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Your data matches 7 different statistics following compositions of up to 3 maps.
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Matching statistic: St000068
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 1
[1,0,1,0]
=> [2,1] => [2,1] => ([],2)
=> 2
[1,1,0,0]
=> [1,2] => [1,2] => ([(0,1)],2)
=> 1
[1,0,1,0,1,0]
=> [3,2,1] => [2,3,1] => ([(1,2)],3)
=> 2
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => ([],3)
=> 3
[1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => ([(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,3,1,2] => ([(2,3)],4)
=> 3
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => ([(2,3)],4)
=> 3
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,4,2,1] => ([(2,3)],4)
=> 3
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => ([],4)
=> 4
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,4,3,1,2] => ([(3,4)],5)
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,5,3,2,1] => ([(3,4)],5)
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => ([],5)
=> 5
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [5,4,1,3,2] => ([(2,3),(2,4)],5)
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
Description
The number of minimal elements in a poset.
Matching statistic: St000542
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 75%
Mp00239: Permutations —Corteel⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> [1] => [1] => 1
[1,0,1,0]
=> [2,1] => [2,1] => 2
[1,1,0,0]
=> [1,2] => [1,2] => 1
[1,0,1,0,1,0]
=> [3,2,1] => [2,3,1] => 2
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => 3
[1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => 2
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [3,4,1,2] => 2
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,3,1,2] => 3
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,3,4,1] => 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,4,3,1] => 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => 3
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,4,2,1] => 3
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => 4
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,4,1,3] => 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,3,1,4] => 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,1,4] => 3
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,1,2,4] => 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [3,4,5,1,2] => 2
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,3,5,1,2] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [3,5,4,1,2] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,5,3,1,2] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,4,3,1,2] => 4
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,5,1,2,3] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,4,1,2,3] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,4,1,5,2] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [3,5,1,4,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,3,1,4,2] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,4,5,1] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,3,5,4,1] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,5,3,4,1] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 3
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [3,4,5,2,1] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [4,3,5,2,1] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,5,4,2,1] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,5,3,2,1] => 4
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => 5
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,5,1,3,2] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [5,4,1,3,2] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [3,5,1,2,4] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [3,4,1,2,5] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,3,1,2,5] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,3,5,1,4] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,3,4,1,5] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [2,4,3,1,5] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => 3
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,2,1] => [5,6,7,4,1,2,3] => ? = 3
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,2,1] => [6,7,5,4,1,2,3] => ? = 4
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [7,6,5,4,1,2,3] => ? = 5
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,4,2,1] => [6,7,5,1,2,3,4] => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,5,2,1] => [4,5,6,1,7,2,3] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,2,1] => [4,6,7,1,5,2,3] => ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,7,2,1] => [6,4,7,1,5,2,3] => ? = 3
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,6,2,1] => [4,7,5,1,6,2,3] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,2,1] => [3,7,4,5,6,1,2] => ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => [7,6,3,4,5,1,2] => ? = 4
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,2,3,1] => [6,5,7,4,2,1,3] => ? = 5
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,2,3,1] => [5,7,6,4,2,1,3] => ? = 4
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,2,3,1] => [6,7,5,4,2,1,3] => ? = 5
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,2,3,1] => [7,6,5,4,2,1,3] => ? = 6
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,5,1] => [4,6,7,1,2,3,5] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,2,5,1] => [6,4,7,1,2,3,5] => ? = 3
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,5,3,4,2,6,1] => [3,4,5,6,1,7,2] => ? = 2
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,2,7,1] => [3,7,4,5,1,6,2] => ? = 2
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [6,5,7,2,3,4,1] => [6,7,5,2,3,1,4] => ? = 4
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [4,5,6,2,3,7,1] => [7,5,4,2,1,6,3] => ? = 5
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,7,3,2,4,5,1] => [6,3,7,1,2,4,5] => ? = 3
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [6,7,2,3,4,5,1] => [7,6,1,2,3,4,5] => ? = 3
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [7,5,2,3,4,6,1] => [5,6,1,2,3,7,4] => ? = 2
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,5,6,1] => [2,3,4,5,6,7,1] => ? = 2
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,2,3,4,6,7,1] => [2,3,4,7,5,6,1] => ? = 2
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,7,1] => [2,3,7,4,5,6,1] => ? = 2
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [7,2,3,4,5,6,1] => ? = 3
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,1,2] => [6,5,7,4,1,3,2] => ? = 4
[1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,1,2] => [5,7,6,4,1,3,2] => ? = 3
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,1,2] => [6,7,5,4,1,3,2] => ? = 4
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,1,2] => [7,6,5,4,1,3,2] => ? = 5
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,7,1,2] => [7,6,4,1,5,3,2] => ? = 4
[1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,5,1,2] => [3,4,5,6,7,2,1] => ? = 3
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,1,2] => [6,7,3,4,5,2,1] => ? = 4
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,1,2] => [7,6,3,4,5,2,1] => ? = 5
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,1,3] => [4,5,6,7,2,3,1] => ? = 3
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,2,1,3] => [5,4,6,7,2,3,1] => ? = 4
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,2,1,3] => [5,6,4,7,2,3,1] => ? = 4
[1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,2,1,3] => [6,5,4,7,2,3,1] => ? = 5
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,2,1,3] => [5,6,7,4,2,3,1] => ? = 4
[1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,2,1,3] => [6,5,7,4,2,3,1] => ? = 5
[1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,2,1,3] => [5,7,6,4,2,3,1] => ? = 4
[1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,2,1,3] => [6,7,5,4,2,3,1] => ? = 5
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,2,1,3] => [7,6,5,4,2,3,1] => ? = 6
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1,7] => [4,5,6,1,2,3,7] => ? = 2
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,2,1,7] => [5,4,6,1,2,3,7] => ? = 3
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,2,1,6] => [4,7,5,1,2,3,6] => ? = 2
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,2,1,7] => [5,6,4,1,2,3,7] => ? = 3
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,2,1,7] => [6,5,4,1,2,3,7] => ? = 4
[1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,2,1,5] => [3,4,6,7,1,5,2] => ? = 2
Description
The number of left-to-right-minima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a left-to-right-minimum if there does not exist a j < i such that $\sigma_j < \sigma_i$.
