => [1,5,4,3,2] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,3,2,5] => [1,5,4,3,2] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,3,5,2] => [1,5,4,3,2] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,5,2,3] => [1,5,4,3,2] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,5,3,2] => [1,5,4,3,2] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,2,3,4] => [1,5,4,3,2] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,2,4,3] => [1,5,4,3,2] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,3,2,4] => [1,5,4,3,2] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,3,4,2] => [1,5,4,3,2] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,4,2,3] => [1,5,4,3,2] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,4,3,2] => [1,5,4,3,2] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,1,3,4,5] => [2,1,5,4,3] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,1,3,5,4] => [2,1,5,4,3] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,1,4,3,5] => [2,1,5,4,3] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,4,1,5] => [2,5,4,1,3] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 0
[2,3,5,1,4] => [2,5,4,1,3] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 0
[2,4,3,1,5] => [2,5,4,1,3] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 0
[2,4,5,1,3] => [2,5,4,1,3] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 0
[2,5,3,1,4] => [2,5,4,1,3] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 0
[2,5,4,1,3] => [2,5,4,1,3] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 0
[3,2,4,1,5] => [3,2,5,1,4] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 1
[3,2,5,1,4] => [3,2,5,1,4] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 1
[3,4,2,1,5] => [3,5,2,1,4] => [4,1,3,5,2] => ([(0,4),(1,2),(1,3),(3,4)],5)
=> 1
[3,4,5,1,2] => [3,5,4,1,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 0
[3,5,2,1,4] => [3,5,2,1,4] => [4,1,3,5,2] => ([(0,4),(1,2),(1,3),(3,4)],5)
=> 1
[3,5,4,1,2] => [3,5,4,1,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 0
[4,2,3,1,5] => [4,2,5,1,3] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[4,2,5,1,3] => [4,2,5,1,3] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[4,3,5,1,2] => [4,3,5,1,2] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 1
[4,5,1,2,3] => [4,5,1,3,2] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> 0
[4,5,1,3,2] => [4,5,1,3,2] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> 0
[4,5,2,1,3] => [4,5,2,1,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> 0
[4,5,3,1,2] => [4,5,3,1,2] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 0
[5,2,3,1,4] => [5,2,4,1,3] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> 1
[5,2,4,1,3] => [5,2,4,1,3] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> 1
[5,3,2,1,4] => [5,3,2,1,4] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> 0
[5,3,4,1,2] => [5,3,4,1,2] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> 1
[5,4,2,1,3] => [5,4,2,1,3] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> 0
[5,4,3,1,2] => [5,4,3,1,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> 0
[2,3,4,5,1,6] => [2,6,5,4,1,3] => [1,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 0
[2,3,4,6,1,5] => [2,6,5,4,1,3] => [1,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 0
[2,3,5,4,1,6] => [2,6,5,4,1,3] => [1,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 0
[2,3,5,6,1,4] => [2,6,5,4,1,3] => [1,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 0
[2,3,6,4,1,5] => [2,6,5,4,1,3] => [1,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 0
[2,3,6,5,1,4] => [2,6,5,4,1,3] => [1,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 0
[2,4,3,5,1,6] => [2,6,5,4,1,3] => [1,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 0
[2,4,3,6,1,5] => [2,6,5,4,1,3] => [1,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 0
[2,4,5,3,1,6] => [2,6,5,4,1,3] => [1,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 0
[2,4,5,6,1,3] => [2,6,5,4,1,3] => [1,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 0
[2,4,6,3,1,5] => [2,6,5,4,1,3] => [1,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 0
[2,4,6,5,1,3] => [2,6,5,4,1,3] => [1,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 0
[2,5,3,4,1,6] => [2,6,5,4,1,3] => [1,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 0
[2,5,3,6,1,4] => [2,6,5,4,1,3] => [1,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 0
[2,5,4,3,1,6] => [2,6,5,4,1,3] => [1,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 0
Description
The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.
