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Your data matches 73 different statistics following compositions of up to 3 maps.
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Matching statistic: St001516
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
St001516: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 2
[2,1] => 2
[1,2,3] => 2
[1,3,2] => 2
[2,1,3] => 2
[2,3,1] => 2
[3,1,2] => 2
[3,2,1] => 2
[1,2,3,4] => 3
[1,2,4,3] => 2
[1,3,2,4] => 1
[1,3,4,2] => 2
[1,4,2,3] => 1
[1,4,3,2] => 3
[2,1,3,4] => 2
[2,1,4,3] => 3
[2,3,1,4] => 1
[2,3,4,1] => 3
[2,4,1,3] => 1
[2,4,3,1] => 2
[3,1,2,4] => 2
[3,1,4,2] => 1
[3,2,1,4] => 3
[3,2,4,1] => 1
[3,4,1,2] => 3
[3,4,2,1] => 2
[4,1,2,3] => 3
[4,1,3,2] => 1
[4,2,1,3] => 2
[4,2,3,1] => 1
[4,3,1,2] => 2
[4,3,2,1] => 3
[1,2,3,4,5] => 4
[1,2,3,5,4] => 3
[1,2,4,3,5] => 2
[1,2,4,5,3] => 2
[1,2,5,3,4] => 2
[1,2,5,4,3] => 3
[1,3,2,4,5] => 2
[1,3,2,5,4] => 2
[1,3,4,2,5] => 1
[1,3,4,5,2] => 3
[1,3,5,2,4] => 0
[1,3,5,4,2] => 2
[1,4,2,3,5] => 1
[1,4,2,5,3] => 0
[1,4,3,2,5] => 2
[1,4,3,5,2] => 2
[1,4,5,2,3] => 2
[1,4,5,3,2] => 3
Description
The number of cyclic bonds of a permutation.
Matching statistic: St000668
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 83%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 83%
Values
[1,2] => [1,1]
=> [1]
=> [1]
=> ? ∊ {2,2}
[2,1] => [2]
=> []
=> []
=> ? ∊ {2,2}
[1,2,3] => [1,1,1]
=> [1,1]
=> [2]
=> 2
[1,3,2] => [2,1]
=> [1]
=> [1]
=> ? ∊ {2,2,2,2,2}
[2,1,3] => [2,1]
=> [1]
=> [1]
=> ? ∊ {2,2,2,2,2}
[2,3,1] => [3]
=> []
=> []
=> ? ∊ {2,2,2,2,2}
[3,1,2] => [3]
=> []
=> []
=> ? ∊ {2,2,2,2,2}
[3,2,1] => [2,1]
=> [1]
=> [1]
=> ? ∊ {2,2,2,2,2}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [2]
=> 2
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [2]
=> 2
[1,3,4,2] => [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3}
[1,4,2,3] => [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [2]
=> 2
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [2]
=> 2
[2,1,4,3] => [2,2]
=> [2]
=> [1,1]
=> 1
[2,3,1,4] => [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3}
[2,3,4,1] => [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3}
[2,4,1,3] => [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3}
[2,4,3,1] => [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3}
[3,1,2,4] => [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3}
[3,1,4,2] => [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [2]
=> 2
[3,2,4,1] => [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3}
[3,4,1,2] => [2,2]
=> [2]
=> [1,1]
=> 1
[3,4,2,1] => [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3}
[4,1,2,3] => [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3}
[4,1,3,2] => [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3}
[4,2,1,3] => [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3}
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [2]
=> 2
[4,3,1,2] => [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3}
[4,3,2,1] => [2,2]
=> [2]
=> [1,1]
=> 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [2,1]
=> 2
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[1,3,4,5,2] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[1,3,5,2,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[1,4,2,5,3] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [2,1]
=> 2
[1,4,5,3,2] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[1,5,2,3,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[1,5,4,2,3] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [2,1]
=> 2
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [2,1]
=> 2
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [2,1]
=> 2
[2,1,4,5,3] => [3,2]
=> [2]
=> [1,1]
=> 1
[2,1,5,3,4] => [3,2]
=> [2]
=> [1,1]
=> 1
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [2,1]
=> 2
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[2,3,1,5,4] => [3,2]
=> [2]
=> [1,1]
=> 1
[2,3,4,1,5] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[2,3,4,5,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[2,3,5,1,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[2,3,5,4,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[2,4,1,3,5] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[2,4,1,5,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[2,4,3,5,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[2,4,5,1,3] => [3,2]
=> [2]
=> [1,1]
=> 1
[2,4,5,3,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[2,5,1,3,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[2,5,1,4,3] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[2,5,3,1,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[2,5,4,1,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[2,5,4,3,1] => [3,2]
=> [2]
=> [1,1]
=> 1
[3,1,2,4,5] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[3,1,2,5,4] => [3,2]
=> [2]
=> [1,1]
=> 1
[3,1,4,2,5] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[3,1,4,5,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[3,1,5,2,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[3,1,5,4,2] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [2,1]
=> 2
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[3,2,4,5,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[3,2,5,1,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[3,2,5,4,1] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[3,4,1,2,5] => [2,2,1]
=> [2,1]
=> [2,1]
=> 2
[3,4,1,5,2] => [3,2]
=> [2]
=> [1,1]
=> 1
[3,4,2,1,5] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[3,4,2,5,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[3,4,5,1,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[3,4,5,2,1] => [3,2]
=> [2]
=> [1,1]
=> 1
[3,5,2,1,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[3,5,2,4,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
Description
The least common multiple of the parts of the partition.