Matching statistic: St000541
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000541: Permutations ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 75%
Mp00239: Permutations —Corteel⟶ Permutations
St000541: Permutations ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> [1] => [1] => ? = 1 - 1
[1,0,1,0]
=> [2,1] => [2,1] => 1 = 2 - 1
[1,1,0,0]
=> [1,2] => [1,2] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [3,2,1] => [2,3,1] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => 2 = 3 - 1
[1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [3,4,1,2] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,3,1,2] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,3,4,1] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,4,3,1] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,4,2,1] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,4,1,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,3,1,4] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,1,4] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,1,2,4] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [3,4,5,1,2] => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,3,5,1,2] => 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [3,5,4,1,2] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,5,3,1,2] => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,4,3,1,2] => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,5,1,2,3] => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,4,1,2,3] => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,4,1,5,2] => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [3,5,1,4,2] => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,3,1,4,2] => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,4,5,1] => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,3,5,4,1] => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,5,3,4,1] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [3,4,5,2,1] => 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [4,3,5,2,1] => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,5,4,2,1] => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,5,3,2,1] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,5,1,3,2] => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [5,4,1,3,2] => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [3,5,1,2,4] => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [3,4,1,2,5] => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,3,1,2,5] => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,3,5,1,4] => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,3,4,1,5] => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [2,4,3,1,5] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => 2 = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [4,5,2,3,1] => 2 = 3 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,2,1] => [5,6,7,4,1,2,3] => ? = 3 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,2,1] => [6,7,5,4,1,2,3] => ? = 4 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [7,6,5,4,1,2,3] => ? = 5 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,4,2,1] => [6,7,5,1,2,3,4] => ? = 3 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,5,2,1] => [4,5,6,1,7,2,3] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,2,1] => [4,6,7,1,5,2,3] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,7,2,1] => [6,4,7,1,5,2,3] => ? = 3 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,6,2,1] => [4,7,5,1,6,2,3] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,2,1] => [3,7,4,5,6,1,2] => ? = 2 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => [7,6,3,4,5,1,2] => ? = 4 - 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,2,3,1] => [6,5,7,4,2,1,3] => ? = 5 - 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,2,3,1] => [5,7,6,4,2,1,3] => ? = 4 - 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,2,3,1] => [6,7,5,4,2,1,3] => ? = 5 - 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,2,3,1] => [7,6,5,4,2,1,3] => ? = 6 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,5,1] => [4,6,7,1,2,3,5] => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,2,5,1] => [6,4,7,1,2,3,5] => ? = 3 - 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,5,3,4,2,6,1] => [3,4,5,6,1,7,2] => ? = 2 - 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,2,7,1] => [3,7,4,5,1,6,2] => ? = 2 - 1
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [6,5,7,2,3,4,1] => [6,7,5,2,3,1,4] => ? = 4 - 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [4,5,6,2,3,7,1] => [7,5,4,2,1,6,3] => ? = 5 - 1
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,7,3,2,4,5,1] => [6,3,7,1,2,4,5] => ? = 3 - 1
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [6,7,2,3,4,5,1] => [7,6,1,2,3,4,5] => ? = 3 - 1
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [7,5,2,3,4,6,1] => [5,6,1,2,3,7,4] => ? = 2 - 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,5,6,1] => [2,3,4,5,6,7,1] => ? = 2 - 1
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,2,3,4,6,7,1] => [2,3,4,7,5,6,1] => ? = 2 - 1
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,7,1] => [2,3,7,4,5,6,1] => ? = 2 - 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [7,2,3,4,5,6,1] => ? = 3 - 1
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,1,2] => [6,5,7,4,1,3,2] => ? = 4 - 1
[1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,1,2] => [5,7,6,4,1,3,2] => ? = 3 - 1
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,1,2] => [6,7,5,4,1,3,2] => ? = 4 - 1
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,1,2] => [7,6,5,4,1,3,2] => ? = 5 - 1
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,7,1,2] => [7,6,4,1,5,3,2] => ? = 4 - 1
[1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,5,1,2] => [3,4,5,6,7,2,1] => ? = 3 - 1
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,1,2] => [6,7,3,4,5,2,1] => ? = 4 - 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,1,2] => [7,6,3,4,5,2,1] => ? = 5 - 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,1,3] => [4,5,6,7,2,3,1] => ? = 3 - 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,2,1,3] => [5,4,6,7,2,3,1] => ? = 4 - 1
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,2,1,3] => [5,6,4,7,2,3,1] => ? = 4 - 1
[1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,2,1,3] => [6,5,4,7,2,3,1] => ? = 5 - 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,2,1,3] => [5,6,7,4,2,3,1] => ? = 4 - 1
[1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,2,1,3] => [6,5,7,4,2,3,1] => ? = 5 - 1
[1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,2,1,3] => [5,7,6,4,2,3,1] => ? = 4 - 1
[1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,2,1,3] => [6,7,5,4,2,3,1] => ? = 5 - 1
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,2,1,3] => [7,6,5,4,2,3,1] => ? = 6 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1,7] => [4,5,6,1,2,3,7] => ? = 2 - 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,2,1,7] => [5,4,6,1,2,3,7] => ? = 3 - 1
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,2,1,6] => [4,7,5,1,2,3,6] => ? = 2 - 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,2,1,7] => [5,6,4,1,2,3,7] => ? = 3 - 1
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,2,1,7] => [6,5,4,1,2,3,7] => ? = 4 - 1
Description
The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right.
