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Your data matches 35 different statistics following compositions of up to 3 maps.
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Matching statistic: St001809
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
St001809: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1
[1,0,1,0]
=> 1
[1,1,0,0]
=> 2
[1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> 4
[1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> 3
[1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> 6
[1,0,1,1,0,0,1,0]
=> 4
[1,0,1,1,0,1,0,0]
=> 4
[1,0,1,1,1,0,0,0]
=> 5
[1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> 5
[1,1,1,0,0,0,1,0]
=> 3
[1,1,1,0,0,1,0,0]
=> 3
[1,1,1,0,1,0,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> 8
[1,0,1,0,1,1,0,0,1,0]
=> 6
[1,0,1,0,1,1,0,1,0,0]
=> 6
[1,0,1,0,1,1,1,0,0,0]
=> 7
[1,0,1,1,0,0,1,0,1,0]
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> 7
[1,0,1,1,1,0,0,0,1,0]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> 5
[1,0,1,1,1,0,1,0,0,0]
=> 5
[1,0,1,1,1,1,0,0,0,0]
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> 7
[1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> 7
[1,1,0,1,1,0,0,0,1,0]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> 5
[1,1,0,1,1,0,1,0,0,0]
=> 5
[1,1,0,1,1,1,0,0,0,0]
=> 6
Description
The index of the step at the first peak of maximal height in a Dyck path.
Matching statistic: St000454
(load all 82 compositions to match this statistic)
(load all 82 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00117: Graphs —Ore closure⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 58%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00117: Graphs —Ore closure⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 58%
Values
[1,0]
=> [1] => ([],1)
=> ([],1)
=> 0 = 1 - 1
[1,0,1,0]
=> [1,2] => ([],2)
=> ([],2)
=> 0 = 1 - 1
[1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> ([],3)
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 4 - 1
[1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> ([],4)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {3,4,5,5,6} - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {3,4,5,5,6} - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {3,4,5,5,6} - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,4,5,5,6} - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,4,5,5,6} - 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> ([],5)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {2,3,3,4,4,4,5,5,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {2,3,3,4,4,4,5,5,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,3,3,4,4,4,5,5,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,3,4,4,4,5,5,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,3,4,4,4,5,5,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {2,3,3,4,4,4,5,5,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {2,3,3,4,4,4,5,5,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {2,3,3,4,4,4,5,5,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,3,3,4,4,4,5,5,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,3,4,4,4,5,5,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,3,4,4,4,5,5,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,3,4,4,4,5,5,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => ([(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)
=> ? ∊ {2,3,3,4,4,4,5,5,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => ([(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)
=> ? ∊ {2,3,3,4,4,4,5,5,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,3,4,4,4,5,5,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,3,4,4,4,5,5,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,3,3,4,4,4,5,5,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(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)
=> ? ∊ {2,3,3,4,4,4,5,5,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => ([(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)
=> 3 = 4 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => ([(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)
=> 4 = 5 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => ([(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)
=> 3 = 4 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => ([(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)
=> ? ∊ {2,3,3,4,4,4,5,5,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => ([(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)
=> 4 = 5 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => ([(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)
=> 4 = 5 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => ([(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)
=> 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => ([],6)
=> ([],6)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => ([(4,5)],6)
=> ([(4,5)],6)
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => ([(4,5)],6)
=> ([(4,5)],6)
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(4,5)],6)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,5,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,6,4,5,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => ([(4,5)],6)
=> ([(4,5)],6)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,3,2,5,6,4] => ([(1,2),(3,5),(4,5)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,4,2,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,2,6] => ([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,4,6,5,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,5,4,2,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,5,4,6,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,3,6,4,5,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,6,5,4,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,3,5,2,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,3,5,6,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,4,3,6,5,2] => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,5,3,4,6,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,4,3,6,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,3,5,6,4] => ([(1,2),(3,5),(4,5)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => ([(1,2),(3,5),(4,5)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,1,4,5,6,3] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,5,3] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,5,4,6,3] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6] => ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001232
(load all 37 compositions to match this statistic)
(load all 37 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 58%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 58%
Values
[1,0]
=> [1] => [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1] => [1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0]
=> [2] => [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1] => [1,0,1,0,1,0]
=> ? = 4 - 1
[1,0,1,1,0,0]
=> [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0]
=> [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [3] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> ? ∊ {2,3,5,5,6} - 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> ? ∊ {2,3,5,5,6} - 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> ? ∊ {2,3,5,5,6} - 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> ? ∊ {2,3,5,5,6} - 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> ? ∊ {2,3,5,5,6} - 1
[1,1,0,1,1,0,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4 = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4 = 5 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,5,6,6,6,6,6,7,7,7,7,8} - 1
[1,1,1,0,1,1,0,0,0,0]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 6 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 6 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5 = 6 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 6 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 6 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 6 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5 = 6 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5 = 6 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10} - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5 = 6 - 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5 = 6 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 6 - 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 6 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 6 - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000777