Matching statistic: St000708
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 83%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 83%
Values
[1,2] => [1,1]
=> [1]
=> [1]
=> ? ∊ {2,2}
[2,1] => [2]
=> []
=> []
=> ? ∊ {2,2}
[1,2,3] => [1,1,1]
=> [1,1]
=> [2]
=> 2
[1,3,2] => [2,1]
=> [1]
=> [1]
=> ? ∊ {2,2,2,2,2}
[2,1,3] => [2,1]
=> [1]
=> [1]
=> ? ∊ {2,2,2,2,2}
[2,3,1] => [3]
=> []
=> []
=> ? ∊ {2,2,2,2,2}
[3,1,2] => [3]
=> []
=> []
=> ? ∊ {2,2,2,2,2}
[3,2,1] => [2,1]
=> [1]
=> [1]
=> ? ∊ {2,2,2,2,2}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [2]
=> 2
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [2]
=> 2
[1,3,4,2] => [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3}
[1,4,2,3] => [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [2]
=> 2
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [2]
=> 2
[2,1,4,3] => [2,2]
=> [2]
=> [1,1]
=> 1
[2,3,1,4] => [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3}
[2,3,4,1] => [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3}
[2,4,1,3] => [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3}
[2,4,3,1] => [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3}
[3,1,2,4] => [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3}
[3,1,4,2] => [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [2]
=> 2
[3,2,4,1] => [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3}
[3,4,1,2] => [2,2]
=> [2]
=> [1,1]
=> 1
[3,4,2,1] => [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3}
[4,1,2,3] => [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3}
[4,1,3,2] => [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3}
[4,2,1,3] => [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3}
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [2]
=> 2
[4,3,1,2] => [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3}
[4,3,2,1] => [2,2]
=> [2]
=> [1,1]
=> 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [2,1]
=> 2
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[1,3,4,5,2] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[1,3,5,2,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[1,4,2,5,3] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [2,1]
=> 2
[1,4,5,3,2] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[1,5,2,3,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[1,5,4,2,3] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [2,1]
=> 2
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [2,1]
=> 2
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [2,1]
=> 2
[2,1,4,5,3] => [3,2]
=> [2]
=> [1,1]
=> 1
[2,1,5,3,4] => [3,2]
=> [2]
=> [1,1]
=> 1
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [2,1]
=> 2
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[2,3,1,5,4] => [3,2]
=> [2]
=> [1,1]
=> 1
[2,3,4,1,5] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[2,3,4,5,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[2,3,5,1,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[2,3,5,4,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[2,4,1,3,5] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[2,4,1,5,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[2,4,3,5,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[2,4,5,1,3] => [3,2]
=> [2]
=> [1,1]
=> 1
[2,4,5,3,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[2,5,1,3,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[2,5,1,4,3] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[2,5,3,1,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[2,5,4,1,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[2,5,4,3,1] => [3,2]
=> [2]
=> [1,1]
=> 1
[3,1,2,4,5] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[3,1,2,5,4] => [3,2]
=> [2]
=> [1,1]
=> 1
[3,1,4,2,5] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[3,1,4,5,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[3,1,5,2,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[3,1,5,4,2] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [2,1]
=> 2
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[3,2,4,5,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[3,2,5,1,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[3,2,5,4,1] => [3,1,1]
=> [1,1]
=> [2]
=> 2
[3,4,1,2,5] => [2,2,1]
=> [2,1]
=> [2,1]
=> 2
[3,4,1,5,2] => [3,2]
=> [2]
=> [1,1]
=> 1
[3,4,2,1,5] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[3,4,2,5,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[3,4,5,1,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[3,4,5,2,1] => [3,2]
=> [2]
=> [1,1]
=> 1
[3,5,2,1,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[3,5,2,4,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
Description
The product of the parts of an integer partition.