For a permutation $\pi$ of length $n$, this is the number of indices $2 \leq j \leq n$ such that for all $1 \leq i < j$, the pair $(i,j)$ is an inversion of $\pi$.
Matching statistic: St000314
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 75%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> [1] => [1] => [1] => 1
[1,0,1,0]
=> [2,1] => [2,1] => [1,2] => 2
[1,1,0,0]
=> [1,2] => [1,2] => [2,1] => 1
[1,0,1,0,1,0]
=> [3,2,1] => [2,3,1] => [2,1,3] => 2
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => [1,2,3] => 3
[1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => [1,3,2] => 2
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => [2,3,1] => 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [3,2,1] => 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [3,4,1,2] => [2,1,4,3] => 2
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,3,1,2] => [1,2,4,3] => 3
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,3,4,1] => [3,2,1,4] => 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,4,3,1] => [3,1,2,4] => 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => [1,3,2,4] => 3
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,4,2,1] => [2,1,3,4] => 3
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => [1,2,3,4] => 4
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,4,1,3] => [3,1,4,2] => 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,3,1,4] => [3,2,4,1] => 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,1,4] => [2,3,4,1] => 3
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => [1,4,3,2] => 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,1,2,4] => [2,4,3,1] => 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [3,4,5,1,2] => [3,2,1,5,4] => 2
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,3,5,1,2] => [2,3,1,5,4] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [3,5,4,1,2] => [3,1,2,5,4] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,5,3,1,2] => [2,1,3,5,4] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,4,3,1,2] => [1,2,3,5,4] => 4
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,5,1,2,3] => [2,1,5,4,3] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,4,1,2,3] => [1,2,5,4,3] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,4,1,5,2] => [3,2,5,1,4] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [3,5,1,4,2] => [3,1,5,2,4] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,3,1,4,2] => [1,3,5,2,4] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,4,5,1] => [4,3,2,1,5] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,3,5,4,1] => [4,3,1,2,5] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,5,3,4,1] => [4,1,3,2,5] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [1,4,3,2,5] => 3
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [3,4,5,2,1] => [3,2,1,4,5] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [4,3,5,2,1] => [2,3,1,4,5] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,5,4,2,1] => [3,1,2,4,5] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,5,3,2,1] => [2,1,3,4,5] => 4
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => 5
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,5,1,3,2] => [2,1,5,3,4] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [5,4,1,3,2] => [1,2,5,3,4] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [3,5,1,2,4] => [3,1,5,4,2] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [3,4,1,2,5] => [3,2,5,4,1] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,3,1,2,5] => [2,3,5,4,1] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,3,5,1,4] => [4,3,1,5,2] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,3,4,1,5] => [4,3,2,5,1] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [2,4,3,1,5] => [4,2,3,5,1] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [2,4,3,5,1] => 3
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,2,1] => [5,6,7,4,1,2,3] => [3,2,1,4,7,6,5] => ? = 3
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,2,1] => [6,7,5,4,1,2,3] => [2,1,3,4,7,6,5] => ? = 4
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [7,6,5,4,1,2,3] => [1,2,3,4,7,6,5] => ? = 5
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,4,2,1] => [6,7,5,1,2,3,4] => [2,1,3,7,6,5,4] => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,5,2,1] => [4,5,6,1,7,2,3] => [4,3,2,7,1,6,5] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,2,1] => [4,6,7,1,5,2,3] => [4,2,1,7,3,6,5] => ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,7,2,1] => [6,4,7,1,5,2,3] => [2,4,1,7,3,6,5] => ? = 3
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,6,2,1] => [4,7,5,1,6,2,3] => [4,1,3,7,2,6,5] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,2,1] => [3,7,4,5,6,1,2] => [5,1,4,3,2,7,6] => ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => [7,6,3,4,5,1,2] => [1,2,5,4,3,7,6] => ? = 4
