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Your data matches 79 different statistics following compositions of up to 3 maps.
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Matching statistic: St000245
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Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [] => 0
[1,0,1,0]
=> [1,2] => [1] => 0
[1,1,0,0]
=> [2,1] => [1] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2] => 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1] => 0
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => 0
[1,1,1,0,0,0]
=> [3,2,1] => [2,1] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => 2
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,2] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,1] => 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,3,1] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1] => 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => 0
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,2,1] => 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4] => 3
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,3] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,4,2] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,3,2] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,3] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,3,4,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,4,3,1] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,4,3,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,4,3,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,4,3,1] => 1
Description
The number of ascents of a permutation.
Matching statistic: St000786
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 70% ●values known / values provided: 95%●distinct values known / distinct values provided: 70%
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 70% ●values known / values provided: 95%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> [1] => [] => ([],0)
=> ? = 0 + 1
[1,0,1,0]
=> [1,2] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,0,0]
=> [2,1] => [1] => ([],1)
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2] => ([],2)
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,2] => ([],2)
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => ([],3)
=> 3 = 2 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => ([],3)
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4] => ([],4)
=> 4 = 3 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4] => ([],4)
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3] => ([(2,3)],4)
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => ([(2,3)],4)
=> 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4] => ([(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,4] => ([(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4] => ([(2,3)],4)
=> 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,4] => ([(2,3)],4)
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,8,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,8,9,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10,9,8,7,6,5,4,3,2] => [1,9,8,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,9,8,7,6,5,4,3,2,1,11] => [10,9,8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 0 + 1
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,8,7,9,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,7,9,8,6,5,4,3,2] => [1,7,8,6,5,4,3,2] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,9,10,8,7,6,5,4,3,2] => [1,9,8,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,10,9,8,7,6,5,4,3,2] => [1,10,9,8,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [7,6,5,4,3,2,1,8,9] => [7,6,5,4,3,2,1,8] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [8,7,6,5,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,9] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [8,7,6,5,4,3,2,1,9,10] => [8,7,6,5,4,3,2,1,9] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8] => ([],8)
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,9,8] => [1,2,3,4,5,6,7,8] => ([],8)
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,8,9,7] => [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? = 6 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,9,4] => [1,2,3,5,6,7,8,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9] => ([],9)
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,10,9] => [1,2,3,4,5,6,7,8,9] => ([],9)
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,6,9,8,7] => [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,7,9,10,8] => [1,2,3,4,5,6,7,9,8] => ([(7,8)],9)
=> ? = 7 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => [2,3,4,5,6,7,8,9,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,9,11,10] => [1,2,3,4,5,6,7,8,9,10] => ([],10)
=> ? = 9 + 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,9,10,1] => [3,2,4,5,6,7,8,9,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 6 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [2,3,4,5,6,7,8,9,10,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 8 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6,9,8] => [1,3,2,5,4,7,6,8] => ([(2,7),(3,6),(4,5)],8)
=> ? = 4 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8] => ([(6,7)],8)
=> ? = 6 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,8,7,9] => [2,1,4,3,6,5,8,7] => ([(0,7),(1,6),(2,5),(3,4)],8)
=> ? = 3 + 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6,7,8,9] => [3,2,1,4,5,6,7,8] => ([(5,6),(5,7),(6,7)],8)
=> ? = 5 + 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7,8,9] => [4,3,2,1,5,6,7,8] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 + 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1,6,7,8,9] => [5,4,3,2,1,6,7,8] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1,7,8,9] => [6,5,4,3,2,1,7,8] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9,10] => [2,1,3,4,5,6,7,8,9] => ([(7,8)],9)
=> ? = 7 + 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6,7,8,9,10] => [3,2,1,4,5,6,7,8,9] => ([(6,7),(6,8),(7,8)],9)
=> ? = 6 + 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7,8,9,10] => [4,3,2,1,5,6,7,8,9] => ([(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 + 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1,6,7,8,9,10] => [5,4,3,2,1,6,7,8,9] => ([(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 4 + 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1,7,8,9,10] => [6,5,4,3,2,1,7,8,9] => ([(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1,8,9,10] => [7,6,5,4,3,2,1,8,9] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,5,6,7,1,8,9] => [2,3,4,5,6,7,1,8] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,6,7,8,9,1,10] => [2,3,4,5,6,7,8,9,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 7 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9,10,11] => [2,1,3,4,5,6,7,8,9,10] => ([(8,9)],10)
=> ? = 8 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,9,8,7,6,5,4,3,2,1,12,11] => [10,9,8,7,6,5,4,3,2,1,11] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 1 + 1
Description
The maximal number of occurrences of a colour in a proper colouring of a graph.
To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the largest part occurring in any of these partitions.
For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes, $[2,2,2]$ and $[3,2,1]$. Therefore, the statistic on this graph is $3$.