Matching statistic: St001632
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 83%
Mp00310: Permutations —toric promotion⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 83%
Values
[1,2] => [1,2] => [1,2] => ([(0,1)],2)
=> 1 = 2 - 1
[2,1] => [1,2] => [1,2] => ([(0,1)],2)
=> 1 = 2 - 1
[1,2,3] => [1,2,3] => [3,2,1] => ([],3)
=> ? ∊ {2,2,2,2,2,2} - 1
[1,3,2] => [1,2,3] => [3,2,1] => ([],3)
=> ? ∊ {2,2,2,2,2,2} - 1
[2,1,3] => [1,2,3] => [3,2,1] => ([],3)
=> ? ∊ {2,2,2,2,2,2} - 1
[2,3,1] => [1,2,3] => [3,2,1] => ([],3)
=> ? ∊ {2,2,2,2,2,2} - 1
[3,1,2] => [1,3,2] => [2,3,1] => ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2} - 1
[3,2,1] => [1,3,2] => [2,3,1] => ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2} - 1
[1,2,3,4] => [1,2,3,4] => [4,2,3,1] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[1,2,4,3] => [1,2,3,4] => [4,2,3,1] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[1,3,2,4] => [1,2,3,4] => [4,2,3,1] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[1,3,4,2] => [1,2,3,4] => [4,2,3,1] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[1,4,2,3] => [1,2,4,3] => [4,3,1,2] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[1,4,3,2] => [1,2,4,3] => [4,3,1,2] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[2,1,3,4] => [1,2,3,4] => [4,2,3,1] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[2,1,4,3] => [1,2,3,4] => [4,2,3,1] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[2,3,1,4] => [1,2,3,4] => [4,2,3,1] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[2,3,4,1] => [1,2,3,4] => [4,2,3,1] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[2,4,1,3] => [1,2,4,3] => [4,3,1,2] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[3,1,2,4] => [1,3,2,4] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 0 = 1 - 1
[3,1,4,2] => [1,3,4,2] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[3,2,1,4] => [1,3,2,4] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 0 = 1 - 1
[3,2,4,1] => [1,3,4,2] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[3,4,1,2] => [1,3,2,4] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 0 = 1 - 1
[3,4,2,1] => [1,3,2,4] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 0 = 1 - 1
[4,1,2,3] => [1,4,3,2] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[4,1,3,2] => [1,4,2,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[4,2,1,3] => [1,4,3,2] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[4,2,3,1] => [1,4,2,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[4,3,1,2] => [1,4,2,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[4,3,2,1] => [1,4,2,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4} - 1
[1,2,3,5,4] => [1,2,3,4,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4} - 1
[1,2,4,3,5] => [1,2,3,4,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4} - 1
[1,2,4,5,3] => [1,2,3,4,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4} - 1
[1,2,5,3,4] => [1,2,3,5,4] => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4} - 1
[1,2,5,4,3] => [1,2,3,5,4] => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4} - 1
[1,3,2,4,5] => [1,2,3,4,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4} - 1
[1,3,2,5,4] => [1,2,3,4,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4} - 1
[1,3,4,2,5] => [1,2,3,4,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4} - 1
[1,3,4,5,2] => [1,2,3,4,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4} - 1
[1,3,5,2,4] => [1,2,3,5,4] => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4} - 1
[1,3,5,4,2] => [1,2,3,5,4] => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4} - 1
[1,4,2,3,5] => [1,2,4,3,5] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4} - 1
[1,4,2,5,3] => [1,2,4,5,3] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4} - 1
[1,4,3,2,5] => [1,2,4,3,5] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4} - 1
[1,4,3,5,2] => [1,2,4,5,3] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4} - 1
[1,4,5,2,3] => [1,2,4,3,5] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4} - 1
[1,4,5,3,2] => [1,2,4,3,5] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4} - 1
[1,5,2,3,4] => [1,2,5,4,3] => [5,4,1,3,2] => ([(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4} - 1
[1,5,2,4,3] => [1,2,5,3,4] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4} - 1
[1,5,3,2,4] => [1,2,5,4,3] => [5,4,1,3,2] => ([(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4} - 1
[1,5,3,4,2] => [1,2,5,3,4] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4} - 1
[1,5,4,2,3] => [1,2,5,3,4] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4} - 1
[1,5,4,3,2] => [1,2,5,3,4] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4} - 1
[2,1,3,4,5] => [1,2,3,4,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4} - 1
[2,1,3,5,4] => [1,2,3,4,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4} - 1
[3,1,2,4,5] => [1,3,2,4,5] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> 0 = 1 - 1
[3,1,2,5,4] => [1,3,2,4,5] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> 0 = 1 - 1
[3,1,4,2,5] => [1,3,4,2,5] => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> 1 = 2 - 1
[3,1,5,2,4] => [1,3,5,4,2] => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[3,1,5,4,2] => [1,3,5,2,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> 0 = 1 - 1
[3,2,1,4,5] => [1,3,2,4,5] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> 0 = 1 - 1
[3,2,1,5,4] => [1,3,2,4,5] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> 0 = 1 - 1
[3,2,4,1,5] => [1,3,4,2,5] => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> 1 = 2 - 1
[3,2,5,1,4] => [1,3,5,4,2] => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[3,2,5,4,1] => [1,3,5,2,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> 0 = 1 - 1
[3,4,1,2,5] => [1,3,2,4,5] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> 0 = 1 - 1
[3,4,1,5,2] => [1,3,2,4,5] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> 0 = 1 - 1
[3,4,2,1,5] => [1,3,2,4,5] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> 0 = 1 - 1
[3,4,2,5,1] => [1,3,2,4,5] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> 0 = 1 - 1
[3,4,5,1,2] => [1,3,5,2,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> 0 = 1 - 1
[3,4,5,2,1] => [1,3,5,2,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> 0 = 1 - 1
[3,5,1,2,4] => [1,3,2,5,4] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 0 = 1 - 1
[3,5,1,4,2] => [1,3,2,5,4] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 0 = 1 - 1
[3,5,2,1,4] => [1,3,2,5,4] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 0 = 1 - 1
[3,5,2,4,1] => [1,3,2,5,4] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 0 = 1 - 1
[3,5,4,1,2] => [1,3,4,2,5] => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> 1 = 2 - 1
[3,5,4,2,1] => [1,3,4,2,5] => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> 1 = 2 - 1
[4,1,2,3,5] => [1,4,3,2,5] => [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 1 = 2 - 1
[4,1,2,5,3] => [1,4,5,3,2] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2 = 3 - 1
[4,1,3,2,5] => [1,4,2,3,5] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> 1 = 2 - 1
[4,1,5,2,3] => [1,4,2,3,5] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> 1 = 2 - 1
[4,1,5,3,2] => [1,4,3,5,2] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1 = 2 - 1
[4,2,1,3,5] => [1,4,3,2,5] => [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 1 = 2 - 1
[4,2,1,5,3] => [1,4,5,3,2] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2 = 3 - 1
[4,2,3,1,5] => [1,4,2,3,5] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> 1 = 2 - 1
[4,2,5,1,3] => [1,4,2,3,5] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> 1 = 2 - 1
[4,2,5,3,1] => [1,4,3,5,2] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1 = 2 - 1
[4,3,1,2,5] => [1,4,2,3,5] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> 1 = 2 - 1
[4,3,2,1,5] => [1,4,2,3,5] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> 1 = 2 - 1
[4,3,5,1,2] => [1,4,2,3,5] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> 1 = 2 - 1
[4,3,5,2,1] => [1,4,2,3,5] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> 1 = 2 - 1
