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Your data matches 43 different statistics following compositions of up to 3 maps.
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Matching statistic: St001766
St001766: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 2
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 3
[1,3,2,4] => 2
[1,3,4,2] => 4
[1,4,2,3] => 4
[1,4,3,2] => 5
[2,1,3,4] => 0
[2,1,4,3] => 3
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 2
[2,4,3,1] => 2
[3,1,2,4] => 0
[3,1,4,2] => 2
[3,2,1,4] => 0
[3,2,4,1] => 0
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,3,2] => 2
[4,2,1,3] => 0
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 0
[1,2,3,5,4] => 4
[1,2,4,3,5] => 3
[1,2,4,5,3] => 6
[1,2,5,3,4] => 6
[1,2,5,4,3] => 7
[1,3,2,4,5] => 2
[1,3,2,5,4] => 6
[1,3,4,2,5] => 4
[1,3,4,5,2] => 6
[1,3,5,2,4] => 7
[1,3,5,4,2] => 8
[1,4,2,3,5] => 4
[1,4,2,5,3] => 7
[1,4,3,2,5] => 5
[1,4,3,5,2] => 7
[1,4,5,2,3] => 7
Description
The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation.
Precisely the 132-avoiding permutations have only one associated reduced pipe dream.
Matching statistic: St000422
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00117: Graphs —Ore closure⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 43%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00117: Graphs —Ore closure⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 43%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 0
[1,2] => [2] => ([],2)
=> ([],2)
=> 0
[2,1] => [2] => ([],2)
=> ([],2)
=> 0
[1,2,3] => [3] => ([],3)
=> ([],3)
=> 0
[1,3,2] => [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[2,1,3] => [3] => ([],3)
=> ([],3)
=> 0
[2,3,1] => [3] => ([],3)
=> ([],3)
=> 0
[3,1,2] => [3] => ([],3)
=> ([],3)
=> 0
[3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 0
[1,2,3,4] => [4] => ([],4)
=> ([],4)
=> 0
[1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,3,3,4,4,5}
[1,3,2,4] => [1,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,3,4,2] => [1,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,4,2,3] => [1,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,3,3,4,4,5}
[2,1,3,4] => [4] => ([],4)
=> ([],4)
=> 0
[2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,3,3,4,4,5}
[2,3,1,4] => [4] => ([],4)
=> ([],4)
=> 0
[2,3,4,1] => [4] => ([],4)
=> ([],4)
=> 0
[2,4,1,3] => [4] => ([],4)
=> ([],4)
=> 0
[2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,3,3,4,4,5}
[3,1,2,4] => [4] => ([],4)
=> ([],4)
=> 0
[3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,3,3,4,4,5}
[3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,3,3,4,4,5}
[3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,3,3,4,4,5}
[3,4,1,2] => [4] => ([],4)
=> ([],4)
=> 0
[3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,3,3,4,4,5}
[4,1,2,3] => [4] => ([],4)
=> ([],4)
=> 0
[4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,3,3,4,4,5}
[4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,3,3,4,4,5}
[4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,3,3,4,4,5}
[4,3,1,2] => [1,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> 2
[4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,3,3,4,4,5}
[1,2,3,4,5] => [5] => ([],5)
=> ([],5)
=> 0
[1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> 2
[1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[1,3,4,2,5] => [1,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> 2
[1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> 2
[1,3,5,2,4] => [1,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> 2
[1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[1,4,2,3,5] => [1,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> 2
[1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[1,4,3,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[1,4,5,2,3] => [1,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> 2
[1,4,5,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[1,5,2,3,4] => [1,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> 2
[1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[1,5,3,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[1,5,4,2,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 4
[1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[2,1,3,4,5] => [5] => ([],5)
=> ([],5)
=> 0
[2,1,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[2,1,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[2,1,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[2,1,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[2,1,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[2,3,1,4,5] => [5] => ([],5)
=> ([],5)
=> 0
[2,3,1,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[2,3,4,1,5] => [5] => ([],5)
=> ([],5)
=> 0
[2,3,4,5,1] => [5] => ([],5)
=> ([],5)
=> 0
[2,3,5,1,4] => [5] => ([],5)
=> ([],5)
=> 0
[2,3,5,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,4,1,3,5] => [5] => ([],5)
=> ([],5)
=> 0
[2,4,1,5,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[2,4,3,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[2,4,3,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[2,4,5,1,3] => [5] => ([],5)
=> ([],5)
=> 0
[2,4,5,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,5,1,3,4] => [5] => ([],5)
=> ([],5)
=> 0
[2,5,1,4,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,5,3,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[2,5,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,5,4,1,3] => [2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[2,5,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[3,1,2,4,5] => [5] => ([],5)
=> ([],5)
=> 0
[3,1,2,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[3,1,4,2,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[3,1,4,5,2] => [2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[3,1,5,2,4] => [2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[3,1,5,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[3,2,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[3,2,1,5,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[3,2,4,1,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[3,2,4,5,1] => [2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[3,2,5,1,4] => [2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[3,2,5,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9}
[3,4,1,2,5] => [5] => ([],5)