Matching statistic: St000672
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [] => 0
[1,0,1,0]
=> [1,2] => [1] => 0
[1,1,0,0]
=> [2,1] => [1] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2] => 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1] => 0
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => 0
[1,1,1,0,0,0]
=> [3,2,1] => [2,1] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => 2
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,2] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,1] => 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,3,1] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1] => 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => 0
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,2,1] => 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4] => 3
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,3] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,4,2] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,3,2] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,3] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,3,4,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,4,3,1] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,4,3,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,4,3,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,4,3,1] => 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,5] => [1,2,3,4,6,7,5] => ? = 5
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,4,7,6,8,5] => [1,2,3,4,7,6,5] => ? = 4
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,4,7,8,6,5] => [1,2,3,4,7,6,5] => ? = 4
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,4,8,7,6,5] => [1,2,3,4,7,6,5] => ? = 4
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,3,5,4,6,7,8] => [1,2,3,5,4,6,7] => ? = 5
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,3,5,4,7,6,8] => [1,2,3,5,4,7,6] => ? = 4
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,4] => [1,2,3,5,6,7,4] => ? = 5
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,3,5,6,8,7,4] => [1,2,3,5,6,7,4] => ? = 5
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,3,5,7,6,8,4] => [1,2,3,5,7,6,4] => ? = 4
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,3,5,7,8,6,4] => [1,2,3,5,7,6,4] => ? = 4
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,3,5,8,7,6,4] => [1,2,3,5,7,6,4] => ? = 4
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,3,6,5,7,8,4] => [1,2,3,6,5,7,4] => ? = 4
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,3,6,5,8,7,4] => [1,2,3,6,5,7,4] => ? = 4
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,3,6,7,5,8,4] => [1,2,3,6,7,5,4] => ? = 4
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,3,6,7,8,5,4] => [1,2,3,6,7,5,4] => ? = 4
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,3,6,8,7,5,4] => [1,2,3,6,7,5,4] => ? = 4
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,3,7,6,5,8,4] => [1,2,3,7,6,5,4] => ? = 3
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,3,7,6,8,5,4] => [1,2,3,7,6,5,4] => ? = 3
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,7,8,6,5,4] => [1,2,3,7,6,5,4] => ? = 3
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,8,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,6,5,7,8] => [1,2,4,3,6,5,7] => ? = 4
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,2,4,3,6,7,8,5] => [1,2,4,3,6,7,5] => ? = 4
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,2,4,3,7,6,8,5] => [1,2,4,3,7,6,5] => ? = 3
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,2,4,3,8,7,6,5] => [1,2,4,3,7,6,5] => ? = 3
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,3] => [1,2,4,5,6,7,3] => ? = 5
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,2,4,6,5,3,7,8] => [1,2,4,6,5,3,7] => ? = 4
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,2,4,6,7,8,5,3] => [1,2,4,6,7,5,3] => ? = 4
[1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,2,4,8,7,6,5,3] => [1,2,4,7,6,5,3] => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,2,5,4,6,7,8,3] => [1,2,5,4,6,7,3] => ? = 4
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,2,5,6,4,7,8,3] => [1,2,5,6,4,7,3] => ? = 4
[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,8,4,3] => [1,2,5,6,7,4,3] => ? = 4
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,2,6,5,4,7,8,3] => [1,2,6,5,4,7,3] => ? = 3
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,2,6,7,8,5,4,3] => [1,2,6,7,5,4,3] => ? = 3
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,5,7,6,8] => [1,3,2,4,5,7,6] => ? = 4
[1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,3,2,4,6,7,8,5] => [1,3,2,4,6,7,5] => ? = 4
[1,0,1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,3,2,4,7,6,8,5] => [1,3,2,4,7,6,5] => ? = 3
[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,8,7,6,5] => [1,3,2,4,7,6,5] => ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,5,4,6,7,8] => [1,3,2,5,4,6,7] => ? = 4
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,7,6,8] => [1,3,2,5,4,7,6] => ? = 3
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,3,2,8,7,6,5,4] => [1,3,2,7,6,5,4] => ? = 2
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,6,7,5,8] => [1,3,4,2,6,7,5] => ? = 4
[1,0,1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,3,4,2,6,7,8,5] => [1,3,4,2,6,7,5] => ? = 4
[1,0,1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,4,2,7,6,8,5] => [1,3,4,2,7,6,5] => ? = 3
[1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,4,2,8,7,6,5] => [1,3,4,2,7,6,5] => ? = 3
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,2] => [1,3,4,5,6,7,2] => ? = 5
[1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,4,5,6,8,7,2] => [1,3,4,5,6,7,2] => ? = 5
[1,0,1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,3,4,5,7,6,2,8] => [1,3,4,5,7,6,2] => ? = 4
[1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,4,5,7,8,6,2] => [1,3,4,5,7,6,2] => ? = 4
[1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [1,3,4,6,7,5,2,8] => [1,3,4,6,7,5,2] => ? = 4
[1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,4,8,7,6,5,2] => [1,3,4,7,6,5,2] => ? = 3
Description
The number of minimal elements in Bruhat order not less than the permutation.
The minimal elements in question are biGrassmannian, that is
$$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$$
for some $(r,a,b)$.
This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].
Matching statistic: St000053
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 92%●distinct values known / distinct values provided: 70%
Mp00252: Permutations —restriction⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 92%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> [1] => [] => []
=> ? = 0
[1,0,1,0]
=> [1,2] => [1] => [1,0]
=> 0
[1,1,0,0]
=> [2,1] => [1] => [1,0]
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2] => [1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,2] => [1,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1] => [1,1,0,0]
=> 0
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => [1,1,0,0]
=> 0
[1,1,1,0,0,0]
=> [3,2,1] => [2,1] => [1,1,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => [1,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1] => [1,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,1] => [1,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1] => [1,1,1,0,0,0]
=> 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => [1,1,1,0,0,0]
=> 0
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,7,6,5,4,3,2,1,9] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,8,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,8,7,6,5,4,3,2,1,10] => [9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,8,9,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10,9,8,7,6,5,4,3,2] => [1,9,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,9,8,7,6,5,4,3,2,1,11] => [10,9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [7,6,8,5,4,3,2,1,9] => [7,6,8,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,8,7,9,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,7,9,8,6,5,4,3,2] => [1,7,8,6,5,4,3,2] => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 2
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,9,10,8,7,6,5,4,3,2] => [1,9,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,10,9,8,7,6,5,4,3,2] => [1,10,9,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [7,6,5,4,3,2,1,8,9] => [7,6,5,4,3,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [8,7,6,5,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,9] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [8,7,6,5,4,3,2,1,9,10] => [8,7,6,5,4,3,2,1,9] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,1] => [2,3,4,5,6,7,8,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,9,1] => [3,2,4,5,6,7,8,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 5
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,2,5,6,7,8,9,1] => [3,4,2,5,6,7,8,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 5
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,7,6,5,4,3,2,9,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> [8,7,6,5,4,3,9,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0]
=> [8,7,6,5,4,9,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> [8,7,6,5,9,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [8,7,6,9,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [8,7,9,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [8,9,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [9,8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,9,8] => [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,8,9,7] => [1,2,3,4,5,6,8,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 6
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,9,4] => [1,2,3,5,6,7,8,4] => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,10,9] => [1,2,3,4,5,6,7,8,9] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,6,9,8,7] => [1,2,3,4,5,6,8,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 6
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,7,9,10,8] => [1,2,3,4,5,6,7,9,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 7
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => [2,3,4,5,6,7,8,9,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,9,11,10] => [1,2,3,4,5,6,7,8,9,10] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,9,10,1] => [3,2,4,5,6,7,8,9,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [2,3,4,5,6,7,8,9,10,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 8
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6,9,8] => [1,3,2,5,4,7,6,8] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,8,7,9] => [2,1,4,3,6,5,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6,7,8,9] => [3,2,1,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7,8,9] => [4,3,2,1,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1,6,7,8,9] => [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 3
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1,7,8,9] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9,10] => [2,1,3,4,5,6,7,8,9] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6,7,8,9,10] => [3,2,1,4,5,6,7,8,9] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7,8,9,10] => [4,3,2,1,5,6,7,8,9] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1,6,7,8,9,10] => [5,4,3,2,1,6,7,8,9] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1,7,8,9,10] => [6,5,4,3,2,1,7,8,9] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 3
Description
The number of valleys of the Dyck path.