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 50%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1] => ([],1)
=> ([],1)
=> 1
[1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,0]
=> [1,2] => ([],2)
=> ([],1)
=> 1
[1,0,1,0,1,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[1,0,1,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {3,4}
[1,1,0,0,1,0]
=> [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {3,4}
[1,1,0,1,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ([],1)
=> 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,3,3,3,4,5,5,6}
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? ∊ {2,3,3,3,4,5,5,6}
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,3,3,3,4,5,5,6}
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,3,3,3,4,5,5,6}
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,3,3,3,4,5,5,6}
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,3,3,3,4,5,5,6}
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,3,3,3,4,5,5,6}
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,3,3,3,4,5,5,6}
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ([],1)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 5
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,6] => ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,6] => ([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 5
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,6] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,5,6,3] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,5,3,6] => ([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,6,3,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 5
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 5
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 6
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 6
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 6
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 5
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 6
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 5
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 6
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [3,1,5,6,2,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 5
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St000771
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 50%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1] => ([],1)
=> 1
[1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> 1
[1,1,0,0]
=> [2] => ([],2)
=> ? = 2
[1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> ? ∊ {2,3,4}
[1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [3] => ([],3)
=> ? ∊ {2,3,4}
[1,1,1,0,0,0]
=> [3] => ([],3)
=> ? ∊ {2,3,4}
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,4,4,4,5,5,6}
[1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3] => ([(2,3)],4)
=> ? ∊ {2,3,3,4,4,4,5,5,6}
[1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> ? ∊ {2,3,3,4,4,4,5,5,6}
[1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {2,3,3,4,4,4,5,5,6}
[1,1,0,1,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [4] => ([],4)
=> ? ∊ {2,3,3,4,4,4,5,5,6}
[1,1,0,1,1,0,0,0]
=> [4] => ([],4)
=> ? ∊ {2,3,3,4,4,4,5,5,6}
[1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> ? ∊ {2,3,3,4,4,4,5,5,6}
[1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> ? ∊ {2,3,3,4,4,4,5,5,6}
[1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ? ∊ {2,3,3,4,4,4,5,5,6}
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [5] => ([],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,0,1,1,0,0,0]
=> [5] => ([],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [5] => ([],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,1,0,1,0,0,0]
=> [5] => ([],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,1,1,0,0,0,0]
=> [5] => ([],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,1,0,0,1,0,1,0,0]
=> [5] => ([],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,0,0,1,1,0,0,0]
=> [5] => ([],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,1,0,1,0,0,1,0,0]
=> [5] => ([],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,0,1,0,1,0,0,0]
=> [5] => ([],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,0,1,1,0,0,0,0]
=> [5] => ([],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,1,1,0,0,0,1,0,0]
=> [5] => ([],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,1,0,0,1,0,0,0]
=> [5] => ([],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,1,0,1,0,0,0,0]
=> [5] => ([],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,1,1,0,0,0,0,0]
=> [5] => ([],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [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)
=> 3
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $2$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St000456
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ([],1)
=> ? = 1
[1,0,1,0]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[1,1,0,0]
=> [1,2] => [1,2] => ([],2)
=> ? = 2
[1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {2,2,4}
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,0,0,1,0]
=> [3,1,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {2,2,4}
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {2,2,4}
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {2,3,3,3,4,4,5,5,6}
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {2,3,3,3,4,4,5,5,6}
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,3,4,4,5,5,6}
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {2,3,3,3,4,4,5,5,6}
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {2,3,3,3,4,4,5,5,6}
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,3,4,4,5,5,6}
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {2,3,3,3,4,4,5,5,6}
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {2,3,3,3,4,4,5,5,6}
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {2,3,3,3,4,4,5,5,6}
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,7,7,8}
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,7,7,8}
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,7,7,8}
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,7,7,8}
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,7,7,8}
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,7,7,8}
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,7,7,8}
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,7,7,8}
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,7,7,8}
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,7,7,8}
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,7,7,8}
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 6
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,7,7,8}
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,7,7,8}
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,7,7,8}
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,7,7,8}
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,7,7,8}
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,7,7,8}
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,7,7,8}
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,7,7,8}
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,7,7,8}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,7,7,8}
[1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,7,7,8}
[1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,7,7,8}
[1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,7,7,8}
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,7,7,8}
[1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,7,7,8}
[1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,7,7,8}
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,7,7,7,7,8}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6)