=> ([],5)
=> 0
[3,4,5,1,2] => [5] => ([],5)
=> ([],5)
=> 0
[3,4,5,2,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[3,5,1,2,4] => [5] => ([],5)
=> ([],5)
=> 0
[3,5,1,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[3,5,2,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[4,1,2,3,5] => [5] => ([],5)
=> ([],5)
=> 0
Description
The energy of a graph, if it is integral.
The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3].
The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.
Matching statistic: St001645
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00264: Graphs —delete endpoints⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 36%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00264: Graphs —delete endpoints⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 36%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,2] => [2] => ([],2)
=> ([],2)
=> ? = 0 + 1
[2,1] => [1,1] => ([(0,1)],2)
=> ([],1)
=> 1 = 0 + 1
[1,2,3] => [3] => ([],3)
=> ([],3)
=> ? ∊ {0,0,0} + 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> ([],1)
=> 1 = 0 + 1
[2,1,3] => [1,2] => ([(1,2)],3)
=> ([],2)
=> ? ∊ {0,0,0} + 1
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> ([],1)
=> 1 = 0 + 1
[3,1,2] => [1,2] => ([(1,2)],3)
=> ([],2)
=> ? ∊ {0,0,0} + 1
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,2,3,4] => [4] => ([],4)
=> ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,3,4,4,5} + 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> 1 = 0 + 1
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,3,4,4,5} + 1
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> 1 = 0 + 1
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,3,4,4,5} + 1
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,3,4,4,5} + 1
[2,1,3,4] => [1,3] => ([(2,3)],4)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,3,4,4,5} + 1
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,3,4,4,5} + 1
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> 1 = 0 + 1
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,3,4,4,5} + 1
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,3,4,4,5} + 1
[3,1,2,4] => [1,3] => ([(2,3)],4)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,3,4,4,5} + 1
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,3,4,4,5} + 1
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,3,4,4,5} + 1
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,3,4,4,5} + 1
[4,1,2,3] => [1,3] => ([(2,3)],4)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,3,4,4,5} + 1
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,3,4,4,5} + 1
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,3,4,4,5} + 1
[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 = 3 + 1
[1,2,3,4,5] => [5] => ([],5)
=> ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> 1 = 0 + 1
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> 1 = 0 + 1
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> 1 = 0 + 1
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,5,3,2,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,5,3,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,5,4,2,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[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)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[2,1,3,4,5] => [1,4] => ([(3,4)],5)
=> ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[2,1,3,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,1,4,3,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[2,1,4,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,1,5,3,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[2,3,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[2,3,1,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[2,3,4,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> 1 = 0 + 1
[2,3,5,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[2,4,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[2,4,1,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[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)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,1,2,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,1,4,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,2,1,5,4] => [1,1,2,1] => ([(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 + 1
[3,2,4,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[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)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[4,1,2,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,1,3,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,2,1,5,3] => [1,1,2,1] => ([(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 + 1
[4,2,3,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,3,1,5,2] => [1,1,2,1] => ([(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 + 1
[4,3,2,5,1] => [1,1,2,1] => ([(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 + 1
[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)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[5,1,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[5,1,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[5,2,1,4,3] => [1,1,2,1] => ([(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 + 1
[5,2,3,4,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[5,3,1,4,2] => [1,1,2,1] => ([(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 + 1
[5,3,2,4,1] => [1,1,2,1] => ([(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 + 1
[5,4,1,3,2] => [1,1,2,1] => ([(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 + 1
[5,4,2,3,1] => [1,1,2,1] => ([(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 + 1
[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 = 4 + 1
[1,2,3,4,6,5] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> 1 = 0 + 1
[1,2,3,5,6,4] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> 1 = 0 + 1
[1,2,4,5,6,3] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> 1 = 0 + 1
[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)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,3,4,5,6,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> 1 = 0 + 1
[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)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,4,3,2,6,5] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
Description
The pebbling number of a connected graph.
