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Your data matches 36 different statistics following compositions of up to 3 maps.
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Matching statistic: St001389
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001389: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 1
[1,2] => [2]
=> 2
[2,1] => [1,1]
=> 1
[1,2,3] => [3]
=> 3
[1,3,2] => [2,1]
=> 2
[2,1,3] => [2,1]
=> 2
[2,3,1] => [2,1]
=> 2
[3,1,2] => [2,1]
=> 2
[3,2,1] => [1,1,1]
=> 1
[1,2,3,4] => [4]
=> 4
[1,2,4,3] => [3,1]
=> 3
[1,3,2,4] => [3,1]
=> 3
[1,3,4,2] => [3,1]
=> 3
[1,4,2,3] => [3,1]
=> 3
[1,4,3,2] => [2,1,1]
=> 2
[2,1,3,4] => [3,1]
=> 3
[2,1,4,3] => [2,2]
=> 3
[2,3,1,4] => [3,1]
=> 3
[2,3,4,1] => [3,1]
=> 3
[2,4,1,3] => [2,2]
=> 3
[2,4,3,1] => [2,1,1]
=> 2
[3,1,2,4] => [3,1]
=> 3
[3,1,4,2] => [2,2]
=> 3
[3,2,1,4] => [2,1,1]
=> 2
[3,2,4,1] => [2,1,1]
=> 2
[3,4,1,2] => [2,2]
=> 3
[3,4,2,1] => [2,1,1]
=> 2
[4,1,2,3] => [3,1]
=> 3
[4,1,3,2] => [2,1,1]
=> 2
[4,2,1,3] => [2,1,1]
=> 2
[4,2,3,1] => [2,1,1]
=> 2
[4,3,1,2] => [2,1,1]
=> 2
[4,3,2,1] => [1,1,1,1]
=> 1
[1,2,3,4,5] => [5]
=> 5
[1,2,3,5,4] => [4,1]
=> 4
[1,2,4,3,5] => [4,1]
=> 4
[1,2,4,5,3] => [4,1]
=> 4
[1,2,5,3,4] => [4,1]
=> 4
[1,2,5,4,3] => [3,1,1]
=> 3
[1,3,2,4,5] => [4,1]
=> 4
[1,3,2,5,4] => [3,2]
=> 5
[1,3,4,2,5] => [4,1]
=> 4
[1,3,4,5,2] => [4,1]
=> 4
[1,3,5,2,4] => [3,2]
=> 5
[1,3,5,4,2] => [3,1,1]
=> 3
[1,4,2,3,5] => [4,1]
=> 4
[1,4,2,5,3] => [3,2]
=> 5
[1,4,3,2,5] => [3,1,1]
=> 3
[1,4,3,5,2] => [3,1,1]
=> 3
[1,4,5,2,3] => [3,2]
=> 5
Description
The number of partitions of the same length below the given integer partition.
For a partition $\lambda_1 \geq \dots \lambda_k > 0$, this number is
$$ \det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$$
Matching statistic: St000108
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000108: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000108: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> []
=> 1
[1,2] => [2]
=> []
=> 1
[2,1] => [1,1]
=> [1]
=> 2
[1,2,3] => [3]
=> []
=> 1
[1,3,2] => [2,1]
=> [1]
=> 2
[2,1,3] => [2,1]
=> [1]
=> 2
[2,3,1] => [2,1]
=> [1]
=> 2
[3,1,2] => [2,1]
=> [1]
=> 2
[3,2,1] => [1,1,1]
=> [1,1]
=> 3
[1,2,3,4] => [4]
=> []
=> 1
[1,2,4,3] => [3,1]
=> [1]
=> 2
[1,3,2,4] => [3,1]
=> [1]
=> 2
[1,3,4,2] => [3,1]
=> [1]
=> 2
[1,4,2,3] => [3,1]
=> [1]
=> 2
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 3
[2,1,3,4] => [3,1]
=> [1]
=> 2
[2,1,4,3] => [2,2]
=> [2]
=> 3
[2,3,1,4] => [3,1]
=> [1]
=> 2
[2,3,4,1] => [3,1]
=> [1]
=> 2
[2,4,1,3] => [2,2]
=> [2]
=> 3
[2,4,3,1] => [2,1,1]
=> [1,1]
=> 3
[3,1,2,4] => [3,1]
=> [1]
=> 2
[3,1,4,2] => [2,2]
=> [2]
=> 3
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 3
[3,2,4,1] => [2,1,1]
=> [1,1]
=> 3
[3,4,1,2] => [2,2]
=> [2]
=> 3
[3,4,2,1] => [2,1,1]
=> [1,1]
=> 3
[4,1,2,3] => [3,1]
=> [1]
=> 2
[4,1,3,2] => [2,1,1]
=> [1,1]
=> 3
[4,2,1,3] => [2,1,1]
=> [1,1]
=> 3
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 3
[4,3,1,2] => [2,1,1]
=> [1,1]
=> 3
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> 4
[1,2,3,4,5] => [5]
=> []
=> 1
[1,2,3,5,4] => [4,1]
=> [1]
=> 2
[1,2,4,3,5] => [4,1]
=> [1]
=> 2
[1,2,4,5,3] => [4,1]
=> [1]
=> 2