Matching statistic: St001068
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 92%●distinct values known / distinct values provided: 70%
Mp00252: Permutations —restriction⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 92%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> [1] => [] => []
=> ? = 0 + 1
[1,0,1,0]
=> [1,2] => [1] => [1,0]
=> 1 = 0 + 1
[1,1,0,0]
=> [2,1] => [1] => [1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1] => [1,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => [1,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,1] => [1,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1] => [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,1] => [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,7,6,5,4,3,2,1,9] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,8,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,8,7,6,5,4,3,2,1,10] => [9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,8,9,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10,9,8,7,6,5,4,3,2] => [1,9,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,9,8,7,6,5,4,3,2,1,11] => [10,9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [7,6,8,5,4,3,2,1,9] => [7,6,8,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,8,7,9,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,7,9,8,6,5,4,3,2] => [1,7,8,6,5,4,3,2] => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,9,10,8,7,6,5,4,3,2] => [1,9,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,10,9,8,7,6,5,4,3,2] => [1,10,9,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [7,6,5,4,3,2,1,8,9] => [7,6,5,4,3,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [8,7,6,5,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,9] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [8,7,6,5,4,3,2,1,9,10] => [8,7,6,5,4,3,2,1,9] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,1] => [2,3,4,5,6,7,8,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6 + 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,9,1] => [3,2,4,5,6,7,8,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 5 + 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,2,5,6,7,8,9,1] => [3,4,2,5,6,7,8,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 5 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,7,6,5,4,3,2,9,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> [8,7,6,5,4,3,9,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0]
=> [8,7,6,5,4,9,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> [8,7,6,5,9,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [8,7,6,9,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [8,7,9,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [8,9,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [9,8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,9,8] => [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,8,9,7] => [1,2,3,4,5,6,8,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 6 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,9,4] => [1,2,3,5,6,7,8,4] => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,10,9] => [1,2,3,4,5,6,7,8,9] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,6,9,8,7] => [1,2,3,4,5,6,8,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,7,9,10,8] => [1,2,3,4,5,6,7,9,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 7 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => [2,3,4,5,6,7,8,9,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,9,11,10] => [1,2,3,4,5,6,7,8,9,10] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9 + 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,9,10,1] => [3,2,4,5,6,7,8,9,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [2,3,4,5,6,7,8,9,10,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 8 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6,9,8] => [1,3,2,5,4,7,6,8] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,8,7,9] => [2,1,4,3,6,5,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3 + 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6,7,8,9] => [3,2,1,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5 + 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7,8,9] => [4,3,2,1,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1,6,7,8,9] => [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1,7,8,9] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 2 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9,10] => [2,1,3,4,5,6,7,8,9] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6,7,8,9,10] => [3,2,1,4,5,6,7,8,9] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6 + 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7,8,9,10] => [4,3,2,1,5,6,7,8,9] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5 + 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1,6,7,8,9,10] => [5,4,3,2,1,6,7,8,9] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1,7,8,9,10] => [6,5,4,3,2,1,7,8,9] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
Matching statistic: St001337
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001337: Graphs ⟶ ℤResult quality: 70% ●values known / values provided: 91%●distinct values known / distinct values provided: 70%
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001337: Graphs ⟶ ℤResult quality: 70% ●values known / values provided: 91%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> [1] => [] => ([],0)
=> ? = 0 + 1
[1,0,1,0]
=> [1,2] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,0,0]
=> [2,1] => [1] => ([],1)
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2] => ([],2)
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,2] => ([],2)
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => ([],3)
=> 3 = 2 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => ([],3)
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4] => ([],4)
=> 4 = 3 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4] => ([],4)
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3] => ([(2,3)],4)
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => ([(2,3)],4)
=> 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4] => ([(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,4] => ([(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4] => ([(2,3)],4)
=> 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,4] => ([(2,3)],4)
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [7,6,5,4,3,2,8,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [7,6,5,4,3,8,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [7,6,5,4,8,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [7,6,5,8,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [7,6,8,5,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [7,8,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,7,6,5,4,3,2,1,9] => [8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,8,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,8,7,6,5,4,3,2,1,10] => [9,8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 0 + 1
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,8,9,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10,9,8,7,6,5,4,3,2] => [1,9,8,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,9,8,7,6,5,4,3,2,1,11] => [10,9,8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 0 + 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [7,6,8,5,4,3,2,1,9] => [7,6,8,5,4,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,8,7,9,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,7,9,8,6,5,4,3,2] => [1,7,8,6,5,4,3,2] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,9,10,8,7,6,5,4,3,2] => [1,9,8,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,10,9,8,7,6,5,4,3,2] => [1,10,9,8,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [7,6,5,4,3,2,1,8,9] => [7,6,5,4,3,2,1,8] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [8,7,6,5,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,9] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [8,7,6,5,4,3,2,1,9,10] => [8,7,6,5,4,3,2,1,9] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,1] => [2,3,4,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,9,1] => [3,2,4,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 + 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,2,5,6,7,8,9,1] => [3,4,2,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,7,6,5,4,3,2,9,1] => [8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> [8,7,6,5,4,3,9,2,1] => [8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0]