Matching statistic: St000454
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 21%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 21%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,2] => [1,2] => [1,2] => ([],2)
=> 0
[2,1] => [1,2] => [1,2] => ([],2)
=> 0
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
=> 0
[1,3,2] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {0,2}
[2,1,3] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {0,2}
[2,3,1] => [1,2,3] => [1,2,3] => ([],3)
=> 0
[3,1,2] => [1,2,3] => [1,2,3] => ([],3)
=> 0
[3,2,1] => [1,2,3] => [1,2,3] => ([],3)
=> 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[1,3,4,2] => [1,3,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[1,4,3,2] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[2,1,3,4] => [1,3,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,1,4,3] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[2,4,1,3] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[3,1,2,4] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[3,1,4,2] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[3,2,1,4] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[3,2,4,1] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[4,1,3,2] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[4,2,1,3] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,2,4,5,3] => [1,2,4,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,2,5,3,4] => [1,2,5,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,2,5,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,3,2,5,4] => [1,3,2,5,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,3,4,2,5] => [1,3,4,2,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,3,4,5,2] => [1,3,4,5,2] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,3,5,4,2] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,4,2,5,3] => [1,4,2,5,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,4,3,2,5] => [1,4,2,5,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,4,3,5,2] => [1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,4,5,2,3] => [1,4,5,2,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,4,5,3,2] => [1,4,5,2,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,5,2,3,4] => [1,5,2,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,5,2,4,3] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,5,3,2,4] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,5,3,4,2] => [1,5,2,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,5,4,2,3] => [1,5,2,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,5,4,3,2] => [1,5,2,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[2,1,3,4,5] => [1,3,4,5,2] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[2,1,3,5,4] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[2,1,4,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[2,1,4,5,3] => [1,4,5,2,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[2,1,5,3,4] => [1,5,2,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[2,1,5,4,3] => [1,5,2,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[2,3,1,4,5] => [1,4,5,2,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[2,3,1,5,4] => [1,5,2,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[2,3,4,1,5] => [1,5,2,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[2,3,5,1,4] => [1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[2,3,5,4,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[2,4,1,3,5] => [1,3,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[2,4,1,5,3] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[2,4,3,1,5] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[2,4,3,5,1] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[2,4,5,1,3] => [1,3,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[2,5,1,3,4] => [1,3,4,2,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,5,1,4,3] => [1,4,2,5,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,5,3,1,4] => [1,4,2,5,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[3,1,4,2,5] => [1,4,2,5,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[3,2,5,1,4] => [1,4,2,5,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[3,5,4,1,2] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[3,5,4,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[4,1,2,3,5] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[4,2,3,5,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[4,3,5,1,2] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[4,3,5,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[4,5,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[5,1,3,4,2] => [1,3,4,2,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[5,2,1,3,4] => [1,3,4,2,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
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: St000456
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 50%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1] => ([],1)
=> ? = 0
[1,2] => [1,2] => [2] => ([],2)
=> ? ∊ {0,0}
[2,1] => [1,2] => [2] => ([],2)
=> ? ∊ {0,0}
[1,2,3] => [1,2,3] => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,2}
[1,3,2] => [1,2,3] => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,2}
[2,1,3] => [1,2,3] => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,2}
[2,3,1] => [1,2,3] => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,2}
[3,1,2] => [1,3,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,2}
[3,2,1] => [1,3,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,2}