[1,2,5,3,4] => [4,1]
=> [1]
=> 2
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 3
[1,3,2,4,5] => [4,1]
=> [1]
=> 2
[1,3,2,5,4] => [3,2]
=> [2]
=> 3
[1,3,4,2,5] => [4,1]
=> [1]
=> 2
[1,3,4,5,2] => [4,1]
=> [1]
=> 2
[1,3,5,2,4] => [3,2]
=> [2]
=> 3
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 3
[1,4,2,3,5] => [4,1]
=> [1]
=> 2
[1,4,2,5,3] => [3,2]
=> [2]
=> 3
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 3
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 3
[1,4,5,2,3] => [3,2]
=> [2]
=> 3
Description
The number of partitions contained in the given partition.
Matching statistic: St000420
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000420: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 100%●distinct values known / distinct values provided: 88%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000420: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 100%●distinct values known / distinct values provided: 88%
Values
[1] => [1]
=> []
=> []
=> ? = 1
[1,2] => [2]
=> []
=> []
=> ? = 1
[2,1] => [1,1]
=> [1]
=> [1,0,1,0]
=> 2
[1,2,3] => [3]
=> []
=> []
=> ? = 1
[1,3,2] => [2,1]
=> [1]
=> [1,0,1,0]
=> 2
[2,1,3] => [2,1]
=> [1]
=> [1,0,1,0]
=> 2
[2,3,1] => [2,1]
=> [1]
=> [1,0,1,0]
=> 2
[3,1,2] => [2,1]
=> [1]
=> [1,0,1,0]
=> 2
[3,2,1] => [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[1,2,3,4] => [4]
=> []
=> []
=> ? = 1
[1,2,4,3] => [3,1]
=> [1]
=> [1,0,1,0]
=> 2
[1,3,2,4] => [3,1]
=> [1]
=> [1,0,1,0]
=> 2
[1,3,4,2] => [3,1]
=> [1]
=> [1,0,1,0]
=> 2
[1,4,2,3] => [3,1]
=> [1]
=> [1,0,1,0]
=> 2
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[2,1,3,4] => [3,1]
=> [1]
=> [1,0,1,0]
=> 2
[2,1,4,3] => [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[2,3,1,4] => [3,1]
=> [1]
=> [1,0,1,0]
=> 2
[2,3,4,1] => [3,1]
=> [1]
=> [1,0,1,0]
=> 2
[2,4,1,3] => [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[3,1,2,4] => [3,1]
=> [1]
=> [1,0,1,0]
=> 2
[3,1,4,2] => [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[3,4,1,2] => [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[4,1,2,3] => [3,1]
=> [1]
=> [1,0,1,0]
=> 2
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 4
[1,2,3,4,5] => [5]
=> []
=> []
=> ? = 1
[1,2,3,5,4] => [4,1]
=> [1]
=> [1,0,1,0]
=> 2
[1,2,4,3,5] => [4,1]
=> [1]
=> [1,0,1,0]
=> 2
[1,2,4,5,3] => [4,1]
=> [1]
=> [1,0,1,0]
=> 2
[1,2,5,3,4] => [4,1]
=> [1]
=> [1,0,1,0]
=> 2
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[1,3,2,4,5] => [4,1]
=> [1]
=> [1,0,1,0]
=> 2
[1,3,2,5,4] => [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[1,3,4,2,5] => [4,1]
=> [1]
=> [1,0,1,0]
=> 2
[1,3,4,5,2] => [4,1]
=> [1]
=> [1,0,1,0]
=> 2
[1,3,5,2,4] => [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[1,4,2,3,5] => [4,1]
=> [1]
=> [1,0,1,0]
=> 2
[1,4,2,5,3] => [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[1,4,5,2,3] => [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[1,5,2,3,4] => [4,1]
=> [1]
=> [1,0,1,0]
=> 2
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[1,5,3,4,2] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[1,2,3,4,5,6] => [6]
=> []
=> []
=> ? = 1
[1,2,3,4,5,6,7] => [7]
=> []
=> []
=> ? = 1
Description
The number of Dyck paths that are weakly above a Dyck path.