=> [8,7,6,5,4,9,3,2,1] => [8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> [8,7,6,5,9,4,3,2,1] => [8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [8,7,6,9,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [8,7,9,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [8,9,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [9,8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8] => ([],8)
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,9,8] => [1,2,3,4,5,6,7,8] => ([],8)
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,8,9,7] => [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? = 6 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,9,4] => [1,2,3,5,6,7,8,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9] => ([],9)
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,10,9] => [1,2,3,4,5,6,7,8,9] => ([],9)
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,6,9,8,7] => [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,7,9,10,8] => [1,2,3,4,5,6,7,9,8] => ([(7,8)],9)
=> ? = 7 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => [2,3,4,5,6,7,8,9,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,9,11,10] => [1,2,3,4,5,6,7,8,9,10] => ([],10)
=> ? = 9 + 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,9,10,1] => [3,2,4,5,6,7,8,9,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 6 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [2,3,4,5,6,7,8,9,10,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 8 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6,9,8] => [1,3,2,5,4,7,6,8] => ([(2,7),(3,6),(4,5)],8)
=> ? = 4 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8] => ([(6,7)],8)
=> ? = 6 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,8,7,9] => [2,1,4,3,6,5,8,7] => ([(0,7),(1,6),(2,5),(3,4)],8)
=> ? = 3 + 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6,7,8,9] => [3,2,1,4,5,6,7,8] => ([(5,6),(5,7),(6,7)],8)
=> ? = 5 + 1
Description
The upper domination number of a graph.
This is the maximum cardinality of a minimal dominating set of $G$.
The smallest graph with different upper irredundance number and upper domination number has eight vertices. It is obtained from the disjoint union of two copies of $K_4$ by joining three of the four vertices of the first with three of the four vertices of the second. For bipartite graphs the two parameters always coincide [1].
Matching statistic: St001338
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001338: Graphs ⟶ ℤResult quality: 70% ●values known / values provided: 91%●distinct values known / distinct values provided: 70%
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001338: Graphs ⟶ ℤResult quality: 70% ●values known / values provided: 91%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> [1] => [] => ([],0)
=> ? = 0 + 1
[1,0,1,0]
=> [1,2] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,0,0]
=> [2,1] => [1] => ([],1)
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2] => ([],2)
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,2] => ([],2)
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => ([],3)
=> 3 = 2 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => ([],3)
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4] => ([],4)
=> 4 = 3 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4] => ([],4)
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3] => ([(2,3)],4)
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => ([(2,3)],4)
=> 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4] => ([(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,4] => ([(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4] => ([(2,3)],4)
=> 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,4] => ([(2,3)],4)
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [7,6,5,4,3,2,8,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [7,6,5,4,3,8,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [7,6,5,4,8,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [7,6,5,8,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [7,6,8,5,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [7,8,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,7,6,5,4,3,2,1,9] => [8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,8,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,8,7,6,5,4,3,2,1,10] => [9,8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 0 + 1
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,8,9,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10,9,8,7,6,5,4,3,2] => [1,9,8,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,9,8,7,6,5,4,3,2,1,11] => [10,9,8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 0 + 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [7,6,8,5,4,3,2,1,9] => [7,6,8,5,4,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,8,7,9,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,7,9,8,6,5,4,3,2] => [1,7,8,6,5,4,3,2] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,9,10,8,7,6,5,4,3,2] => [1,9,8,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,10,9,8,7,6,5,4,3,2] => [1,10,9,8,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [7,6,5,4,3,2,1,8,9] => [7,6,5,4,3,2,1,8] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [8,7,6,5,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,9] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [8,7,6,5,4,3,2,1,9,10] => [8,7,6,5,4,3,2,1,9] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,1] => [2,3,4,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,9,1] => [3,2,4,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 + 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,2,5,6,7,8,9,1] => [3,4,2,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,7,6,5,4,3,2,9,1] => [8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> [8,7,6,5,4,3,9,2,1] => [8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0]
=> [8,7,6,5,4,9,3,2,1] => [8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> [8,7,6,5,9,4,3,2,1] => [8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [8,7,6,9,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [8,7,9,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [8,9,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [9,8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8] => ([],8)
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,9,8] => [1,2,3,4,5,6,7,8] => ([],8)
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,8,9,7] => [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? = 6 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,9,4] => [1,2,3,5,6,7,8,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9] => ([],9)
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,10,9] => [1,2,3,4,5,6,7,8,9] => ([],9)
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,6,9,8,7] => [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,7,9,10,8] => [1,2,3,4,5,6,7,9,8] => ([(7,8)],9)
=> ? = 7 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => [2,3,4,5,6,7,8,9,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,9,11,10] => [1,2,3,4,5,6,7,8,9,10] => ([],10)
=> ? = 9 + 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,9,10,1] => [3,2,4,5,6,7,8,9,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 6 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [2,3,4,5,6,7,8,9,10,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 8 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6,9,8] => [1,3,2,5,4,7,6,8] => ([(2,7),(3,6),(4,5)],8)
=> ? = 4 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8] => ([(6,7)],8)
=> ? = 6 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,8,7,9] => [2,1,4,3,6,5,8,7] => ([(0,7),(1,6),(2,5),(3,4)],8)
=> ? = 3 + 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6,7,8,9] => [3,2,1,4,5,6,7,8] => ([(5,6),(5,7),(6,7)],8)
=> ? = 5 + 1
Description
The upper irredundance number of a graph.