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[1,2,4,3] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[1,3,2,4] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[1,3,4,2] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[1,4,2,3] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[1,4,3,2] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[2,1,3,4] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[2,1,4,3] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[2,3,1,4] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[2,3,4,1] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[2,4,1,3] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[2,4,3,1] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[3,1,2,4] => [1,3,2,4] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[3,1,4,2] => [1,3,4,2] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[3,2,1,4] => [1,3,2,4] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[3,2,4,1] => [1,3,4,2] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[3,4,1,2] => [1,3,2,4] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[3,4,2,1] => [1,3,2,4] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[4,1,2,3] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,3,2] => [1,4,2,3] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[4,2,1,3] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,2,3,1] => [1,4,2,3] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[4,3,1,2] => [1,4,2,3] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[4,3,2,1] => [1,4,2,3] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,3,3,4,4,5}
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,2,3,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,2,4,3,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,2,4,5,3] => [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,2,5,3,4] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,2,5,4,3] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,3,2,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,3,2,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,3,4,2,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,3,4,5,2] => [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,3,5,2,4] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,3,5,4,2] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,4,2,3,5] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,4,2,5,3] => [1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,4,3,2,5] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,4,3,5,2] => [1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,4,5,2,3] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,4,5,3,2] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,5,2,3,4] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,5,2,4,3] => [1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,5,3,2,4] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,5,1,3,4] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,5,3,1,4] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,1,5,2,4] => [1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,2,5,1,4] => [1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[4,1,2,5,3] => [1,4,5,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[4,2,1,5,3] => [1,4,5,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,1,2,3,4] => [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[5,1,4,2,3] => [1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,2,1,3,4] => [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[5,2,4,1,3] => [1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,4,1,3,2] => [1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,4,2,3,1] => [1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,4,3,1,2] => [1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,4,3,2,1] => [1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,6,3,4,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,2,6,4,3,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,3,6,2,4,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,3,6,4,2,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,4,2,6,3,5] => [1,2,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,4,3,6,2,5] => [1,2,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,5,2,3,6,4] => [1,2,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,5,3,2,6,4] => [1,2,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,6,2,3,4,5] => [1,2,6,5,4,3] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,6,2,5,3,4] => [1,2,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,6,3,2,4,5] => [1,2,6,5,4,3] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,6,3,5,2,4] => [1,2,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,6,5,2,4,3] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,6,5,3,4,2] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,6,5,4,2,3] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,6,5,4,3,2] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,1,6,3,4,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[2,1,6,4,3,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[2,3,6,1,4,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[2,3,6,4,1,5] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[2,4,1,6,3,5] => [1,2,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,4,3,6,1,5] => [1,2,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,5,1,3,6,4] => [1,2,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,5,3,1,6,4] => [1,2,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,6,1,3,4,5] => [1,2,6,5,4,3] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[2,6,1,5,3,4] => [1,2,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,6,3,1,4,5] => [1,2,6,5,4,3] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[2,6,3,5,1,4] => [1,2,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,6,5,1,4,3] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,6,5,3,4,1] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,6,5,4,1,3] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,6,5,4,3,1] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
Description
The monochromatic index of a connected graph.
This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path.