Matching statistic: St000777
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 75%
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 75%
Values
[1] => [1] => [1] => ([],1)
=> 1
[1,2] => [2] => [2] => ([],2)
=> ? ∊ {1,2}
[2,1] => [2] => [2] => ([],2)
=> ? ∊ {1,2}
[1,2,3] => [3] => [3] => ([],3)
=> ? ∊ {1,2,2,2,2}
[1,3,2] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 3
[2,1,3] => [3] => [3] => ([],3)
=> ? ∊ {1,2,2,2,2}
[2,3,1] => [3] => [3] => ([],3)
=> ? ∊ {1,2,2,2,2}
[3,1,2] => [3] => [3] => ([],3)
=> ? ∊ {1,2,2,2,2}
[3,2,1] => [2,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,2,2,2,2}
[1,2,3,4] => [4] => [4] => ([],4)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[1,2,4,3] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[1,3,2,4] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,3,4,2] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,4,2,3] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,4,3,2] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,1,3,4] => [4] => [4] => ([],4)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[2,1,4,3] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[2,3,1,4] => [4] => [4] => ([],4)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[2,3,4,1] => [4] => [4] => ([],4)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[2,4,1,3] => [4] => [4] => ([],4)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[2,4,3,1] => [3,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[3,1,2,4] => [4] => [4] => ([],4)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[3,1,4,2] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[3,2,1,4] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[3,2,4,1] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[3,4,1,2] => [4] => [4] => ([],4)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[3,4,2,1] => [3,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[4,1,2,3] => [4] => [4] => ([],4)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[4,1,3,2] => [3,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[4,2,1,3] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[4,2,3,1] => [3,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4}
[4,3,1,2] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[4,3,2,1] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,2,3,4,5] => [5] => [5] => ([],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,2,3,5,4] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,2,4,3,5] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,2,4,5,3] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,2,5,3,4] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,2,5,4,3] => [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,3,2,4,5] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,3,2,5,4] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,3,4,2,5] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,3,4,5,2] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,3,5,2,4] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,3,5,4,2] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,2,3,5] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,4,2,5,3] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,4,3,2,5] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,4,3,5,2] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,4,5,2,3] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,4,5,3,2] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,5,2,3,4] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,5,2,4,3] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,5,3,2,4] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,5,3,4,2] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,5,4,2,3] => [1,1,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,5,4,3,2] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[2,1,3,4,5] => [5] => [5] => ([],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,1,3,5,4] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,1,4,3,5] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,1,4,5,3] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,1,5,3,4] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,1,5,4,3] => [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,3,1,4,5] => [5] => [5] => ([],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,3,1,5,4] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,3,4,1,5] => [5] => [5] => ([],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,3,4,5,1] => [5] => [5] => ([],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,3,5,1,4] => [5] => [5] => ([],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,3,5,4,1] => [4,1] => [1,4] => ([(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,4,1,3,5] => [5] => [5] => ([],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,4,1,5,3] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,4,3,1,5] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,4,3,5,1] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,4,5,1,3] => [5] => [5] => ([],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,4,5,3,1] => [4,1] => [1,4] => ([(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,5,1,3,4] => [5] => [5] => ([],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[4,3,1,2,5] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[4,3,1,5,2] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[4,3,2,1,5] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[4,3,2,5,1] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[4,3,5,1,2] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[4,3,5,2,1] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[5,3,1,2,4] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[5,3,1,4,2] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[5,3,2,1,4] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[5,3,2,4,1] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[5,3,4,1,2] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[5,3,4,2,1] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[5,4,1,2,3] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[5,4,1,3,2] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[5,4,2,1,3] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[5,4,2,3,1] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[5,4,3,1,2] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[5,4,3,2,1] => [1,2,1,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,2,4,5,6] => [1,5] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3