[4,5,1,2,3] => [1,4,2,5,3] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 1 - 1
[4,5,1,3,2] => [1,4,3,2,5] => [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 1 = 2 - 1
[4,5,2,1,3] => [1,4,2,5,3] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 1 - 1
[4,5,2,3,1] => [1,4,3,2,5] => [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 1 = 2 - 1
[4,5,3,1,2] => [1,4,2,5,3] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 1 - 1
[4,5,3,2,1] => [1,4,2,5,3] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 1 - 1
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: St000772
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 67%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 67%
Values
[1,2] => [1] => [1] => ([],1)
=> 1 = 2 - 1
[2,1] => [1] => [1] => ([],1)
=> 1 = 2 - 1
[1,2,3] => [1,2] => [2] => ([],2)
=> ? ∊ {2,2,2} - 1
[1,3,2] => [1,2] => [2] => ([],2)
=> ? ∊ {2,2,2} - 1
[2,1,3] => [2,1] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[2,3,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[3,1,2] => [1,2] => [2] => ([],2)
=> ? ∊ {2,2,2} - 1
[3,2,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,2,3,4] => [1,2,3] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3} - 1
[1,2,4,3] => [1,2,3] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3} - 1
[1,3,2,4] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,3,4,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,4,2,3] => [1,2,3] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3} - 1
[1,4,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,1,3,4] => [2,1,3] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3} - 1
[2,1,4,3] => [2,1,3] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3} - 1
[2,3,1,4] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,3,4,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,4,1,3] => [2,1,3] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3} - 1
[2,4,3,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[3,1,2,4] => [3,1,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3} - 1
[3,1,4,2] => [3,1,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3} - 1
[3,2,1,4] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[3,2,4,1] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[3,4,1,2] => [3,1,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3} - 1
[3,4,2,1] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[4,1,2,3] => [1,2,3] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3} - 1
[4,1,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[4,2,1,3] => [2,1,3] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3} - 1
[4,2,3,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[4,3,1,2] => [3,1,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3} - 1
[4,3,2,1] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,2,3,4,5] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4} - 1
[1,2,3,5,4] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4} - 1
[1,2,4,3,5] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,2,4,5,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,2,5,3,4] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4} - 1
[1,2,5,4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,3,2,4,5] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4} - 1
[1,3,2,5,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4} - 1
[1,3,4,2,5] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,3,4,5,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,3,5,2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4} - 1
[1,3,5,4,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,4,2,3,5] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4} - 1
[1,4,2,5,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4} - 1
[1,4,3,2,5] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,4,3,5,2] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,4,5,2,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4} - 1
[1,4,5,3,2] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,5,2,3,4] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4} - 1
[1,5,2,4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,5,3,2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4} - 1
[1,5,3,4,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,5,4,2,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4} - 1
[1,5,4,3,2] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,1,3,4,5] => [2,1,3,4] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4} - 1
[2,1,3,5,4] => [2,1,3,4] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4} - 1
[2,1,4,3,5] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,1,4,5,3] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,1,5,3,4] => [2,1,3,4] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4} - 1
[2,1,5,4,3] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,3,1,4,5] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4} - 1
[2,3,1,5,4] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4} - 1
[2,3,4,1,5] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,3,4,5,1] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,3,5,1,4] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4} - 1
[2,3,5,4,1] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,4,1,3,5] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4} - 1
[2,4,1,5,3] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4} - 1
[2,4,3,1,5] => [2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,4,3,5,1] => [2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,4,5,1,3] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4} - 1
[2,4,5,3,1] => [2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,5,1,3,4] => [2,1,3,4] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4} - 1
[2,5,1,4,3] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,5,3,1,4] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4} - 1
[2,5,3,4,1] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,5,4,1,3] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4} - 1
[2,5,4,3,1] => [2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,1,2,4,5] => [3,1,2,4] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4} - 1
[3,1,2,5,4] => [3,1,2,4] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4} - 1
[3,1,4,2,5] => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,1,4,5,2] => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,1,5,2,4] => [3,1,2,4] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4} - 1
[3,1,5,4,2] => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,2,1,4,5] => [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4} - 1
[3,2,1,5,4] => [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4} - 1
[3,2,4,1,5] => [3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,2,4,5,1] => [3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,2,5,1,4] => [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4} - 1
[3,2,5,4,1] => [3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,4,1,2,5] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4} - 1
[3,4,1,5,2] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4} - 1
[3,4,2,1,5] => [3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,4,2,5,1] => [3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,4,5,1,2] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4} - 1
[3,4,5,2,1] => [3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,5,1,2,4] => [3,1,2,4] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4} - 1
[3,5,2,1,4] => [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4} - 1
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: St000993
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 83%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 83%
Values
[1,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {2,2}
[2,1] => [2,1] => [2]
=> []
=> ? ∊ {2,2}
[1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 2
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {2,2,2,2,2}
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {2,2,2,2,2}
[2,3,1] => [3,2,1] => [3]
=> []
=> ? ∊ {2,2,2,2,2}
[3,1,2] => [3,2,1] => [3]
=> []
=> ? ∊ {2,2,2,2,2}