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,2,3,1] => [6,5,7,4,2,1,3] => [2,3,1,4,6,7,5] => ? = 5
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,2,3,1] => [5,7,6,4,2,1,3] => [3,1,2,4,6,7,5] => ? = 4
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,2,3,1] => [6,7,5,4,2,1,3] => [2,1,3,4,6,7,5] => ? = 5
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,2,3,1] => [7,6,5,4,2,1,3] => [1,2,3,4,6,7,5] => ? = 6
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,5,1] => [4,6,7,1,2,3,5] => [4,2,1,7,6,5,3] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,2,5,1] => [6,4,7,1,2,3,5] => [2,4,1,7,6,5,3] => ? = 3
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,5,3,4,2,6,1] => [3,4,5,6,1,7,2] => [5,4,3,2,7,1,6] => ? = 2
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,2,7,1] => [3,7,4,5,1,6,2] => [5,1,4,3,7,2,6] => ? = 2
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [6,5,7,2,3,4,1] => [6,7,5,2,3,1,4] => [2,1,3,6,5,7,4] => ? = 4
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [4,5,6,2,3,7,1] => [7,5,4,2,1,6,3] => [1,3,4,6,7,2,5] => ? = 5
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,7,3,2,4,5,1] => [6,3,7,1,2,4,5] => [2,5,1,7,6,4,3] => ? = 3
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [6,7,2,3,4,5,1] => [7,6,1,2,3,4,5] => [1,2,7,6,5,4,3] => ? = 3
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [7,5,2,3,4,6,1] => [5,6,1,2,3,7,4] => [3,2,7,6,5,1,4] => ? = 2
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,5,6,1] => [2,3,4,5,6,7,1] => [6,5,4,3,2,1,7] => ? = 2
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,2,3,4,6,7,1] => [2,3,4,7,5,6,1] => [6,5,4,1,3,2,7] => ? = 2
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,7,1] => [2,3,7,4,5,6,1] => [6,5,1,4,3,2,7] => ? = 2
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [7,2,3,4,5,6,1] => [1,6,5,4,3,2,7] => ? = 3
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,1,2] => [6,5,7,4,1,3,2] => [2,3,1,4,7,5,6] => ? = 4
[1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,1,2] => [5,7,6,4,1,3,2] => [3,1,2,4,7,5,6] => ? = 3
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,1,2] => [6,7,5,4,1,3,2] => [2,1,3,4,7,5,6] => ? = 4
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,1,2] => [7,6,5,4,1,3,2] => [1,2,3,4,7,5,6] => ? = 5
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,7,1,2] => [7,6,4,1,5,3,2] => [1,2,4,7,3,5,6] => ? = 4
[1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,5,1,2] => [3,4,5,6,7,2,1] => [5,4,3,2,1,6,7] => ? = 3
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,1,2] => [6,7,3,4,5,2,1] => [2,1,5,4,3,6,7] => ? = 4
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,1,2] => [7,6,3,4,5,2,1] => [1,2,5,4,3,6,7] => ? = 5
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,1,3] => [4,5,6,7,2,3,1] => [4,3,2,1,6,5,7] => ? = 3
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,2,1,3] => [5,4,6,7,2,3,1] => [3,4,2,1,6,5,7] => ? = 4
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,2,1,3] => [5,6,4,7,2,3,1] => [3,2,4,1,6,5,7] => ? = 4
[1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,2,1,3] => [6,5,4,7,2,3,1] => [2,3,4,1,6,5,7] => ? = 5
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,2,1,3] => [5,6,7,4,2,3,1] => [3,2,1,4,6,5,7] => ? = 4
[1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,2,1,3] => [6,5,7,4,2,3,1] => [2,3,1,4,6,5,7] => ? = 5
[1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,2,1,3] => [5,7,6,4,2,3,1] => [3,1,2,4,6,5,7] => ? = 4
[1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,2,1,3] => [6,7,5,4,2,3,1] => [2,1,3,4,6,5,7] => ? = 5
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,2,1,3] => [7,6,5,4,2,3,1] => [1,2,3,4,6,5,7] => ? = 6
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1,7] => [4,5,6,1,2,3,7] => [4,3,2,7,6,5,1] => ? = 2
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,2,1,7] => [5,4,6,1,2,3,7] => [3,4,2,7,6,5,1] => ? = 3
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,2,1,6] => [4,7,5,1,2,3,6] => [4,1,3,7,6,5,2] => ? = 2
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,2,1,7] => [5,6,4,1,2,3,7] => [3,2,4,7,6,5,1] => ? = 3
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,2,1,7] => [6,5,4,1,2,3,7] => [2,3,4,7,6,5,1] => ? = 4
[1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,2,1,5] => [3,4,6,7,1,5,2] => [5,4,2,1,7,3,6] => ? = 2
Description
The number of left-to-right-maxima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a '''left-to-right-maximum''' if there does not exist a $j < i$ such that $\sigma_j > \sigma_i$.
This is also the number of weak exceedences of a permutation that are not mid-points of a decreasing subsequence of length 3, see [1] for more on the later description.