[1,3,2,4,6,5] => [1,3,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,3,2,5,4,6] => [1,2,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,3,2,5,6,4] => [1,2,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,3,2,6,4,5] => [1,2,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,3,2,6,5,4] => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St001232
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 88%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 88%
Values
[1] => [1]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
[1,2] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[2,1] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,2,3] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,3,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[3,1,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[3,2,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[1,2,3,4] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,2,4,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,3,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[2,1,3,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {3,3,3,3} - 1
[2,3,1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,3,4,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,4,1,3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {3,3,3,3} - 1
[2,4,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[3,1,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[3,1,4,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {3,3,3,3} - 1
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[3,4,1,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {3,3,3,3} - 1
[3,4,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[4,1,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[4,1,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[4,2,1,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[4,3,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[4,3,2,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,2,3,4,5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,2,3,5,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,2,4,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,2,4,5,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,2,5,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,3,2,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,3,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[1,3,4,2,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,3,4,5,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,3,5,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,4,2,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,4,2,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[1,4,3,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,4,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[1,4,5,3,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,5,2,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,5,3,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,5,4,2,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,5,4,3,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
[2,1,3,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[2,1,3,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[2,1,4,3,5] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[2,1,4,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[2,1,5,3,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[2,1,5,4,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[2,3,1,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[2,3,5,1,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[2,4,1,3,5] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[2,4,1,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[2,4,5,1,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[2,5,1,3,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[2,5,1,4,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[2,5,4,1,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[3,1,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[3,1,4,2,5] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[3,1,4,5,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[3,1,5,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[3,1,5,4,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[3,2,1,5,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[3,2,5,1,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[3,2,5,4,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[3,4,1,2,5] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[3,4,1,5,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[3,4,5,1,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[3,5,1,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[3,5,1,4,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[3,5,2,1,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[3,5,2,4,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[3,5,4,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[4,1,2,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[4,1,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[4,1,5,3,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[4,2,1,5,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[4,2,5,1,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[4,2,5,3,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[4,3,1,5,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[4,3,5,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[4,5,1,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[4,5,1,3,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[4,5,2,1,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[4,5,2,3,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[4,5,3,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000771
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 75%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 75%
Values
[1] => => [1] => ([],1)
=> 1
[1,2] => 1 => [1,1] => ([(0,1)],2)
=> 1
[2,1] => 0 => [2] => ([],2)
=> ? = 2
[1,2,3] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,3,2] => 10 => [1,2] => ([(1,2)],3)
=> ? ∊ {2,2,2,3}
[2,1,3] => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[2,3,1] => 00 => [3] => ([],3)
=> ? ∊ {2,2,2,3}
[3,1,2] => 00 => [3] => ([],3)
=> ? ∊ {2,2,2,3}
[3,2,1] => 00 => [3] => ([],3)
=> ? ∊ {2,2,2,3}