For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St001232
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 64%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 64%
Values
[1] => [1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,2] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[2,1] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 0 + 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0} + 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0} + 1
[2,1,3] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0} + 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0} + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,5} + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,5} + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5 = 4 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,5} + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,5} + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,5} + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,5} + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,5} + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,5} + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,5} + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,5} + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,5} + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,5} + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,5} + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,5} + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,5} + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,5} + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,5} + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,4,5} + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 7 = 6 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 7 = 6 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 5 = 4 + 1
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9} + 1
[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 7 = 6 + 1
[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 7 = 6 + 1
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[1,3,2,5,6,4] => [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> 7 = 6 + 1
[1,3,5,6,2,4] => [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,3,5,6,4,2] => [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> 9 = 8 + 1
[1,5,2,3,6,4] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[1,5,2,4,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> 9 = 8 + 1
[1,5,2,4,6,3] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[1,5,3,2,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> 9 = 8 + 1
[1,5,3,2,6,4] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> 9 = 8 + 1
[1,5,3,4,6,2] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[1,5,4,2,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> 9 = 8 + 1
[1,5,4,2,6,3] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> 9 = 8 + 1
[1,5,4,3,6,2] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[2,1,5,3,4,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> 7 = 6 + 1
[2,1,5,3,6,4] => [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[2,1,5,4,3,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> 7 = 6 + 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000455
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 15%●distinct values known / distinct values provided: 7%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 15%●distinct values known / distinct values provided: 7%
Values
[1] => [1] => [1] => ([],1)
=> ? = 0
[1,2] => [1,2] => [2] => ([],2)
=> ? ∊ {0,0}
[2,1] => [2,1] => [2] => ([],2)
=> ? ∊ {0,0}
[1,2,3] => [1,2,3] => [3] => ([],3)
=> ? ∊ {0,2}
[1,3,2] => [1,3,2] => [1,2] => ([(1,2)],3)
=> 0
[2,1,3] => [2,1,3] => [3] => ([],3)
=> ? ∊ {0,2}
[2,3,1] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 0
[3,1,2] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 0
[3,2,1] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 0
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,4,4,5}
[1,2,4,3] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[1,3,2,4] => [1,3,2,4] => [1,3] => ([(2,3)],4)
=> 0
[1,3,4,2] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,4,4,5}
[1,4,2,3] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,4,4,5}
[1,4,3,2] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,4,4,5}
[2,1,3,4] => [2,1,3,4] => [4] => ([],4)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,4,4,5}
[2,1,4,3] => [2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[2,3,1,4] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[2,3,4,1] => [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[2,4,1,3] => [3,4,1,2] => [4] => ([],4)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,4,4,5}
[2,4,3,1] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,4,4,5}
[3,1,2,4] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[3,1,4,2] => [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[3,2,1,4] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[3,2,4,1] => [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[3,4,1,2] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,4,4,5}
[3,4,2,1] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,4,4,5}
[4,1,2,3] => [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[4,1,3,2] => [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[4,2,1,3] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,4,4,5}
[4,2,3,1] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,4,4,5}
[4,3,1,2] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,4,4,5}
[4,3,2,1] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,4,4,5}
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,2,3,5,4] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,2,4,3,5] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[1,2,4,5,3] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,2,5,3,4] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,2,5,4,3] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,3,2,4,5] => [1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> 0
[1,3,2,5,4] => [1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,3,4,2,5] => [1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,3,4,5,2] => [1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,3,5,2,4] => [1,4,5,2,3] => [1,4] => ([(3,4)],5)
=> 0
[1,3,5,4,2] => [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,4,2,3,5] => [1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,4,2,5,3] => [1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,4,3,2,5] => [1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,4,3,5,2] => [1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,4,5,2,3] => [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,4,5,3,2] => [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,5,2,3,4] => [1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,5,2,4,3] => [1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,5,3,2,4] => [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,5,3,4,2] => [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,5,4,2,3] => [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,5,4,3,2] => [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[2,1,3,4,5] => [2,1,3,4,5] => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[2,1,3,5,4] => [2,1,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[2,1,4,3,5] => [2,1,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[2,1,4,5,3] => [2,1,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[2,1,5,3,4] => [2,1,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[2,1,5,4,3] => [2,1,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[2,3,1,4,5] => [3,2,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