Matching statistic: St000991
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 75%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> [1] => [1] => [1] => 1
[1,0,1,0]
=> [2,1] => [2,1] => [1,2] => 2
[1,1,0,0]
=> [1,2] => [1,2] => [2,1] => 1
[1,0,1,0,1,0]
=> [3,2,1] => [2,3,1] => [1,3,2] => 2
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => [1,2,3] => 3
[1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => [2,1,3] => 2
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => [3,1,2] => 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [3,2,1] => 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [3,4,1,2] => [2,1,4,3] => 2
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,3,1,2] => [2,1,3,4] => 3
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,3,4,1] => [1,4,3,2] => 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,4,3,1] => [1,3,4,2] => 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => [1,3,2,4] => 3
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,4,2,1] => [1,2,4,3] => 3
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => [1,2,3,4] => 4
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,4,1,3] => [3,1,4,2] => 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,3,1,4] => [4,1,3,2] => 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,1,4] => [4,1,2,3] => 3
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => [3,2,1,4] => 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,1,2,4] => [4,2,1,3] => 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [4,3,1,2] => 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [3,4,5,1,2] => [2,1,5,4,3] => 2
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,3,5,1,2] => [2,1,5,3,4] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [3,5,4,1,2] => [2,1,4,5,3] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,5,3,1,2] => [2,1,3,5,4] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,4,3,1,2] => [2,1,3,4,5] => 4
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,5,1,2,3] => [3,2,1,5,4] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,4,1,2,3] => [3,2,1,4,5] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,4,1,5,2] => [2,5,1,4,3] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [3,5,1,4,2] => [2,4,1,5,3] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,3,1,4,2] => [2,4,1,3,5] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,4,5,1] => [1,5,4,3,2] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,3,5,4,1] => [1,4,5,3,2] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,5,3,4,1] => [1,4,3,5,2] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [1,4,3,2,5] => 3
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [3,4,5,2,1] => [1,2,5,4,3] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [4,3,5,2,1] => [1,2,5,3,4] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,5,4,2,1] => [1,2,4,5,3] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,5,3,2,1] => [1,2,3,5,4] => 4
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => 5
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,5,1,3,2] => [2,3,1,5,4] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [5,4,1,3,2] => [2,3,1,4,5] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [3,5,1,2,4] => [4,2,1,5,3] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [3,4,1,2,5] => [5,2,1,4,3] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,3,1,2,5] => [5,2,1,3,4] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,3,5,1,4] => [4,1,5,3,2] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,3,4,1,5] => [5,1,4,3,2] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [2,4,3,1,5] => [5,1,3,4,2] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [5,1,3,2,4] => 3
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,2,1] => [5,6,7,4,1,2,3] => [3,2,1,4,7,6,5] => ? = 3
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,2,1] => [6,7,5,4,1,2,3] => [3,2,1,4,5,7,6] => ? = 4
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [7,6,5,4,1,2,3] => [3,2,1,4,5,6,7] => ? = 5
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,4,2,1] => [6,7,5,1,2,3,4] => [4,3,2,1,5,7,6] => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,5,2,1] => [4,5,6,1,7,2,3] => [3,2,7,1,6,5,4] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,2,1] => [4,6,7,1,5,2,3] => [3,2,5,1,7,6,4] => ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,7,2,1] => [6,4,7,1,5,2,3] => [3,2,5,1,7,4,6] => ? = 3
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,6,2,1] => [4,7,5,1,6,2,3] => [3,2,6,1,5,7,4] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,2,1] => [3,7,4,5,6,1,2] => [2,1,6,5,4,7,3] => ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => [7,6,3,4,5,1,2] => [2,1,5,4,3,6,7] => ? = 4
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,2,3,1] => [6,5,7,4,2,1,3] => [3,1,2,4,7,5,6] => ? = 5
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,2,3,1] => [5,7,6,4,2,1,3] => [3,1,2,4,6,7,5] => ? = 4
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,2,3,1] => [6,7,5,4,2,1,3] => [3,1,2,4,5,7,6] => ? = 5
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,2,3,1] => [7,6,5,4,2,1,3] => [3,1,2,4,5,6,7] => ? = 6
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,5,1] => [4,6,7,1,2,3,5] => [5,3,2,1,7,6,4] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,2,5,1] => [6,4,7,1,2,3,5] => [5,3,2,1,7,4,6] => ? = 3
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,5,3,4,2,6,1] => [3,4,5,6,1,7,2] => [2,7,1,6,5,4,3] => ? = 2
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,2,7,1] => [3,7,4,5,1,6,2] => [2,6,1,5,4,7,3] => ? = 2
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [6,5,7,2,3,4,1] => [6,7,5,2,3,1,4] => [4,1,3,2,5,7,6] => ? = 4
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [4,5,6,2,3,7,1] => [7,5,4,2,1,6,3] => [3,6,1,2,4,5,7] => ? = 5
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,7,3,2,4,5,1] => [6,3,7,1,2,4,5] => [5,4,2,1,7,3,6] => ? = 3
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [6,7,2,3,4,5,1] => [7,6,1,2,3,4,5] => [5,4,3,2,1,6,7] => ? = 3