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5] => [3,2,4,1,6,5] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5] => [2,1,5,4,6,3] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5] => [4,2,5,1,6,3] => ([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5] => [3,2,5,4,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => [2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10}
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,5,6] => [3,2,4,1,5,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10}
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => [2,1,5,4,3,6] => ([(1,2),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,5,3,6] => [4,2,5,1,3,6] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,4,5,1,3,6] => [3,2,5,4,1,6] => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10}
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,5,6,3] => [4,2,6,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> 4
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,5,1,6,3] => [5,2,6,4,1,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => [3,2,6,4,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [3,2,5,6,1,4] => ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> 3
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,5,1,6,3,4] => [5,2,4,6,1,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,5,6,1,3,4] => [3,2,4,6,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 3
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,5,1,6,4] => [5,2,3,6,1,4] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,6,1,4] => [4,2,3,6,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,6,1,3,4,5] => [3,2,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,3,6,1,4,5] => [4,2,3,5,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,4,6,1,5] => [5,2,3,4,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,4,6,2,5] => [3,5,1,4,6,2] => ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [3,4,1,6,2,5] => [4,5,3,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,4,6,1,2,5] => [2,5,3,4,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [3,1,4,5,6,2] => [3,6,1,4,5,2] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [3,4,1,5,6,2] => [4,6,3,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 6
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,4,5,1,6,2] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 9
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [2,6,3,4,5,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,5,2,6,4] => [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 3
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [3,5,1,2,6,4] => [2,5,3,6,1,4] => ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,1,5,6,2,4] => [3,4,1,6,5,2] => ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [3,5,1,6,2,4] => [5,4,3,6,1,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 7
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,5,6,1,2,4] => [2,4,3,6,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 3
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [3,6,1,2,4,5] => [2,4,3,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
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: St001498
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 33%
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> ? = 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {1,2}
[1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {1,2}
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {1,2,2,3,4}
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {1,2,2,3,4}
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {1,2,2,3,4}
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {1,2,2,3,4}
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {1,2,2,3,4}
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,2,2,3,3,3,4,4,4,5,5,6}
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,2,2,3,3,3,4,4,4,5,5,6}
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,2,2,3,3,3,4,4,4,5,5,6}
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,2,2,3,3,3,4,4,4,5,5,6}
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,2,2,3,3,3,4,4,4,5,5,6}
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,2,2,3,3,3,4,4,4,5,5,6}
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,2,2,3,3,3,4,4,4,5,5,6}
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,2,2,3,3,3,4,4,4,5,5,6}
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,2,2,3,3,3,4,4,4,5,5,6}
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,2,2,3,3,3,4,4,4,5,5,6}
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,2,2,3,3,3,4,4,4,5,5,6}
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,2,2,3,3,3,4,4,4,5,5,6}
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,2,2,3,3,3,4,4,4,5,5,6}
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 3
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 3
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 3
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 3
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 5
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 5
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 5
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> 4
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 5
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> 4
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 5
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 5
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 5
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> 4
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 5
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> 4
Description
The normalised height of a Nakayama algebra with magnitude 1.
We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
Matching statistic: St001645
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 58%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 58%
Values
[1,0]
=> [1] => ([],1)
=> ([],1)
=> 1
[1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,0]
=> [1,2] => ([],2)
=> ([],1)
=> 1
[1,0,1,0,1,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[1,1,0,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 4
[1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ([],1)
=> 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,4,4,5,5,6}
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,4,4,5,5,6}
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,4,4,5,5,6}
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,4,4,5,5,6}
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,4,4,5,5,6}
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,4,4,5,5,6}
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ([],1)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => ([(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)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => ([(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)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => ([(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)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,2,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,2,1] => ([(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),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,2,1] => ([(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),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,4,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,6,3,2,4,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,5,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,6,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [4,5,3,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,5,1] => ([(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),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [5,3,4,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [6,4,2,3,5,1] => ([(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),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [5,4,2,3,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,5,2,3,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [6,3,2,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10}
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 2
Description
The pebbling number of a connected graph.