[2,3,1,5,4] => [3,2,1,5,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[2,3,4,1,5] => [4,2,3,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[2,3,4,5,1] => [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[2,3,5,1,4] => [4,2,5,1,3] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[2,3,5,4,1] => [5,2,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[2,4,1,3,5] => [3,4,1,2,5] => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[2,4,1,5,3] => [3,5,1,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[2,4,3,1,5] => [4,3,2,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[2,4,3,5,1] => [5,3,2,4,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[2,4,5,1,3] => [4,5,3,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[2,4,5,3,1] => [5,4,3,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[2,5,1,3,4] => [3,5,1,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[2,5,1,4,3] => [3,5,1,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[2,5,3,1,4] => [4,5,3,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[2,5,3,4,1] => [5,4,3,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[2,5,4,1,3] => [4,5,3,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[2,5,4,3,1] => [5,4,3,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[3,1,2,4,5] => [3,2,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[3,1,2,5,4] => [3,2,1,5,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[3,1,4,2,5] => [4,2,3,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[3,1,4,5,2] => [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[3,1,5,2,4] => [4,2,5,1,3] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[3,1,5,4,2] => [5,2,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[3,2,1,4,5] => [3,2,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[3,2,1,5,4] => [3,2,1,5,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[3,2,4,1,5] => [4,2,3,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[3,2,4,5,1] => [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[3,2,5,1,4] => [4,2,5,1,3] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[3,5,1,2,4] => [4,5,3,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[3,5,2,1,4] => [4,5,3,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[4,1,2,3,5] => [4,2,3,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[4,1,2,5,3] => [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[4,1,3,2,5] => [4,2,3,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[4,1,3,5,2] => [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[5,1,2,3,4] => [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[5,1,2,4,3] => [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
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.
Matching statistic: St000936
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000936: Integer partitions ⟶ ℤResult quality: 7% ●values known / values provided: 15%●distinct values known / distinct values provided: 7%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000936: Integer partitions ⟶ ℤResult quality: 7% ●values known / values provided: 15%●distinct values known / distinct values provided: 7%
Values
[1] => [1]
=> []
=> ?
=> ? = 0
[1,2] => [1,1]
=> [1]
=> []
=> ? ∊ {0,0}
[2,1] => [2]
=> []
=> ?
=> ? ∊ {0,0}
[1,2,3] => [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,2}
[1,3,2] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,2}
[2,1,3] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,2}
[2,3,1] => [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,2}
[3,1,2] => [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,2}
[3,2,1] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,2}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[2,3,4,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[2,4,1,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[2,4,3,1] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[3,1,4,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[3,2,4,1] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[3,4,2,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[4,1,2,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[4,1,3,2] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[4,2,1,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[4,3,1,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[4,3,2,1] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,3,5,2,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,4,2,5,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,4,5,3,2] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,4,2,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,2,3,1,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,2,3,4,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,5,4,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,5,6,4] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,6,4,5] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,6,5,4] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3,6,5] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,2,4,5,3,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,4,6,5,3] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,5,3,4,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,5,4,3,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,5,4,6,3] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,5,6,3,4] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,2,6,3,5,4] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,6,4,3,5] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,6,4,5,3] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,6,5,4,3] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,3,2,4,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,3,2,4,6,5] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,3,2,5,4,6] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,3,2,6,5,4] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,3,4,2,5,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,5,4,2,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,6,4,5,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,2,3,5,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,3,2,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,4,3,2,6,5] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,4,3,5,2,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,3,6,5,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,5,2,3,6] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,4,6,2,5,3] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,5,2,4,3,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,3,2,4,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,3,4,2,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,5,3,4,6,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,3,6,2,4] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,5,4,3,2,6] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
Description
The number of even values of the symmetric group character corresponding to the partition.
For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugace class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $4$.
It is shown in [1] that the sum of the values of the statistic over all partitions of a given size is even.
Matching statistic: St000938
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000938: Integer partitions ⟶ ℤResult quality: 7% ●values known / values provided: 15%●distinct values known / distinct values provided: 7%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000938: Integer partitions ⟶ ℤResult quality: 7% ●values known / values provided: 15%●distinct values known / distinct values provided: 7%
Values
[1] => [1]
=> []
=> ?
=> ? = 0
[1,2] => [1,1]
=> [1]
=> []
=> ? ∊ {0,0}
[2,1] => [2]
=> []
=> ?