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [7,5,2,3,4,6,1] => [5,6,1,2,3,7,4] => [4,7,3,2,1,6,5] => ? = 2
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,5,6,1] => [2,3,4,5,6,7,1] => [1,7,6,5,4,3,2] => ? = 2
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,2,3,4,6,7,1] => [2,3,4,7,5,6,1] => [1,6,5,7,4,3,2] => ? = 2
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,7,1] => [2,3,7,4,5,6,1] => [1,6,5,4,7,3,2] => ? = 2
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [7,2,3,4,5,6,1] => [1,6,5,4,3,2,7] => ? = 3
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,1,2] => [6,5,7,4,1,3,2] => [2,3,1,4,7,5,6] => ? = 4
[1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,1,2] => [5,7,6,4,1,3,2] => [2,3,1,4,6,7,5] => ? = 3
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,1,2] => [6,7,5,4,1,3,2] => [2,3,1,4,5,7,6] => ? = 4
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,1,2] => [7,6,5,4,1,3,2] => [2,3,1,4,5,6,7] => ? = 5
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,7,1,2] => [7,6,4,1,5,3,2] => [2,3,5,1,4,6,7] => ? = 4
[1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,5,1,2] => [3,4,5,6,7,2,1] => [1,2,7,6,5,4,3] => ? = 3
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,1,2] => [6,7,3,4,5,2,1] => [1,2,5,4,3,7,6] => ? = 4
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,1,2] => [7,6,3,4,5,2,1] => [1,2,5,4,3,6,7] => ? = 5
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,1,3] => [4,5,6,7,2,3,1] => [1,3,2,7,6,5,4] => ? = 3
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,2,1,3] => [5,4,6,7,2,3,1] => [1,3,2,7,6,4,5] => ? = 4
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,2,1,3] => [5,6,4,7,2,3,1] => [1,3,2,7,4,6,5] => ? = 4
[1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,2,1,3] => [6,5,4,7,2,3,1] => [1,3,2,7,4,5,6] => ? = 5
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,2,1,3] => [5,6,7,4,2,3,1] => [1,3,2,4,7,6,5] => ? = 4
[1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,2,1,3] => [6,5,7,4,2,3,1] => [1,3,2,4,7,5,6] => ? = 5
[1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,2,1,3] => [5,7,6,4,2,3,1] => [1,3,2,4,6,7,5] => ? = 4
[1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,2,1,3] => [6,7,5,4,2,3,1] => [1,3,2,4,5,7,6] => ? = 5
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,2,1,3] => [7,6,5,4,2,3,1] => [1,3,2,4,5,6,7] => ? = 6
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1,7] => [4,5,6,1,2,3,7] => [7,3,2,1,6,5,4] => ? = 2
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,2,1,7] => [5,4,6,1,2,3,7] => [7,3,2,1,6,4,5] => ? = 3
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,2,1,6] => [4,7,5,1,2,3,6] => [6,3,2,1,5,7,4] => ? = 2
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,2,1,7] => [5,6,4,1,2,3,7] => [7,3,2,1,4,6,5] => ? = 3
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,2,1,7] => [6,5,4,1,2,3,7] => [7,3,2,1,4,5,6] => ? = 4
[1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,2,1,5] => [3,4,6,7,1,5,2] => [2,5,1,7,6,4,3] => ? = 2
Description
The number of right-to-left minima of a permutation.
For the number of left-to-right maxima, see [[St000314]].
Matching statistic: St000061
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000061: Binary trees ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 75%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000061: Binary trees ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> [1] => [1] => [.,.]
=> ? = 1
[1,0,1,0]
=> [2,1] => [2,1] => [[.,.],.]
=> 2
[1,1,0,0]
=> [1,2] => [1,2] => [.,[.,.]]
=> 1
[1,0,1,0,1,0]
=> [3,2,1] => [2,3,1] => [[.,.],[.,.]]
=> 2
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => [[[.,.],.],.]
=> 3
[1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => [[.,[.,.]],.]
=> 2
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 2
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,3,1,2] => [[[.,[.,.]],.],.]
=> 3
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> 3
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,4,2,1] => [[[.,.],.],[.,.]]
=> 3
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => [[[[.,.],.],.],.]
=> 4
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> 3
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => [[.,[.,[.,.]]],.]
=> 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,1,2,4] => [[.,[.,.]],[.,.]]
=> 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [3,5,4,1,2] => [[.,[.,.]],[[.,.],.]]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,5,1,2,3] => [[.,[.,[.,.]]],[.,.]]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [3,5,1,4,2] => [[.,[.,.]],[[.,.],.]]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,3,1,4,2] => [[[.,[.,.]],[.,.]],.]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,5,3,4,1] => [[.,.],[[.,[.,.]],.]]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,5,4,2,1] => [[[.,.],.],[[.,.],.]]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> 5
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,5,1,3,2] => [[.,[[.,.],.]],[.,.]]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [5,4,1,3,2] => [[[.,[[.,.],.]],.],.]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,3,1,2,5] => [[[.,[.,.]],.],[.,.]]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [2,4,3,1,5] => [[.,.],[[.,.],[.,.]]]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [4,5,2,3,1] => [[[.,.],[.,.]],[.,.]]
=> 3
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,2,1] => [5,6,7,4,1,2,3] => [[[.,[.,[.,.]]],.],[.,[.,.]]]
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,2,1] => [6,7,5,4,1,2,3] => [[[[.,[.,[.,.]]],.],.],[.,.]]
=> ? = 4
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [7,6,5,4,1,2,3] => [[[[[.,[.,[.,.]]],.],.],.],.]
=> ? = 5
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,4,2,1] => [6,7,5,1,2,3,4] => [[[.,[.,[.,[.,.]]]],.],[.,.]]
=> ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,5,2,1] => [4,5,6,1,7,2,3] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,2,1] => [4,6,7,1,5,2,3] => [[.,[.,[.,.]]],[[.,.],[.,.]]]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,7,2,1] => [6,4,7,1,5,2,3] => [[[.,[.,[.,.]]],[.,.]],[.,.]]