[1,2,3,4] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,2,4,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,3,2,4] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,3,4,2] => 100 => [1,3] => ([(2,3)],4)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,4,2,3] => 100 => [1,3] => ([(2,3)],4)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,4,3,2] => 100 => [1,3] => ([(2,3)],4)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,1,3,4] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,3,1,4] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,3,4,1] => 000 => [4] => ([],4)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,4,1,3] => 000 => [4] => ([],4)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[2,4,3,1] => 000 => [4] => ([],4)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,1,2,4] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,1,4,2] => 000 => [4] => ([],4)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,2,1,4] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => 000 => [4] => ([],4)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,4,1,2] => 000 => [4] => ([],4)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[3,4,2,1] => 000 => [4] => ([],4)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[4,1,2,3] => 000 => [4] => ([],4)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[4,1,3,2] => 000 => [4] => ([],4)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[4,2,1,3] => 000 => [4] => ([],4)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[4,2,3,1] => 000 => [4] => ([],4)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[4,3,1,2] => 000 => [4] => ([],4)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[4,3,2,1] => 000 => [4] => ([],4)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,2,3,4,5] => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,2,3,5,4] => 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,2,4,3,5] => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,4,5,3] => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,2,5,3,4] => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,2,5,4,3] => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,3,2,4,5] => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,5,4] => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,3,4,2,5] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => 1000 => [1,4] => ([(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,3,5,2,4] => 1000 => [1,4] => ([(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,3,5,4,2] => 1000 => [1,4] => ([(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,4,2,3,5] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,5,3] => 1000 => [1,4] => ([(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,4,3,2,5] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,5,2] => 1000 => [1,4] => ([(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,4,5,2,3] => 1000 => [1,4] => ([(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,4,5,3,2] => 1000 => [1,4] => ([(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,5,2,3,4] => 1000 => [1,4] => ([(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,5,2,4,3] => 1000 => [1,4] => ([(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,5,3,2,4] => 1000 => [1,4] => ([(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,5,3,4,2] => 1000 => [1,4] => ([(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,5,4,2,3] => 1000 => [1,4] => ([(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,5,4,3,2] => 1000 => [1,4] => ([(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,1,3,4,5] => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,3,5,4] => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,1,4,3,5] => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,1,4,5,3] => 0100 => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,1,5,3,4] => 0100 => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,1,5,4,3] => 0100 => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,3,1,4,5] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,3,1,5,4] => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,3,4,1,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[2,3,4,5,1] => 0000 => [5] => ([],5)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,3,5,1,4] => 0000 => [5] => ([],5)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,3,5,4,1] => 0000 => [5] => ([],5)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,4,1,3,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[2,4,1,5,3] => 0000 => [5] => ([],5)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,4,3,1,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[3,1,2,4,5] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,1,4,2,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[3,2,1,4,5] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,2,4,1,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[3,4,1,2,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[3,4,2,1,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[4,1,2,3,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[4,1,3,2,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[4,2,1,3,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[4,2,3,1,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[4,3,1,2,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[4,3,2,1,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,2,3,4,5,6] => 11111 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,2,3,5,4,6] => 11101 => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,2,4,3,5,6] => 11011 => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,2,4,5,3,6] => 11001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,2,5,3,4,6] => 11001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,2,5,4,3,6] => 11001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,3,2,4,5,6] => 10111 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,3,2,5,4,6] => 10101 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,3,4,2,5,6] => 10011 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,3,4,5,2,6] => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,3,5,2,4,6] => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,3,5,4,2,6] => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,4,2,3,5,6] => 10011 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,4,2,5,3,6] => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,4,3,2,5,6] => 10011 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,4,3,5,2,6] => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $2$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St001875
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 25%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 25%
Values
[1] => [1] => ([],1)
=> ([],1)
=> ? = 1
[1,2] => [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,2}
[2,1] => [2,1] => ([],2)
=> ([],1)
=> ? ∊ {1,2}
[1,2,3] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([],1)
=> ? ∊ {1,2,2,2,2,3}