A set $S$ of vertices is irredundant, if there is no vertex in $S$, whose closed neighbourhood is contained in the union of the closed neighbourhoods of the other vertices of $S$.
The upper irredundance number is the largest size of a maximal irredundant set.
The smallest graph with different upper irredundance number and upper domination number [[St001337]] has eight vertices. It is obtained from the disjoint union of two copies of $K_4$ by joining three of the four vertices of the first with three of the four vertices of the second. For bipartite graphs the two parameters always coincide [2].
Matching statistic: St000528
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000528: Posets ⟶ ℤResult quality: 70% ●values known / values provided: 90%●distinct values known / distinct values provided: 70%
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000528: Posets ⟶ ℤResult quality: 70% ●values known / values provided: 90%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> [1] => [] => ([],0)
=> ? = 0 + 1
[1,0,1,0]
=> [1,2] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,0,0]
=> [2,1] => [1] => ([],1)
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2] => ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,2] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1] => ([],2)
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => ([],2)
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,1] => ([],2)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => ([(0,1),(0,2)],3)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,1] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,3,1] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1] => ([],3)
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => ([],3)
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,2,1] => ([],3)
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => ([],3)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,3,5,4,7,6,8] => [1,2,3,5,4,7,6] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
=> ? = 4 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,6,5,7,8] => [1,2,4,3,6,5,7] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
=> ? = 4 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,2,4,3,6,7,8,5] => [1,2,4,3,6,7,5] => ([(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(4,3),(6,1)],7)
=> ? = 4 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,2,4,3,7,6,8,5] => [1,2,4,3,7,6,5] => ([(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,1),(3,2)],7)
=> ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,2,4,3,8,7,6,5] => [1,2,4,3,7,6,5] => ([(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,1),(3,2)],7)
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,2,4,6,5,3,7,8] => [1,2,4,6,5,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ? = 4 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,2,5,4,6,7,8,3] => [1,2,5,4,6,7,3] => ([(0,5),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ? = 4 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,2,5,6,4,7,8,3] => [1,2,5,6,4,7,3] => ([(0,5),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ? = 4 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7,8] => [1,3,2,4,5,6,7] => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 5 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,5,7,6,8] => [1,3,2,4,5,7,6] => ([(0,3),(0,4),(3,6),(4,6),(5,1),(5,2),(6,5)],7)
=> ? = 4 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,3,2,4,6,7,8,5] => [1,3,2,4,6,7,5] => ([(0,3),(0,4),(3,6),(4,6),(5,1),(6,2),(6,5)],7)
=> ? = 4 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,6,7,5,8] => [1,3,4,2,6,7,5] => ([(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(6,1)],7)
=> ? = 4 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,3,4,2,6,7,8,5] => [1,3,4,2,6,7,5] => ([(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(6,1)],7)
=> ? = 4 + 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,4,2,7,6,8,5] => [1,3,4,2,7,6,5] => ([(0,2),(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,1)],7)
=> ? = 3 + 1
[1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,4,2,8,7,6,5] => [1,3,4,2,7,6,5] => ([(0,2),(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,1)],7)
=> ? = 3 + 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,4,3,5,6,7,8,2] => [1,4,3,5,6,7,2] => ([(0,2),(0,3),(0,4),(3,6),(4,6),(5,1),(6,5)],7)
=> ? = 4 + 1
[1,1,0,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,1,7,6,5,8,4,3] => [2,1,7,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6)],7)
=> ? = 1 + 1
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,8,7,6,5,4,3] => [2,1,7,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6)],7)
=> ? = 1 + 1
[1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,5,7,6,8] => [2,3,4,1,5,7,6] => ([(0,6),(1,5),(2,6),(5,2),(6,3),(6,4)],7)
=> ? = 4 + 1
[1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [2,3,4,1,5,7,8,6] => [2,3,4,1,5,7,6] => ([(0,6),(1,5),(2,6),(5,2),(6,3),(6,4)],7)
=> ? = 4 + 1
[1,1,0,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,8,7,6] => [2,3,4,1,5,7,6] => ([(0,6),(1,5),(2,6),(5,2),(6,3),(6,4)],7)
=> ? = 4 + 1
[1,1,0,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,4,5,1,8,7,6] => [2,3,4,5,1,7,6] => ([(0,5),(0,6),(1,4),(2,5),(2,6),(3,2),(4,3)],7)
=> ? = 4 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,6,7,1,8] => [2,3,4,5,6,7,1] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 5 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,1] => [2,3,4,5,6,7,1] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 5 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,5,6,8,7,1] => [2,3,4,5,6,7,1] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 5 + 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,4,2,5,6,7,8,1] => [3,4,2,5,6,7,1] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
=> ? = 4 + 1
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,4,5,2,6,7,8,1] => [3,4,5,2,6,7,1] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
=> ? = 4 + 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,8,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7)],8)
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,8,7,6,5,4,3,2,1,10] => [9,8,7,6,5,4,3,2,1] => ([],9)
=> ? = 0 + 1
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,8,9,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7)],8)
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10,9,8,7,6,5,4,3,2] => [1,9,8,7,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8)],9)
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,9,8,7,6,5,4,3,2,1,11] => [10,9,8,7,6,5,4,3,2,1] => ([],10)
=> ? = 0 + 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [7,6,8,5,4,3,2,1,9] => [7,6,8,5,4,3,2,1] => ([(5,7),(6,7)],8)
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,8,7,9,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7)],8)
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,7,9,8,6,5,4,3,2] => [1,7,8,6,5,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(7,1)],8)
=> ? = 2 + 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,9,10,8,7,6,5,4,3,2] => [1,9,8,7,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8)],9)
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,10,9,8,7,6,5,4,3,2] => [1,10,9,8,7,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9)],10)
=> ? = 1 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [7,6,5,4,3,2,1,8,9] => [7,6,5,4,3,2,1,8] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [8,7,6,5,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,9] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [8,7,6,5,4,3,2,1,9,10] => [8,7,6,5,4,3,2,1,9] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,1] => [2,3,4,5,6,7,8,1] => ([(1,7),(3,4),(4,6),(5,3),(6,2),(7,5)],8)
=> ? = 6 + 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,9,1] => [3,2,4,5,6,7,8,1] => ([(1,7),(2,7),(4,5),(5,3),(6,4),(7,6)],8)
=> ? = 5 + 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,2,5,6,7,8,9,1] => [3,4,2,5,6,7,8,1] => ([(1,7),(2,4),(4,7),(5,3),(6,5),(7,6)],8)
=> ? = 5 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8] => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,9,8] => [1,2,3,4,5,6,7,8] => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,8,9,7] => [1,2,3,4,5,6,8,7] => ([(0,6),(3,5),(4,3),(5,7),(6,4),(7,1),(7,2)],8)
=> ? = 6 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,9,4] => [1,2,3,5,6,7,8,4] => ([(0,6),(3,5),(4,3),(5,1),(6,7),(7,2),(7,4)],8)
=> ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9] => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,10,9] => [1,2,3,4,5,6,7,8,9] => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 8 + 1
Description
The height of a poset.