=> ? = 3
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,6,2,1] => [4,7,5,1,6,2,3] => [[.,[.,[.,.]]],[[.,[.,.]],.]]
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,2,1] => [3,7,4,5,6,1,2] => [[.,[.,.]],[[.,[.,[.,.]]],.]]
=> ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => [7,6,3,4,5,1,2] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> ? = 4
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,2,3,1] => [6,5,7,4,2,1,3] => [[[[[.,.],[.,.]],.],.],[.,.]]
=> ? = 5
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,2,3,1] => [5,7,6,4,2,1,3] => [[[[.,.],[.,.]],.],[[.,.],.]]
=> ? = 4
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,2,3,1] => [6,7,5,4,2,1,3] => [[[[[.,.],[.,.]],.],.],[.,.]]
=> ? = 5
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,2,3,1] => [7,6,5,4,2,1,3] => [[[[[[.,.],[.,.]],.],.],.],.]
=> ? = 6
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,5,1] => [4,6,7,1,2,3,5] => [[.,[.,[.,.]]],[[.,.],[.,.]]]
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,2,5,1] => [6,4,7,1,2,3,5] => [[[.,[.,[.,.]]],[.,.]],[.,.]]
=> ? = 3
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,5,3,4,2,6,1] => [3,4,5,6,1,7,2] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> ? = 2
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,2,7,1] => [3,7,4,5,1,6,2] => [[.,[.,.]],[[.,[.,[.,.]]],.]]
=> ? = 2
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [6,5,7,2,3,4,1] => [6,7,5,2,3,1,4] => [[[[.,.],[.,[.,.]]],.],[.,.]]
=> ? = 4
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [4,5,6,2,3,7,1] => [7,5,4,2,1,6,3] => [[[[[.,.],[.,.]],.],[.,.]],.]
=> ? = 5
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,7,3,2,4,5,1] => [6,3,7,1,2,4,5] => [[[.,[.,.]],[.,[.,.]]],[.,.]]
=> ? = 3
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [6,7,2,3,4,5,1] => [7,6,1,2,3,4,5] => [[[.,[.,[.,[.,[.,.]]]]],.],.]
=> ? = 3
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [7,5,2,3,4,6,1] => [5,6,1,2,3,7,4] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> ? = 2
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,5,6,1] => [2,3,4,5,6,7,1] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 2
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,2,3,4,6,7,1] => [2,3,4,7,5,6,1] => [[.,.],[.,[.,[[.,[.,.]],.]]]]
=> ? = 2
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,7,1] => [2,3,7,4,5,6,1] => [[.,.],[.,[[.,[.,[.,.]]],.]]]
=> ? = 2
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [7,2,3,4,5,6,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> ? = 3
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,1,2] => [6,5,7,4,1,3,2] => [[[[.,[[.,.],.]],.],.],[.,.]]
=> ? = 4
[1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,1,2] => [5,7,6,4,1,3,2] => [[[.,[[.,.],.]],.],[[.,.],.]]
=> ? = 3
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,1,2] => [6,7,5,4,1,3,2] => [[[[.,[[.,.],.]],.],.],[.,.]]
=> ? = 4
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,1,2] => [7,6,5,4,1,3,2] => [[[[[.,[[.,.],.]],.],.],.],.]
=> ? = 5
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,7,1,2] => [7,6,4,1,5,3,2] => [[[[.,[[.,.],.]],[.,.]],.],.]
=> ? = 4
[1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,5,1,2] => [3,4,5,6,7,2,1] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> ? = 3
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,1,2] => [6,7,3,4,5,2,1] => [[[[.,.],.],[.,[.,.]]],[.,.]]
=> ? = 4
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,1,2] => [7,6,3,4,5,2,1] => [[[[[.,.],.],[.,[.,.]]],.],.]
=> ? = 5
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,1,3] => [4,5,6,7,2,3,1] => [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> ? = 3
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,2,1,3] => [5,4,6,7,2,3,1] => [[[[.,.],[.,.]],.],[.,[.,.]]]
=> ? = 4
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,2,1,3] => [5,6,4,7,2,3,1] => [[[[.,.],[.,.]],.],[.,[.,.]]]
=> ? = 4
[1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,2,1,3] => [6,5,4,7,2,3,1] => [[[[[.,.],[.,.]],.],.],[.,.]]
=> ? = 5
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,2,1,3] => [5,6,7,4,2,3,1] => [[[[.,.],[.,.]],.],[.,[.,.]]]
=> ? = 4
[1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,2,1,3] => [6,5,7,4,2,3,1] => [[[[[.,.],[.,.]],.],.],[.,.]]
=> ? = 5
[1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,2,1,3] => [5,7,6,4,2,3,1] => [[[[.,.],[.,.]],.],[[.,.],.]]
=> ? = 4
[1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,2,1,3] => [6,7,5,4,2,3,1] => [[[[[.,.],[.,.]],.],.],[.,.]]
=> ? = 5
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,2,1,3] => [7,6,5,4,2,3,1] => [[[[[[.,.],[.,.]],.],.],.],.]