=> ? ∊ {0,0}
[1,2,3] => [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,2}
[1,3,2] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,2}
[2,1,3] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,2}
[2,3,1] => [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,2}
[3,1,2] => [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,2}
[3,2,1] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,2}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[2,3,4,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[2,4,1,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[2,4,3,1] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[3,1,4,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[3,2,4,1] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[3,4,2,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[4,1,2,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[4,1,3,2] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[4,2,1,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[4,3,1,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[4,3,2,1] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,3,5,2,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,4,2,5,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,4,5,3,2] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,4,2,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,2,3,1,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,2,3,4,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,5,4,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,5,6,4] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,6,4,5] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,6,5,4] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3,6,5] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,2,4,5,3,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,4,6,5,3] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,5,3,4,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,5,4,3,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,5,4,6,3] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,5,6,3,4] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,2,6,3,5,4] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,6,4,3,5] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,6,4,5,3] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,6,5,4,3] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,3,2,4,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,3,2,4,6,5] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,3,2,5,4,6] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,3,2,6,5,4] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,3,4,2,5,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,5,4,2,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,6,4,5,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,2,3,5,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,3,2,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,4,3,2,6,5] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,4,3,5,2,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,3,6,5,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,5,2,3,6] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,4,6,2,5,3] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,5,2,4,3,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,3,2,4,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,3,4,2,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,5,3,4,6,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,3,6,2,4] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,5,4,3,2,6] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
Description
The number of zeros of the symmetric group character corresponding to the partition.
For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugacy class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $2$.
Matching statistic: St000940
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000940: Integer partitions ⟶ ℤResult quality: 7% ●values known / values provided: 15%●distinct values known / distinct values provided: 7%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000940: Integer partitions ⟶ ℤResult quality: 7% ●values known / values provided: 15%●distinct values known / distinct values provided: 7%
Values
[1] => [1]
=> []
=> ?
=> ? = 0
[1,2] => [1,1]
=> [1]
=> []
=> ? ∊ {0,0}
[2,1] => [2]
=> []
=> ?
=> ? ∊ {0,0}
[1,2,3] => [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,2}
[1,3,2] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,2}
[2,1,3] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,2}
[2,3,1] => [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,2}
[3,1,2] => [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,2}
[3,2,1] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,2}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[2,3,4,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[2,4,1,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[2,4,3,1] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[3,1,4,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[3,2,4,1] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[3,4,2,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[4,1,2,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[4,1,3,2] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[4,2,1,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[4,3,1,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[4,3,2,1] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,5}
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,3,5,2,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,4,2,5,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,4,5,3,2] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,4,2,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9}
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,2,3,1,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,2,3,4,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,5,4,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,5,6,4] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,6,4,5] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,6,5,4] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3,6,5] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,2,4,5,3,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,4,6,5,3] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,5,3,4,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,5,4,3,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,5,4,6,3] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,5,6,3,4] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,2,6,3,5,4] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,6,4,3,5] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,6,4,5,3] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,6,5,4,3] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,3,2,4,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,3,2,4,6,5] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,3,2,5,4,6] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,3,2,6,5,4] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,3,4,2,5,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,5,4,2,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,6,4,5,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,2,3,5,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,3,2,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,4,3,2,6,5] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,4,3,5,2,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,3,6,5,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,4,5,2,3,6] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,4,6,2,5,3] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,5,2,4,3,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,3,2,4,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,3,4,2,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,5,3,4,6,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,5,3,6,2,4] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,5,4,3,2,6] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
Description
The number of characters of the symmetric group whose value on the partition is zero.
The maximal value for any given size is recorded in [2].
The following 33 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001964The interval resolution global dimension of a poset. St001877Number of indecomposable injective modules with projective dimension 2. 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. St001875The number of simple modules with projective dimension at most 1. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001330The hat guessing number of a graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. 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. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001902The number of potential covers of a poset. St000181The number of connected components of the Hasse diagram for the poset. St000908The length of the shortest maximal antichain in a poset. St001472The permanent of the Coxeter matrix of the poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St000264The girth of a graph, which is not a tree. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001058The breadth of the ordered tree.
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