[3,2,1] => [3,2,1] => [3]
=> []
=> ? ∊ {2,2,2,2,2}
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 3
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 2
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 2
[1,3,4,2] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3}
[1,4,2,3] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3}
[1,4,3,2] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3}
[2,1,3,4] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 2
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 1
[2,3,1,4] => [3,2,1,4] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3}
[2,3,4,1] => [4,2,3,1] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3}
[2,4,1,3] => [3,4,1,2] => [2,1,1]
=> [1,1]
=> 2
[2,4,3,1] => [4,3,2,1] => [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3}
[3,1,2,4] => [3,2,1,4] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3}
[3,1,4,2] => [4,2,3,1] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3}
[3,2,1,4] => [3,2,1,4] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3}
[3,2,4,1] => [4,2,3,1] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3}
[3,4,1,2] => [4,3,2,1] => [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3}
[3,4,2,1] => [4,3,2,1] => [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3}
[4,1,2,3] => [4,2,3,1] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3}
[4,1,3,2] => [4,2,3,1] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3}
[4,2,1,3] => [4,3,2,1] => [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3}
[4,2,3,1] => [4,3,2,1] => [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3}
[4,3,1,2] => [4,3,2,1] => [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3}
[4,3,2,1] => [4,3,2,1] => [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3}
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 4
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,2,4,5,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 2
[1,2,5,3,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 2
[1,2,5,4,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 1
[1,3,4,2,5] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 2
[1,3,4,5,2] => [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 2
[1,3,5,2,4] => [1,4,5,2,3] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,3,5,4,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[1,4,2,3,5] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 2
[1,4,2,5,3] => [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 2
[1,4,3,2,5] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 2
[1,4,3,5,2] => [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 2
[1,4,5,2,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[1,4,5,3,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[1,5,2,3,4] => [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 2
[1,5,2,4,3] => [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 2
[1,5,3,2,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[1,5,3,4,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[1,5,4,2,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[1,5,4,3,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 3
[2,1,3,5,4] => [2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 1
[2,1,4,3,5] => [2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 1
[2,1,4,5,3] => [2,1,5,4,3] => [3,2]
=> [2]
=> 1
[2,1,5,3,4] => [2,1,5,4,3] => [3,2]
=> [2]
=> 1
[2,1,5,4,3] => [2,1,5,4,3] => [3,2]
=> [2]
=> 1
[2,3,1,4,5] => [3,2,1,4,5] => [3,1,1]
=> [1,1]
=> 2
[2,3,1,5,4] => [3,2,1,5,4] => [3,2]
=> [2]
=> 1
[2,3,4,1,5] => [4,2,3,1,5] => [3,1,1]
=> [1,1]
=> 2
[2,3,4,5,1] => [5,2,3,4,1] => [3,1,1]
=> [1,1]
=> 2
[2,3,5,1,4] => [4,2,5,1,3] => [3,1,1]
=> [1,1]
=> 2
[2,3,5,4,1] => [5,2,4,3,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[2,4,1,3,5] => [3,4,1,2,5] => [2,1,1,1]
=> [1,1,1]
=> 3
[2,4,1,5,3] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 2
[2,4,3,1,5] => [4,3,2,1,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[2,4,3,5,1] => [5,3,2,4,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[2,4,5,1,3] => [4,5,3,1,2] => [3,1,1]
=> [1,1]
=> 2
[2,4,5,3,1] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[2,5,1,3,4] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 2
[2,5,1,4,3] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 2
[2,5,3,1,4] => [4,5,3,1,2] => [3,1,1]
=> [1,1]
=> 2
[2,5,3,4,1] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[2,5,4,1,3] => [4,5,3,1,2] => [3,1,1]
=> [1,1]
=> 2
[2,5,4,3,1] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[3,1,2,4,5] => [3,2,1,4,5] => [3,1,1]
=> [1,1]
=> 2
[3,1,2,5,4] => [3,2,1,5,4] => [3,2]
=> [2]
=> 1
[3,1,4,2,5] => [4,2,3,1,5] => [3,1,1]
=> [1,1]
=> 2
[3,1,4,5,2] => [5,2,3,4,1] => [3,1,1]
=> [1,1]
=> 2
[3,1,5,2,4] => [4,2,5,1,3] => [3,1,1]
=> [1,1]
=> 2
[3,1,5,4,2] => [5,2,4,3,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[3,2,1,4,5] => [3,2,1,4,5] => [3,1,1]
=> [1,1]
=> 2
[3,2,1,5,4] => [3,2,1,5,4] => [3,2]
=> [2]
=> 1
[3,2,4,1,5] => [4,2,3,1,5] => [3,1,1]
=> [1,1]
=> 2
[3,2,5,4,1] => [5,2,4,3,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[3,4,1,2,5] => [4,3,2,1,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[3,4,1,5,2] => [5,3,2,4,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[3,4,2,1,5] => [4,3,2,1,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[3,4,2,5,1] => [5,3,2,4,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[3,4,5,1,2] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[3,4,5,2,1] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[3,5,1,4,2] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[3,5,2,4,1] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[3,5,4,1,2] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
[3,5,4,2,1] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St000259
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 50%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 50%
Values
[1,2] => [1,2] => [1,2] => ([],2)
=> ? ∊ {2,2}
[2,1] => [1,2] => [1,2] => ([],2)
=> ? ∊ {2,2}
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {2,2,2,2}
[1,3,2] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {2,2,2,2}
[2,1,3] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {2,2,2,2}
[2,3,1] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {2,2,2,2}
[3,1,2] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[3,2,1] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3}
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3}
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3}
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3}
[1,4,2,3] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3}
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3}
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3}
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3}
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3}
[3,1,4,2] => [1,3,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3}
[3,2,4,1] => [1,3,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[3,4,1,2] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3}
[3,4,2,1] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3}
[4,1,2,3] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3}
[4,2,1,3] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,2,3,1] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3}
[4,3,1,2] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3}
[4,3,2,1] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3}