Matching statistic: St001603
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 29%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 29%
Values
[1,2] => [1] => [1]
=> []
=> ? ∊ {1,2}
[2,1] => [1] => [1]
=> []
=> ? ∊ {1,2}
[1,2,3] => [1,2] => [1,1]
=> [1]
=> ? ∊ {1,1,2,2,2,3}
[1,3,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {1,1,2,2,2,3}
[2,1,3] => [2,1] => [2]
=> []
=> ? ∊ {1,1,2,2,2,3}
[2,3,1] => [2,1] => [2]
=> []
=> ? ∊ {1,1,2,2,2,3}
[3,1,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {1,1,2,2,2,3}
[3,2,1] => [2,1] => [2]
=> []
=> ? ∊ {1,1,2,2,2,3}
[1,2,3,4] => [1,2,3] => [1,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,4,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,2,4] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,4,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,4,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,4,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,1,3,4] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,1,4,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,3,1,4] => [2,3,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,3,4,1] => [2,3,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,4,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,4,3,1] => [2,3,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[3,1,2,4] => [3,1,2] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[3,1,4,2] => [3,1,2] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[3,2,1,4] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[3,2,4,1] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[3,4,1,2] => [3,1,2] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[3,4,2,1] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[4,1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[4,1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[4,2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[4,2,3,1] => [2,3,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[4,3,1,2] => [3,1,2] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[4,3,2,1] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,3,4,5] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,3,5,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,4,3,5] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,2,4,5,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,2,5,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,5,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,2,4,5] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,2,5,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,4,2,5] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,4,5,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,5,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,3,5,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,2,3,5] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,2,5,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,3,2,5] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,3,5,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,5,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,4,5,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,5,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[1,5,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[5,1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,3,4,5,6] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 3
[1,2,3,4,6,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 3
[1,2,3,5,4,6] => [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,2,3,5,6,4] => [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,2,3,6,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 3
[1,2,3,6,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,2,4,3,5,6] => [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,2,4,3,6,5] => [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,2,4,6,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,2,5,4,3,6] => [1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,2,5,4,6,3] => [1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,2,5,6,4,3] => [1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,2,6,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 3
[1,2,6,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,2,6,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,2,6,5,4,3] => [1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,3,2,4,5,6] => [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,3,2,4,6,5] => [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[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]
=> 1
[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]
=> 1
[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]
=> 1
[1,4,3,2,6,5] => [1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,4,3,6,2,5] => [1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[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]
=> 1
[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]
=> 1
[1,5,3,4,6,2] => [1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,5,3,6,4,2] => [1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> 1
[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]
=> 1
[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]
=> 3
[1,6,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,6,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,6,2,5,4,3] => [1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,6,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
Description
The number of colourings of a polygon such that the multiplicities of a colour are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St000777
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 71%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 71%
Values
[1,2] => 1 => [1,1] => ([(0,1)],2)
=> 2
[2,1] => 0 => [2] => ([],2)
=> ? = 1
[1,2,3] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,3,2] => 10 => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,2,2}
[2,1,3] => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 3
[2,3,1] => 00 => [3] => ([],3)
=> ? ∊ {1,1,2,2}
[3,1,2] => 00 => [3] => ([],3)
=> ? ∊ {1,1,2,2}
[3,2,1] => 00 => [3] => ([],3)
=> ? ∊ {1,1,2,2}
[1,2,3,4] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,2,4,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3}
[1,3,2,4] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,3,4,2] => 100 => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3}
[1,4,2,3] => 100 => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3}
[1,4,3,2] => 100 => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3}
[2,1,3,4] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,1,4,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3}
[2,3,1,4] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,3,4,1] => 000 => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3}
[2,4,1,3] => 000 => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3}
[2,4,3,1] => 000 => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3}
[3,1,2,4] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[3,1,4,2] => 000 => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3}
[3,2,1,4] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[3,2,4,1] => 000 => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3}
[3,4,1,2] => 000 => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3}
[3,4,2,1] => 000 => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3}
[4,1,2,3] => 000 => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3}
[4,1,3,2] => 000 => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3}
[4,2,1,3] => 000 => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3}
[4,2,3,1] => 000 => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3}
[4,3,1,2] => 000 => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3}
[4,3,2,1] => 000 => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3}
[1,2,3,4,5] => 1111 => [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)
=> 2
[1,2,3,5,4] => 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[1,2,4,3,5] => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,2,4,5,3] => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[1,2,5,3,4] => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[1,2,5,4,3] => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[1,3,2,4,5] => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,3,2,5,4] => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[1,3,4,2,5] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,3,4,5,2] => 1000 => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[1,3,5,2,4] => 1000 => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[1,3,5,4,2] => 1000 => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[1,4,2,3,5] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,4,2,5,3] => 1000 => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[1,4,3,2,5] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,4,3,5,2] => 1000 => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[1,4,5,2,3] => 1000 => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[1,4,5,3,2] => 1000 => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[1,5,2,3,4] => 1000 => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[1,5,2,4,3] => 1000 => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[1,5,3,2,4] => 1000 => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[1,5,3,4,2] => 1000 => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[1,5,4,2,3] => 1000 => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[1,5,4,3,2] => 1000 => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,1,3,4,5] => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,3,5,4] => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,1,4,3,5] => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[2,1,4,5,3] => 0100 => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,1,5,3,4] => 0100 => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,1,5,4,3] => 0100 => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,3,1,4,5] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,3,1,5,4] => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,3,4,1,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[2,3,4,5,1] => 0000 => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,3,5,1,4] => 0000 => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,3,5,4,1] => 0000 => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,4,1,3,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[2,4,1,5,3] => 0000 => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,4,3,1,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[3,1,2,4,5] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,1,4,2,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[3,2,1,4,5] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,2,4,1,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[3,4,1,2,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[3,4,2,1,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[4,1,2,3,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[4,1,3,2,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[4,2,1,3,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[4,2,3,1,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[4,3,1,2,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[4,3,2,1,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,2,3,4,5,6] => 11111 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,2,3,5,4,6] => 11101 => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,2,4,3,5,6] => 11011 => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,2,4,5,3,6] => 11001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,2,5,3,4,6] => 11001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,2,5,4,3,6] => 11001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,3,2,4,5,6] => 10111 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,3,2,5,4,6] => 10101 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,3,4,2,5,6] => 10011 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,3,4,5,2,6] => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,3,5,2,4,6] => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,3,5,4,2,6] => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,4,2,3,5,6] => 10011 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,4,2,5,3,6] => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,4,3,2,5,6] => 10011 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,4,3,5,2,6] => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,4,5,2,3,6] => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
The following 25 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. St000478Another weight of a partition according to Alladi. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000993The multiplicity of the largest part of an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001568The smallest positive integer that does not appear twice in the partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001570The minimal number of edges to add to make a graph Hamiltonian. St001875The number of simple modules with projective dimension at most 1. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.
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