[1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([],1)
=> ? ∊ {1,2,2,2,2,3}
[2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> ([],1)
=> ? ∊ {1,2,2,2,2,3}
[2,3,1] => [2,3,1] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,3}
[3,1,2] => [3,1,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,3}
[3,2,1] => [3,2,1] => ([],3)
=> ([],1)
=> ? ∊ {1,2,2,2,2,3}
[1,2,3,4] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[1,2,4,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[1,3,2,4] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[1,3,4,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[1,4,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[1,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[2,1,3,4] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[2,3,1,4] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[2,3,4,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[2,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[2,4,3,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[3,1,2,4] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[3,1,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[3,2,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[3,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[4,1,2,3] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[4,1,3,2] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[4,2,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[4,2,3,1] => [4,2,3,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[4,3,1,2] => [4,3,1,2] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[4,3,2,1] => [4,3,2,1] => ([],4)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[1,2,3,4,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,2,3,5,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,2,4,3,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,2,4,5,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,2,5,3,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,2,5,4,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,3,2,4,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,3,2,5,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,3,4,2,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,3,4,5,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,3,5,2,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,3,5,4,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,4,2,3,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,4,2,5,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,4,3,2,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,4,3,5,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,4,5,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,4,5,3,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,5,2,3,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,5,2,4,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,5,3,2,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,5,3,4,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[3,4,2,5,1] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[3,5,2,4,1] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[4,2,3,5,1] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[4,2,5,3,1] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[4,5,2,3,1] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[4,5,3,1,2] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[5,2,3,1,4] => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[5,2,4,1,3] => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[5,3,1,2,4] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[5,3,1,4,2] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[5,3,4,1,2] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[3,2,4,5,1,6] => [3,2,6,5,1,4] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[3,2,4,6,1,5] => [3,2,6,5,1,4] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[3,2,5,4,1,6] => [3,2,6,5,1,4] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[3,2,5,6,1,4] => [3,2,6,5,1,4] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[3,2,6,4,1,5] => [3,2,6,5,1,4] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[3,2,6,5,1,4] => [3,2,6,5,1,4] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[3,4,2,1,5,6] => [3,6,2,1,5,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[3,4,2,1,6,5] => [3,6,2,1,5,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[3,4,2,5,1,6] => [3,6,2,5,1,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,4,2,6,1,5] => [3,6,2,5,1,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,5,2,1,4,6] => [3,6,2,1,5,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[3,5,2,1,6,4] => [3,6,2,1,5,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[3,5,2,4,1,6] => [3,6,2,5,1,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,5,2,6,1,4] => [3,6,2,5,1,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,6,2,1,4,5] => [3,6,2,1,5,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[3,6,2,1,5,4] => [3,6,2,1,5,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[3,6,2,4,1,5] => [3,6,2,5,1,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,6,2,5,1,4] => [3,6,2,5,1,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,2,3,5,1,6] => [4,2,6,5,1,3] => ([(0,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[4,2,3,6,1,5] => [4,2,6,5,1,3] => ([(0,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[4,2,5,3,1,6] => [4,2,6,5,1,3] => ([(0,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[4,2,5,6,1,3] => [4,2,6,5,1,3] => ([(0,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[4,2,6,3,1,5] => [4,2,6,5,1,3] => ([(0,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[4,2,6,5,1,3] => [4,2,6,5,1,3] => ([(0,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[4,3,1,2,5,6] => [4,3,1,6,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[4,3,1,2,6,5] => [4,3,1,6,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[4,3,1,5,2,6] => [4,3,1,6,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[4,3,1,5,6,2] => [4,3,1,6,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[4,3,1,6,2,5] => [4,3,1,6,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[4,3,1,6,5,2] => [4,3,1,6,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[4,3,5,1,2,6] => [4,3,6,1,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[4,3,5,1,6,2] => [4,3,6,1,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[4,3,5,6,1,2] => [4,3,6,5,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[4,3,6,1,2,5] => [4,3,6,1,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
Description
The number of simple modules with projective dimension at most 1.