Matching statistic: St001875
Values
[1,0]
=> ([],1)
=> ([],1)
=> ? = 1
[1,0,1,0]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,2}
[1,1,0,0]
=> ([],2)
=> ([],1)
=> ? ∊ {1,2}
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> ([],1)
=> ? ∊ {1,2,2,4}
[1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> ([],1)
=> ? ∊ {1,2,2,4}
[1,1,0,1,0,0]
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,4}
[1,1,1,0,0,0]
=> ([],3)
=> ([],1)
=> ? ∊ {1,2,2,4}
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> ([],1)
=> ? ∊ {1,2,2,2,2,3,3,4,4,5,5,6}
[1,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? ∊ {1,2,2,2,2,3,3,4,4,5,5,6}
[1,0,1,1,0,1,0,0]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,3,3,4,4,5,5,6}
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ? ∊ {1,2,2,2,2,3,3,4,4,5,5,6}
[1,1,0,0,1,0,1,0]
=> ([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ? ∊ {1,2,2,2,2,3,3,4,4,5,5,6}
[1,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,3,3,4,4,5,5,6}
[1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,3,3,4,4,5,5,6}
[1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> ([],1)
=> ? ∊ {1,2,2,2,2,3,3,4,4,5,5,6}
[1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> ? ∊ {1,2,2,2,2,3,3,4,4,5,5,6}
[1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> ([],1)
=> ? ∊ {1,2,2,2,2,3,3,4,4,5,5,6}
[1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,3,3,4,4,5,5,6}
[1,1,1,1,0,0,0,0]
=> ([],4)
=> ([],1)
=> ? ∊ {1,2,2,2,2,3,3,4,4,5,5,6}
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,0,0,1,0,1,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,0,0,1,1,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,0,1,0,1,0,1,0]
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,0,1,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,0,0,1,1,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,1,1,0,0,0,0]
=> ([(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,0,0,0,1,0,1,0]
=> ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,0,0,1,0,0,1,0]
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,0,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,0,0,1,1,0,0,0]
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,0,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,0,1,0,1,0,0,0]
=> ([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,1,0,1,1,0,0,0,0]
=> ([(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,1,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,0,0,1,1,0,1,0,0]
=> ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,0,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,1,0,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,0,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,0,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,1,0,0,1,1,0,1,0,0,0]
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,1,0,1,0,0,1,1,0,0,0]
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,1,0,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,1,0,1,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,1,1,0,1,0,1,0,0,0,0]
=> ([(2,5),(3,4),(3,5)],6)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,6),(1,3),(1,6),(3,5),(4,2),(5,4),(6,5)],7)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> ([(0,3),(1,5),(1,6),(3,5),(3,6),(4,2),(5,4),(6,4)],7)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> ([(0,5),(0,6),(1,5),(1,6),(3,4),(4,2),(5,3),(6,4)],7)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,3),(0,6),(1,5),(1,6),(3,5),(4,2),(5,4),(6,4)],7)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,6),(1,5),(2,3),(2,5),(3,6),(5,6),(6,4)],7)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> ([(0,6),(1,2),(2,6),(3,5),(4,5),(6,3),(6,4)],7)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(0,6),(1,5),(1,6),(3,2),(4,2),(5,3),(5,4),(6,3),(6,4)],7)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(4,6),(5,6)],7)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,6),(1,2),(1,6),(2,4),(2,5),(4,3),(5,3),(6,4),(6,5)],7)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> ([(0,6),(1,6),(2,5),(3,5),(4,3),(6,2),(6,4)],7)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1)],7)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> ([(0,3),(1,4),(1,6),(2,5),(3,4),(3,6),(4,2),(6,5)],7)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,2),(2,1)],3)
=> 3
Description
The number of simple modules with projective dimension at most 1.
Matching statistic: St001880
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 42%
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 42%
Values
[1,0]
=> [[1],[]]
=> ([],1)
=> ([],1)
=> ? = 1
[1,0,1,0]
=> [[1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2}
[1,1,0,0]
=> [[2],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,3,4}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,3,4}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,3,4}
[1,1,0,1,0,0]
=> [[3],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,3,4}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,3,4}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,5,6}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,5,6}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,5,6}
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,5,6}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,5,6}
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,5,6}
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,5,6}
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,5,6}
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,5,6}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,5,6}
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,5,6}
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,5,6}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,5,6}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,3,3,3,4,4,4,5,5,6}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,8}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [[4,4,2],[2,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [[4,3,2],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [[4,3,3],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [[4,4,3],[3,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [[4,4,3],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [[4,3,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [[4,4,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [[4,3,2],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1,1],[2,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1,1],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [[3,2,2,2,1],[1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [[3,3,2,2,1],[2,1,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [[4,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [[3,3,3,2,1],[2,2,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [[4,3,2,1],[2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> 6
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [[4,4,2,1],[3,1]]
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 5
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [[5,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [[4,3,3,1],[2,2]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [[4,4,3,1],[3,2]]
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 5
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [[5,3,1],[2]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
The following 25 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000528The height of a poset. St000912The number of maximal antichains in a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000080The rank of the poset. St001782The order of rowmotion on the set of order ideals of a poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000983The length of the longest alternating subword. St000189The number of elements in the poset. St001343The dimension of the reduced incidence algebra of a poset. St001717The largest size of an interval in a poset. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St000656The number of cuts of a poset. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St000455The second largest eigenvalue of a graph if it is integral. St001060The distinguishing index of a graph. St000264The girth of a graph, which is not a tree. St000075The orbit size of a standard tableau under promotion. St000738The first entry in the last row of a standard tableau. St000422The energy of a graph, if it is integral. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.
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