This equals the rank of the poset [[St000080]] plus one.
Matching statistic: St000912
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000912: Posets ⟶ ℤResult quality: 70% ●values known / values provided: 90%●distinct values known / distinct values provided: 70%
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000912: Posets ⟶ ℤResult quality: 70% ●values known / values provided: 90%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> [1] => [] => ([],0)
=> ? = 0 + 1
[1,0,1,0]
=> [1,2] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,0,0]
=> [2,1] => [1] => ([],1)
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2] => ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,2] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1] => ([],2)
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => ([],2)
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,1] => ([],2)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => ([(0,1),(0,2)],3)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,1] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,3,1] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1] => ([],3)
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => ([],3)
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,2,1] => ([],3)
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => ([],3)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,3,5,4,7,6,8] => [1,2,3,5,4,7,6] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
=> ? = 4 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,6,5,7,8] => [1,2,4,3,6,5,7] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
=> ? = 4 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,2,4,3,6,7,8,5] => [1,2,4,3,6,7,5] => ([(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(4,3),(6,1)],7)
=> ? = 4 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,2,4,3,7,6,8,5] => [1,2,4,3,7,6,5] => ([(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,1),(3,2)],7)
=> ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,2,4,3,8,7,6,5] => [1,2,4,3,7,6,5] => ([(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,1),(3,2)],7)
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,2,4,6,5,3,7,8] => [1,2,4,6,5,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ? = 4 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,2,5,4,6,7,8,3] => [1,2,5,4,6,7,3] => ([(0,5),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ? = 4 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,2,5,6,4,7,8,3] => [1,2,5,6,4,7,3] => ([(0,5),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ? = 4 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7,8] => [1,3,2,4,5,6,7] => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 5 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,5,7,6,8] => [1,3,2,4,5,7,6] => ([(0,3),(0,4),(3,6),(4,6),(5,1),(5,2),(6,5)],7)
=> ? = 4 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,3,2,4,6,7,8,5] => [1,3,2,4,6,7,5] => ([(0,3),(0,4),(3,6),(4,6),(5,1),(6,2),(6,5)],7)
=> ? = 4 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,6,7,5,8] => [1,3,4,2,6,7,5] => ([(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(6,1)],7)
=> ? = 4 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,3,4,2,6,7,8,5] => [1,3,4,2,6,7,5] => ([(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(6,1)],7)
=> ? = 4 + 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,4,2,7,6,8,5] => [1,3,4,2,7,6,5] => ([(0,2),(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,1)],7)
=> ? = 3 + 1
[1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,4,2,8,7,6,5] => [1,3,4,2,7,6,5] => ([(0,2),(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,1)],7)
=> ? = 3 + 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,4,3,5,6,7,8,2] => [1,4,3,5,6,7,2] => ([(0,2),(0,3),(0,4),(3,6),(4,6),(5,1),(6,5)],7)
=> ? = 4 + 1
[1,1,0,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,1,7,6,5,8,4,3] => [2,1,7,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6)],7)
=> ? = 1 + 1
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,8,7,6,5,4,3] => [2,1,7,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6)],7)
=> ? = 1 + 1
[1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,5,7,6,8] => [2,3,4,1,5,7,6] => ([(0,6),(1,5),(2,6),(5,2),(6,3),(6,4)],7)
=> ? = 4 + 1
[1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [2,3,4,1,5,7,8,6] => [2,3,4,1,5,7,6] => ([(0,6),(1,5),(2,6),(5,2),(6,3),(6,4)],7)
=> ? = 4 + 1
[1,1,0,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,8,7,6] => [2,3,4,1,5,7,6] => ([(0,6),(1,5),(2,6),(5,2),(6,3),(6,4)],7)
=> ? = 4 + 1
[1,1,0,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,4,5,1,8,7,6] => [2,3,4,5,1,7,6] => ([(0,5),(0,6),(1,4),(2,5),(2,6),(3,2),(4,3)],7)
=> ? = 4 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,6,7,1,8] => [2,3,4,5,6,7,1] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 5 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,1] => [2,3,4,5,6,7,1] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 5 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,5,6,8,7,1] => [2,3,4,5,6,7,1] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 5 + 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,4,2,5,6,7,8,1] => [3,4,2,5,6,7,1] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
=> ? = 4 + 1
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,4,5,2,6,7,8,1] => [3,4,5,2,6,7,1] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
=> ? = 4 + 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,8,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7)],8)
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,8,7,6,5,4,3,2,1,10] => [9,8,7,6,5,4,3,2,1] => ([],9)
=> ? = 0 + 1
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,8,9,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7)],8)
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10,9,8,7,6,5,4,3,2] => [1,9,8,7,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8)],9)
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,9,8,7,6,5,4,3,2,1,11] => [10,9,8,7,6,5,4,3,2,1] => ([],10)
=> ? = 0 + 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [7,6,8,5,4,3,2,1,9] => [7,6,8,5,4,3,2,1] => ([(5,7),(6,7)],8)
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,8,7,9,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7)],8)
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,7,9,8,6,5,4,3,2] => [1,7,8,6,5,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(7,1)],8)
=> ? = 2 + 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,9,10,8,7,6,5,4,3,2] => [1,9,8,7,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8)],9)
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,10,9,8,7,6,5,4,3,2] => [1,10,9,8,7,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9)],10)
=> ? = 1 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [7,6,5,4,3,2,1,8,9] => [7,6,5,4,3,2,1,8] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [8,7,6,5,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,9] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [8,7,6,5,4,3,2,1,9,10] => [8,7,6,5,4,3,2,1,9] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,1] => [2,3,4,5,6,7,8,1] => ([(1,7),(3,4),(4,6),(5,3),(6,2),(7,5)],8)
=> ? = 6 + 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,9,1] => [3,2,4,5,6,7,8,1] => ([(1,7),(2,7),(4,5),(5,3),(6,4),(7,6)],8)
=> ? = 5 + 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,2,5,6,7,8,9,1] => [3,4,2,5,6,7,8,1] => ([(1,7),(2,4),(4,7),(5,3),(6,5),(7,6)],8)
=> ? = 5 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8] => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,9,8] => [1,2,3,4,5,6,7,8] => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,8,9,7] => [1,2,3,4,5,6,8,7] => ([(0,6),(3,5),(4,3),(5,7),(6,4),(7,1),(7,2)],8)
=> ? = 6 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,9,4] => [1,2,3,5,6,7,8,4] => ([(0,6),(3,5),(4,3),(5,1),(6,7),(7,2),(7,4)],8)
=> ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9] => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,10,9] => [1,2,3,4,5,6,7,8,9] => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 8 + 1
Description
The number of maximal antichains in a poset.