Matching statistic: St000454
(load all 45 compositions to match this statistic)
(load all 45 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 88%
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 88%
Values
[1] => [1] => [1] => ([],1)
=> 0 = 1 - 1
[1,2] => [2] => [2] => ([],2)
=> 0 = 1 - 1
[2,1] => [1,1] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,2,3] => [3] => [3] => ([],3)
=> 0 = 1 - 1
[1,3,2] => [2,1] => [1,2] => ([(1,2)],3)
=> 1 = 2 - 1
[2,1,3] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> ? ∊ {2,2} - 1
[2,3,1] => [2,1] => [1,2] => ([(1,2)],3)
=> 1 = 2 - 1
[3,1,2] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> ? ∊ {2,2} - 1
[3,2,1] => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,2,3,4] => [4] => [4] => ([],4)
=> 0 = 1 - 1
[1,2,4,3] => [3,1] => [1,3] => ([(2,3)],4)
=> 1 = 2 - 1
[1,3,2,4] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[1,3,4,2] => [3,1] => [1,3] => ([(2,3)],4)
=> 1 = 2 - 1
[1,4,2,3] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[1,4,3,2] => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[2,1,3,4] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[2,1,4,3] => [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,3,1,4] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[2,3,4,1] => [3,1] => [1,3] => ([(2,3)],4)
=> 1 = 2 - 1
[2,4,1,3] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[2,4,3,1] => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[3,1,2,4] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[3,1,4,2] => [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,2,1,4] => [1,1,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[3,2,4,1] => [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,4,1,2] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[3,4,2,1] => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[4,1,2,3] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[4,1,3,2] => [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[4,2,1,3] => [1,1,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[4,2,3,1] => [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[4,3,1,2] => [1,1,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[4,3,2,1] => [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,2,3,4,5] => [5] => [5] => ([],5)
=> 0 = 1 - 1
[1,2,3,5,4] => [4,1] => [1,4] => ([(3,4)],5)
=> 1 = 2 - 1
[1,2,4,3,5] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[1,2,4,5,3] => [4,1] => [1,4] => ([(3,4)],5)
=> 1 = 2 - 1
[1,2,5,3,4] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[1,2,5,4,3] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[1,3,2,4,5] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[1,3,2,5,4] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[1,3,4,2,5] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[1,3,4,5,2] => [4,1] => [1,4] => ([(3,4)],5)
=> 1 = 2 - 1
[1,3,5,2,4] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[1,3,5,4,2] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[1,4,2,3,5] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[1,4,2,5,3] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[1,4,3,2,5] => [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[1,4,3,5,2] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[1,4,5,2,3] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[1,4,5,3,2] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[1,5,2,3,4] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[1,5,2,4,3] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[1,5,3,2,4] => [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[1,5,3,4,2] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[1,5,4,2,3] => [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[1,5,4,3,2] => [2,1,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[2,1,3,4,5] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,1,3,5,4] => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,1,4,3,5] => [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[2,1,4,5,3] => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,1,5,3,4] => [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[2,1,5,4,3] => [1,2,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[2,3,1,4,5] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[2,3,1,5,4] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[2,3,4,1,5] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[2,3,4,5,1] => [4,1] => [1,4] => ([(3,4)],5)
=> 1 = 2 - 1
[2,3,5,1,4] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[2,3,5,4,1] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[2,4,1,3,5] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[2,4,1,5,3] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[2,4,3,1,5] => [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[2,4,3,5,1] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[2,4,5,1,3] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[2,4,5,3,1] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
[3,1,2,4,5] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,1,2,5,4] => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,1,4,5,2] => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,2,1,4,5] => [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[3,2,1,5,4] => [1,1,2,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[3,2,4,5,1] => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[4,1,2,3,5] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[4,1,2,5,3] => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[4,1,3,5,2] => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[4,2,1,3,5] => [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[4,2,1,5,3] => [1,1,2,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[4,2,3,5,1] => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[4,3,1,2,5] => [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[4,3,1,5,2] => [1,1,2,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[4,3,2,5,1] => [1,1,2,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[5,1,2,3,4] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[5,1,2,4,3] => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[5,1,3,4,2] => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[5,2,1,3,4] => [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[5,2,1,4,3] => [1,1,2,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[5,2,3,4,1] => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[5,3,1,2,4] => [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[5,3,1,4,2] => [1,1,2,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[5,3,2,4,1] => [1,1,2,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[5,4,1,2,3] => [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001645
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 88%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 88%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 1
[1,2] => [2] => ([],2)
=> ([],1)
=> 1
[2,1] => [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,2,3] => [3] => ([],3)
=> ([],1)
=> 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[2,1,3] => [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {2,2}
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[3,1,2] => [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {2,2}
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,3,4] => [4] => ([],4)
=> ([],1)
=> 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,1,3,4] => [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[3,1,2,4] => [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[4,1,2,3] => [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,2,3,4,5] => [5] => ([],5)
=> ([],1)