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[1,2,5,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[1,3,5,2,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[1,4,2,5,3] => [1,2,4,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,4,3,2,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[1,4,3,5,2] => [1,2,4,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,4,5,2,3] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[1,4,5,3,2] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[1,5,2,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,5,2,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[1,5,3,2,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,5,3,4,2] => [1,2,5,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[1,5,4,2,3] => [1,2,5,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[1,5,4,3,2] => [1,2,5,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,1,5,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[2,1,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,3,5,1,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[2,3,5,4,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[2,4,1,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,4,1,5,3] => [1,2,4,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,4,3,1,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,4,3,5,1] => [1,2,4,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,4,5,1,3] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,4,5,3,1] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,5,1,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,5,3,1,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,1,4,5,2] => [1,3,4,5,2] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[3,1,5,2,4] => [1,3,5,4,2] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[3,1,5,4,2] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,2,4,5,1] => [1,3,4,5,2] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[3,2,5,1,4] => [1,3,5,4,2] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[3,2,5,4,1] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,4,5,1,2] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,4,5,2,1] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,5,1,2,4] => [1,3,2,5,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[3,5,1,4,2] => [1,3,2,5,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[3,5,2,1,4] => [1,3,2,5,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[3,5,2,4,1] => [1,3,2,5,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[4,1,2,5,3] => [1,4,5,3,2] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[4,1,3,5,2] => [1,4,5,2,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[4,1,5,3,2] => [1,4,3,5,2] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[4,2,1,5,3] => [1,4,5,3,2] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[4,2,3,5,1] => [1,4,5,2,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[4,2,5,3,1] => [1,4,3,5,2] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[4,3,1,5,2] => [1,4,5,2,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[4,3,2,5,1] => [1,4,5,2,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[5,1,2,3,4] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,1,2,4,3] => [1,5,3,2,4] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,1,4,2,3] => [1,5,3,4,2] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[5,2,1,3,4] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St001060
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 50%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 50%
Values
[1,2] => [1,2] => [1,2] => ([],2)
=> ? ∊ {2,2}
[2,1] => [1,2] => [1,2] => ([],2)
=> ? ∊ {2,2}
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {2,2,2,2,2,2}
[1,3,2] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2}
[2,1,3] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2}
[2,3,1] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {2,2,2,2,2,2}
[3,1,2] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {2,2,2,2,2,2}
[3,2,1] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {2,2,2,2,2,2}
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,3,4,2] => [1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[1,4,2,3] => [1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,4,3,2] => [1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[2,1,3,4] => [1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[2,1,4,3] => [1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[2,3,1,4] => [1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[2,4,1,3] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[3,1,2,4] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[3,1,4,2] => [1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[3,2,1,4] => [1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[3,2,4,1] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[4,1,3,2] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[4,2,1,3] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4}
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4}
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4}
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4}
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4}
[1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4}
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4}
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4}
[1,3,4,2,5] => [1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4}
[1,3,4,5,2] => [1,3,4,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,3,5,2,4] => [1,3,5,2,4] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2] => [1,3,5,2,4] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4}
[1,4,2,5,3] => [1,4,2,5,3] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,2,5] => [1,4,2,5,3] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,5,2] => [1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4}
[1,4,5,2,3] => [1,4,5,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2
[1,4,5,3,2] => [1,4,5,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2
[1,5,2,3,4] => [1,5,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,5,2,4,3] => [1,5,2,4,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,5,3,2,4] => [1,5,2,4,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,5,3,4,2] => [1,5,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,5,4,2,3] => [1,5,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,5,4,3,2] => [1,5,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,3,4,5] => [1,3,4,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 3
[2,1,3,5,4] => [1,3,5,2,4] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,1,4,3,5] => [1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4}
[2,1,4,5,3] => [1,4,5,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2
[2,1,5,3,4] => [1,5,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,5,4,3] => [1,5,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,3,1,4,5] => [1,4,5,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2
[2,3,1,5,4] => [1,5,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,3,4,1,5] => [1,5,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4}
[2,3,5,1,4] => [1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4}
[2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4}
[2,4,1,3,5] => [1,3,5,2,4] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,4,1,5,3] => [1,5,2,4,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,4,3,1,5] => [1,5,2,4,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,4,3,5,1] => [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4}
[2,4,5,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4}
[2,4,5,3,1] => [1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4}
[2,5,1,3,4] => [1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4}
[2,5,1,4,3] => [1,4,2,5,3] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,5,3,1,4] => [1,4,2,5,3] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,5,3,4,1] => [1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4}
[2,5,4,1,3] => [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4}
[2,5,4,3,1] => [1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4}
[3,1,2,4,5] => [1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4}
[3,1,2,5,4] => [1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4}
[3,1,4,2,5] => [1,4,2,5,3] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,1,4,5,2] => [1,4,5,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2
[3,1,5,2,4] => [1,5,2,4,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,1,5,4,2] => [1,5,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,2,1,4,5] => [1,4,5,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2