Matching statistic: St000996
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [] => [] => 0
[1,0,1,0]
=> [1,2] => [1] => [1] => 0
[1,1,0,0]
=> [2,1] => [1] => [1] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2] => [2,1] => 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,2] => [2,1] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1] => [1,2] => 0
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => [1,2] => 0
[1,1,1,0,0,0]
=> [3,2,1] => [2,1] => [1,2] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => [2,3,1] => 2
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => [2,3,1] => 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => [2,1,3] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => [2,1,3] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,2] => [2,1,3] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => [3,2,1] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => [3,2,1] => 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1] => [3,1,2] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,1] => [3,1,2] => 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,3,1] => [3,1,2] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1] => [1,2,3] => 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => [1,2,3] => 0
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,2,1] => [1,2,3] => 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => [1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4] => [2,3,4,1] => 3
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4] => [2,3,4,1] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3] => [2,3,1,4] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => [2,3,1,4] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,3] => [2,3,1,4] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4] => [2,4,3,1] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,4] => [2,4,3,1] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2] => [2,4,1,3] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,2] => [2,4,1,3] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,4,2] => [2,4,1,3] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2] => [2,1,3,4] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,2] => [2,1,3,4] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,3,2] => [2,1,3,4] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,2] => [2,1,3,4] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4] => [3,2,4,1] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,4] => [3,2,4,1] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3] => [3,2,1,4] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => [3,2,1,4] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,3] => [3,2,1,4] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4] => [3,4,2,1] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,4] => [3,4,2,1] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1] => [3,4,1,2] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => [3,4,1,2] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,3,4,1] => [3,4,1,2] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,4,3,1] => [3,1,2,4] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,4,3,1] => [3,1,2,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,4,3,1] => [3,1,2,4] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,4,3,1] => [3,1,2,4] => 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,3,5,4,6,7,8] => [1,2,3,5,4,6,7] => [2,3,4,6,5,7,1] => ? = 5
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,3,5,4,7,6,8] => [1,2,3,5,4,7,6] => [2,3,4,6,5,1,7] => ? = 4
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,3,5,7,6,8,4] => [1,2,3,5,7,6,4] => [2,3,4,6,1,5,7] => ? = 4
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,3,5,7,8,6,4] => [1,2,3,5,7,6,4] => [2,3,4,6,1,5,7] => ? = 4
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,3,5,8,7,6,4] => [1,2,3,5,7,6,4] => [2,3,4,6,1,5,7] => ? = 4
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,3,6,5,7,8,4] => [1,2,3,6,5,7,4] => [2,3,4,7,6,1,5] => ? = 4
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,3,6,5,8,7,4] => [1,2,3,6,5,7,4] => [2,3,4,7,6,1,5] => ? = 4
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,6,5,7,8] => [1,2,4,3,6,5,7] => [2,3,5,4,7,6,1] => ? = 4
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,2,4,3,6,7,8,5] => [1,2,4,3,6,7,5] => [2,3,5,4,7,1,6] => ? = 4
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,2,4,3,7,6,8,5] => [1,2,4,3,7,6,5] => [2,3,5,4,1,6,7] => ? = 3
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,2,4,3,8,7,6,5] => [1,2,4,3,7,6,5] => [2,3,5,4,1,6,7] => ? = 3
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,2,4,6,5,3,7,8] => [1,2,4,6,5,3,7] => [2,3,5,7,6,4,1] => ? = 4
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,2,4,6,7,8,5,3] => [1,2,4,6,7,5,3] => [2,3,5,7,1,4,6] => ? = 4
[1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,2,4,8,7,6,5,3] => [1,2,4,7,6,5,3] => [2,3,5,1,4,6,7] => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,2,5,4,6,7,8,3] => [1,2,5,4,6,7,3] => [2,3,6,5,7,1,4] => ? = 4
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,2,5,6,4,7,8,3] => [1,2,5,6,4,7,3] => [2,3,6,7,5,1,4] => ? = 4
[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,8,4,3] => [1,2,5,6,7,4,3] => [2,3,6,7,1,4,5] => ? = 4
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,2,6,5,4,7,8,3] => [1,2,6,5,4,7,3] => [2,3,7,6,5,1,4] => ? = 3
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,2,6,7,8,5,4,3] => [1,2,6,7,5,4,3] => [2,3,7,1,4,5,6] => ? = 3
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7,8] => [1,3,2,4,5,6,7] => [2,4,3,5,6,7,1] => ? = 5
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,5,7,6,8] => [1,3,2,4,5,7,6] => [2,4,3,5,6,1,7] => ? = 4
[1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,3,2,4,6,7,8,5] => [1,3,2,4,6,7,5] => [2,4,3,5,7,1,6] => ? = 4
[1,0,1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,3,2,4,7,6,8,5] => [1,3,2,4,7,6,5] => [2,4,3,5,1,6,7] => ? = 3