=> 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,5,3,2,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,5,3,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,5,4,2,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,1,3,4,5] => [1,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,1,3,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,1,4,3,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,1,4,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,1,5,3,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,1,5,4,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,3,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,3,1,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,3,4,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[2,3,5,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,4,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,4,1,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,4,3,1,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,4,3,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,4,5,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,5,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[2,5,4,3,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[3,5,4,2,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,5,3,2,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[5,4,3,2,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,2,3,4,5,6] => [6] => ([],6)
=> ([],1)
=> 1
[1,2,3,4,6,5] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 2
[1,2,3,5,6,4] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 2
[1,2,3,6,5,4] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,4,5,6,3] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 2
[1,2,4,6,5,3] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,5,6,4,3] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,6,5,4,3] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,3,4,5,6,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 2
[1,3,4,6,5,2] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,3,5,6,4,2] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,3,6,5,4,2] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,4,5,6,3,2] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,4,6,5,3,2] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,5,6,4,3,2] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,6,5,4,3,2] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[2,1,6,5,4,3] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 2
[2,3,4,6,5,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
Description
The pebbling number of a connected graph.
Matching statistic: St000455
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> ([(0,1)],2)
=> -1 = 1 - 2
[1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
[2,1] => ([],2)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,3,1] => ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,3} - 2
[3,1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,3} - 2
[3,2,1] => ([],3)
=> ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[2,4,3,1] => ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1 = 3 - 2
[3,4,2,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[4,1,3,2] => ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[4,2,3,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[4,3,1,2] => ([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4} - 2
[4,3,2,1] => ([],4)
=> ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 3 - 2
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 2
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 2
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 2
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 2
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 2
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 2
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 3 - 2
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 2
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 2
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 2
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 2
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 2
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 3 - 2
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 2
[1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 3 - 2
[1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 2
[1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 2
[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 2
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 2
[1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 2
[2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 2
[2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 2
[2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 2
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 2
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 3 - 2
[2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 2
[2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 2
[2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 2
[2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 2
[2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 2
[2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 2
[2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 2
[2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 3 - 2
[2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 2
[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 2
[2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 2
[2,5,3,1,4] => ([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 3 - 2
[2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} - 2
[3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
[3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 3 - 2
[3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 3 - 2
[3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 3 - 2
[3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 3 - 2
[4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 3 - 2
[4,1,3,5,2] => ([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 3 - 2
[4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 3 - 2
[4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 3 - 2
[4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[4,5,2,3,1] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 1 = 3 - 2
[4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 1 = 3 - 2
[5,3,4,1,2] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 1 = 3 - 2
[5,4,3,2,1] => ([],5)
=> ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,2,3,5,6,4] => ([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 3 - 2
[1,2,3,6,4,5] => ([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 3 - 2
[1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 1 = 3 - 2
[1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,2),(1,4),(1,6),(2,3),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 3 - 2
[1,2,4,6,5,3] => ([(0,5),(4,2),(4,3),(5,1),(5,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 3 - 2
[1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,2),(1,4),(1,6),(2,3),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 3 - 2
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
The following 26 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000264The girth of a graph, which is not a tree. St001060The distinguishing index of a graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001330The hat guessing number of a graph. St001896The number of right descents of a signed permutations. St001863The number of weak excedances of a signed permutation. St001769The reflection length of a signed permutation. St001864The number of excedances of a signed permutation. St000422The energy of a graph, if it is integral. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001427The number of descents of a signed permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000746The number of pairs with odd minimum in a perfect matching. St000942The number of critical left to right maxima of the parking functions. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001712The number of natural descents of a standard Young tableau. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001624The breadth of a lattice. St001626The number of maximal proper sublattices of a lattice.
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