[3,2,1,5,4] => [1,5,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,2,4,1,5] => [1,5,2,4,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,2,4,5,1] => [1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4}
[3,2,5,1,4] => [1,4,2,5,3] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,2,5,4,1] => [1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4}
[3,4,1,5,2] => [1,5,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,4,2,1,5] => [1,5,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[4,1,3,5,2] => [1,3,5,2,4] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[4,1,5,2,3] => [1,5,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[4,1,5,3,2] => [1,5,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[4,2,1,3,5] => [1,3,5,2,4] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[4,2,1,5,3] => [1,5,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[4,2,3,1,5] => [1,5,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St001200
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 36%●distinct values known / distinct values provided: 33%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 36%●distinct values known / distinct values provided: 33%
Values
[1,2] => 1 => [1] => [1,0]
=> ? ∊ {2,2}
[2,1] => 0 => [1] => [1,0]
=> ? ∊ {2,2}
[1,2,3] => 11 => [2] => [1,1,0,0]
=> ? ∊ {2,2,2,2}
[1,3,2] => 10 => [1,1] => [1,0,1,0]
=> 2
[2,1,3] => 01 => [1,1] => [1,0,1,0]
=> 2
[2,3,1] => 00 => [2] => [1,1,0,0]
=> ? ∊ {2,2,2,2}
[3,1,2] => 00 => [2] => [1,1,0,0]
=> ? ∊ {2,2,2,2}
[3,2,1] => 00 => [2] => [1,1,0,0]
=> ? ∊ {2,2,2,2}
[1,2,3,4] => 111 => [3] => [1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[1,2,4,3] => 110 => [2,1] => [1,1,0,0,1,0]
=> 2
[1,3,2,4] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,3,4,2] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2
[1,4,2,3] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2
[1,4,3,2] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2
[2,1,3,4] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2
[2,1,4,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 3
[2,3,1,4] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2
[2,3,4,1] => 000 => [3] => [1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[2,4,1,3] => 000 => [3] => [1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[2,4,3,1] => 000 => [3] => [1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[3,1,2,4] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2
[3,1,4,2] => 000 => [3] => [1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[3,2,1,4] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2
[3,2,4,1] => 000 => [3] => [1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[3,4,1,2] => 000 => [3] => [1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[3,4,2,1] => 000 => [3] => [1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[4,1,2,3] => 000 => [3] => [1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[4,1,3,2] => 000 => [3] => [1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[4,2,1,3] => 000 => [3] => [1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[4,2,3,1] => 000 => [3] => [1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[4,3,1,2] => 000 => [3] => [1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[4,3,2,1] => 000 => [3] => [1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3}
[1,2,3,4,5] => 1111 => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[1,2,3,5,4] => 1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[1,2,4,3,5] => 1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[1,2,4,5,3] => 1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,2,5,3,4] => 1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,2,5,4,3] => 1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,3,2,4,5] => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[1,3,2,5,4] => 1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3
[1,3,4,2,5] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
[1,3,4,5,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,3,5,2,4] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,3,5,4,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,2,3,5] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
[1,4,2,5,3] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,3,2,5] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
[1,4,3,5,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,5,2,3] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,5,3,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,5,2,3,4] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,5,2,4,3] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,5,3,2,4] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,5,3,4,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,5,4,2,3] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,5,4,3,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[2,1,3,4,5] => 0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[2,1,3,5,4] => 0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
[2,1,4,3,5] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3
[2,1,4,5,3] => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[2,1,5,3,4] => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[2,1,5,4,3] => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[2,3,1,4,5] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,5,4] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[2,3,4,1,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[2,3,4,5,1] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,3,5,1,4] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,3,5,4,1] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,4,1,3,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[2,4,1,5,3] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,4,3,1,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[2,4,3,5,1] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,4,5,1,3] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,4,5,3,1] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,5,1,3,4] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,5,1,4,3] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,5,3,1,4] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,5,3,4,1] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,5,4,1,3] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,5,4,3,1] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[3,1,2,4,5] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,1,2,5,4] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[3,1,4,2,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[3,1,4,5,2] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[3,1,5,2,4] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[3,1,5,4,2] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[3,2,1,4,5] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,2,4,5,1] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[3,2,5,1,4] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[3,2,5,4,1] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[3,4,1,5,2] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[3,4,2,5,1] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[3,4,5,1,2] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[3,4,5,2,1] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[3,5,1,2,4] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[3,5,1,4,2] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[3,5,2,1,4] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[3,5,2,4,1] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[3,5,4,1,2] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[3,5,4,2,1] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
Description
The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001198
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 50%
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 50%
Values
[1,2] => [1,2] => [1,2] => [1,0,1,0]
=> 2
[2,1] => [2,1] => [2,1] => [1,1,0,0]
=> ? = 2
[1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2
[1,3,2] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {2,2,2,2}