[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,8,7,6,5] => [1,3,2,4,7,6,5] => [2,4,3,5,1,6,7] => ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,5,4,6,7,8] => [1,3,2,5,4,6,7] => [2,4,3,6,5,7,1] => ? = 4
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,7,6,8] => [1,3,2,5,4,7,6] => [2,4,3,6,5,1,7] => ? = 3
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,3,2,8,7,6,5,4] => [1,3,2,7,6,5,4] => [2,4,3,1,5,6,7] => ? = 2
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,6,7,5,8] => [1,3,4,2,6,7,5] => [2,4,5,3,7,1,6] => ? = 4
[1,0,1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,3,4,2,6,7,8,5] => [1,3,4,2,6,7,5] => [2,4,5,3,7,1,6] => ? = 4
[1,0,1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,4,2,7,6,8,5] => [1,3,4,2,7,6,5] => [2,4,5,3,1,6,7] => ? = 3
[1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,4,2,8,7,6,5] => [1,3,4,2,7,6,5] => [2,4,5,3,1,6,7] => ? = 3
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,2] => [1,3,4,5,6,7,2] => [2,4,5,6,7,1,3] => ? = 5
[1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,4,5,6,8,7,2] => [1,3,4,5,6,7,2] => [2,4,5,6,7,1,3] => ? = 5
[1,0,1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,3,4,5,7,6,2,8] => [1,3,4,5,7,6,2] => [2,4,5,6,1,3,7] => ? = 4
[1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,4,5,7,8,6,2] => [1,3,4,5,7,6,2] => [2,4,5,6,1,3,7] => ? = 4
[1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [1,3,4,6,7,5,2,8] => [1,3,4,6,7,5,2] => [2,4,5,7,1,3,6] => ? = 4
[1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,4,8,7,6,5,2] => [1,3,4,7,6,5,2] => [2,4,5,1,3,6,7] => ? = 3
[1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,3,5,6,7,8,4,2] => [1,3,5,6,7,4,2] => [2,4,6,7,1,3,5] => ? = 4
[1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,3,5,7,8,6,4,2] => [1,3,5,7,6,4,2] => [2,4,6,1,3,5,7] => ? = 3
[1,0,1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,3,6,5,7,8,4,2] => [1,3,6,5,7,4,2] => [2,4,7,6,1,3,5] => ? = 3
[1,0,1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [1,4,3,2,6,7,8,5] => [1,4,3,2,6,7,5] => [2,5,4,3,7,1,6] => ? = 3
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,7,6,5,8] => [1,4,3,2,7,6,5] => [2,5,4,3,1,6,7] => ? = 2
[1,0,1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,4,3,2,7,6,8,5] => [1,4,3,2,7,6,5] => [2,5,4,3,1,6,7] => ? = 2
[1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,4,3,2,8,7,6,5] => [1,4,3,2,7,6,5] => [2,5,4,3,1,6,7] => ? = 2
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,4,3,5,6,7,8,2] => [1,4,3,5,6,7,2] => [2,5,4,6,7,1,3] => ? = 4
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,4,3,6,5,8,7,2] => [1,4,3,6,5,7,2] => [2,5,4,7,6,1,3] => ? = 3
[1,0,1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,4,3,7,8,6,5,2] => [1,4,3,7,6,5,2] => [2,5,4,1,3,6,7] => ? = 2
[1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,4,3,8,7,6,5,2] => [1,4,3,7,6,5,2] => [2,5,4,1,3,6,7] => ? = 2
[1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,4,5,3,6,7,8,2] => [1,4,5,3,6,7,2] => [2,5,6,4,7,1,3] => ? = 4
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,6,5,4,3,8,7,2] => [1,6,5,4,3,7,2] => [2,7,6,5,4,1,3] => ? = 2
Description
The number of exclusive left-to-right maxima of a permutation.
This is the number of left-to-right maxima that are not right-to-left minima.
The following 69 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000702The number of weak deficiencies of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000292The number of ascents of a binary word. St000159The number of distinct parts of the integer partition. St000308The height of the tree associated to a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000470The number of runs in a permutation. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000542The number of left-to-right-minima of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000482The (zero)-forcing number of a graph. St001298The number of repeated entries in the Lehmer code of a permutation. St000021The number of descents of a permutation. St000204The number of internal nodes of a binary tree. St000155The number of exceedances (also excedences) of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000314The number of left-to-right-maxima of a permutation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000080The rank of the poset. St000316The number of non-left-to-right-maxima of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000213The number of weak exceedances (also weak excedences) of a permutation. St000325The width of the tree associated to a permutation. St000991The number of right-to-left minima of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000836The number of descents of distance 2 of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000312The number of leaves in a graph. St000742The number of big ascents of a permutation after prepending zero. St000711The number of big exceedences of a permutation. St000619The number of cyclic descents of a permutation. St000837The number of ascents of distance 2 of a permutation. St000024The number of double up and double down steps of a Dyck path. St000636The hull number of a graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001883The mutual visibility number of a graph. St000710The number of big deficiencies of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001737The number of descents of type 2 in a permutation. St001864The number of excedances of a signed permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000236The number of cyclical small weak excedances. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001960The number of descents of a permutation minus one if its first entry is not one. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001964The interval resolution global dimension of a poset. St001556The number of inversions of the third entry of a permutation.
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