[2,1,3] => [2,1,3] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[2,3,1] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {2,2,2,2}
[3,1,2] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {2,2,2,2}
[3,2,1] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {2,2,2,2}
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3}
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2
[1,3,4,2] => [1,4,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3}
[1,4,2,3] => [1,4,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3}
[1,4,3,2] => [1,4,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3}
[2,1,3,4] => [2,1,3,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [3,2,1,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[2,3,4,1] => [4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3}
[2,4,1,3] => [3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[2,4,3,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3}
[3,1,2,4] => [3,2,1,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[3,1,4,2] => [4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3}
[3,2,1,4] => [3,2,1,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[3,2,4,1] => [4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3}
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3}
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3}
[4,1,2,3] => [4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3}
[4,1,3,2] => [4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3}
[4,2,1,3] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3}
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3}
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3}
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3}
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[1,2,4,3,5] => [1,2,4,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,2,4,5,3] => [1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[1,2,5,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,3,2,5,4] => [1,3,2,5,4] => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,3,4,5,2] => [1,5,3,4,2] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[1,3,5,2,4] => [1,4,5,2,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,3,5,4,2] => [1,5,4,3,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,2,5,3] => [1,5,3,4,2] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,3,5,2] => [1,5,3,4,2] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[1,4,5,2,3] => [1,5,4,3,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[1,4,5,3,2] => [1,5,4,3,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[1,5,2,3,4] => [1,5,3,4,2] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[1,5,2,4,3] => [1,5,3,4,2] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[1,5,3,2,4] => [1,5,4,3,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[1,5,3,4,2] => [1,5,4,3,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[1,5,4,2,3] => [1,5,4,3,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,1,3,4,5] => [2,1,3,4,5] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 3
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[2,1,4,3,5] => [2,1,4,3,5] => [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> 2
[2,1,4,5,3] => [2,1,5,4,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,1,5,3,4] => [2,1,5,4,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,1,5,4,3] => [2,1,5,4,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,3,1,4,5] => [3,2,1,4,5] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[2,3,1,5,4] => [3,2,1,5,4] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> 2
[2,3,4,1,5] => [4,2,3,1,5] => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> 2
[2,3,4,5,1] => [5,2,3,4,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,3,5,1,4] => [4,2,5,1,3] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[2,3,5,4,1] => [5,2,4,3,1] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,4,1,3,5] => [3,4,1,2,5] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[2,4,1,5,3] => [3,5,1,4,2] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,4,3,1,5] => [4,3,2,1,5] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[2,4,3,5,1] => [5,3,2,4,1] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,4,5,1,3] => [4,5,3,1,2] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[2,4,5,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,5,1,3,4] => [3,5,1,4,2] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,5,1,4,3] => [3,5,1,4,2] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,5,3,1,4] => [4,5,3,1,2] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[2,5,3,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[2,5,4,1,3] => [4,5,3,1,2] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[2,5,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[3,1,2,4,5] => [3,2,1,4,5] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[3,1,2,5,4] => [3,2,1,5,4] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> 2
[3,1,4,2,5] => [4,2,3,1,5] => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> 2
[3,1,4,5,2] => [5,2,3,4,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4}
[3,1,5,2,4] => [4,2,5,1,3] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[3,2,1,4,5] => [3,2,1,4,5] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[3,2,1,5,4] => [3,2,1,5,4] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> 2
[3,2,4,1,5] => [4,2,3,1,5] => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> 2
[3,2,5,1,4] => [4,2,5,1,3] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[3,4,1,2,5] => [4,3,2,1,5] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[3,4,2,1,5] => [4,3,2,1,5] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[3,5,1,2,4] => [4,5,3,1,2] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[3,5,2,1,4] => [4,5,3,1,2] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[4,1,2,3,5] => [4,2,3,1,5] => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> 2
[4,1,3,2,5] => [4,2,3,1,5] => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> 2
[4,2,1,3,5] => [4,3,2,1,5] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[4,2,3,1,5] => [4,3,2,1,5] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[4,3,1,2,5] => [4,3,2,1,5] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[4,3,2,1,5] => [4,3,2,1,5] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 2
Description
The 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 $eA$.
The following 63 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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 $eA$. St000681The Grundy value of Chomp on Ferrers diagrams. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000937The number of positive values of the symmetric group character corresponding to the partition. St001568The smallest positive integer that does not appear twice in the partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000939The number of characters of the symmetric group whose value on the partition is positive. St000264The girth of a graph, which is not a tree. St001964The interval resolution global dimension of a poset. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000260The radius of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000454The largest eigenvalue of a graph if it is integral. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000455The second largest eigenvalue of a graph if it is integral. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St000022The number of fixed points of a permutation. St000731The number of double exceedences of a permutation. St001330The hat guessing number of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001779The order of promotion on the set of linear extensions of a poset. St000632The jump number of the poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St000068The number of minimal elements in a poset. St001624The breadth of a lattice. St000447The number of pairs of vertices of a graph with distance 3. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001570The minimal number of edges to add to make a graph Hamiltonian.
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