=> ? = 6
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1,7] => [4,5,6,1,2,3,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,2,1,7] => [5,4,6,1,2,3,7] => [[[.,[.,[.,.]]],.],[.,[.,.]]]
=> ? = 3
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,2,1,6] => [4,7,5,1,2,3,6] => [[.,[.,[.,.]]],[[.,[.,.]],.]]
=> ? = 2
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,2,1,7] => [5,6,4,1,2,3,7] => [[[.,[.,[.,.]]],.],[.,[.,.]]]
=> ? = 3
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,2,1,7] => [6,5,4,1,2,3,7] => [[[[.,[.,[.,.]]],.],.],[.,.]]
=> ? = 4
Description
The number of nodes on the left branch of a binary tree.
Also corresponds to [[/StatisticsDatabase/St000011/|ST000011]] after applying the [[/BinaryTrees#Maps|Tamari bijection]] between binary trees and Dyck path.
Matching statistic: St001816
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001816: Standard tableaux ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 38%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001816: Standard tableaux ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 38%
Values
[1,0]
=> []
=> []
=> ?
=> ? = 1 - 2
[1,0,1,0]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 0 = 2 - 2
[1,1,0,0]
=> []
=> []
=> ?
=> ? = 1 - 2
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0 = 2 - 2
[1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 1 = 3 - 2
[1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0 = 2 - 2
[1,1,0,1,0,0]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 0 = 2 - 2
[1,1,1,0,0,0]
=> []
=> []
=> ?
=> ? = 1 - 2
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 2 - 2
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> ? = 3 - 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> ? = 2 - 2
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 2 - 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> ? = 3 - 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 2 = 4 - 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> ? = 2 - 2
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 1 = 3 - 2
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0 = 2 - 2
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,0]
=> []
=> []
=> ?
=> ? = 1 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [[1,3,4,5,7,8,12],[2,6,9,10,11,13,14]]
=> ? = 2 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [[1,2,3,5,6,10],[4,7,8,9,11,12]]
=> ? = 3 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [[1,3,5,6,7,8,12],[2,4,9,10,11,13,14]]
=> ? = 2 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[1,3,4,5,6,10],[2,7,8,9,11,12]]
=> ? = 3 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[1,2,3,4,8],[5,6,7,9,10]]
=> ? = 4 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [[1,3,4,5,7,10,12],[2,6,8,9,11,13,14]]
=> ? = 2 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [[1,2,3,5,8,10],[4,6,7,9,11,12]]
=> ? = 3 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [[1,3,5,6,7,10,12],[2,4,8,9,11,13,14]]
=> ? = 2 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [[1,3,4,5,8,10],[2,6,7,9,11,12]]
=> ? = 2 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[1,2,3,6,8],[4,5,7,9,10]]
=> ? = 3 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,3,5,7,8,10,12],[2,4,6,9,11,13,14]]
=> ? = 2 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,3,5,6,8,10],[2,4,7,9,11,12]]
=> ? = 2 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> ? = 2 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> ? = 3 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[1,3,4,5,7,8],[2,6,9,10,11,12]]
=> ? = 3 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[1,2,3,5,6],[4,7,8,9,10]]
=> ? = 4 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[1,3,5,6,7,8],[2,4,9,10,11,12]]
=> ? = 3 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 4 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> ? = 5 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [[1,3,4,5,7,10],[2,6,8,9,11,12]]
=> ? = 2 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[1,2,3,5,8],[4,6,7,9,10]]
=> ? = 3 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [[1,3,5,6,7,10],[2,4,8,9,11,12]]
=> ? = 2 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 2 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> ? = 3 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,7,8,10],[2,4,6,9,11,12]]
=> ? = 2 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> ? = 2 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 2 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 1 = 3 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 3 - 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> ? = 4 - 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 3 - 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> ? = 3 - 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 2 = 4 - 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> ? = 2 - 2
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> ? = 2 - 2
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0 = 2 - 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 1 = 3 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 0 = 2 - 2
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ?
=> ? = 1 - 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [[1,3,4,5,7,8,9,12,16],[2,6,10,11,13,14,15,17,18]]
=> ? = 2 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [[1,2,3,5,6,7,10,14],[4,8,9,11,12,13,15,16]]
=> ? = 3 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [[1,3,5,6,7,8,9,12,16],[2,4,10,11,13,14,15,17,18]]
=> ? = 2 - 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [[1,3,4,5,6,7,10,14],[2,8,9,11,12,13,15,16]]
=> ? = 3 - 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [[1,2,3,4,5,8,12],[6,7,9,10,11,13,14]]
=> ? = 4 - 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 1 = 3 - 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 2 = 4 - 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0 = 2 - 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 1 = 3 - 2
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0 = 2 - 2
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 0 = 2 - 2
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 1 = 3 - 2
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 2 = 4 - 2
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0 = 2 - 2
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 1 = 3 - 2
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0 = 2 - 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 0 = 2 - 2
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 1 = 3 - 2
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 2 = 4 - 2
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0 = 2 - 2
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 1 = 3 - 2
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0 = 2 - 2
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 0 = 2 - 2
Description
Eigenvalues of the top-to-random operator acting on a simple module.
These eigenvalues are given in [1] and [3].
The simple module of the symmetric group indexed by a partition $\lambda$ has dimension equal to the number of standard tableaux of shape $\lambda$. Hence, the eigenvalues of any linear operator defined on this module can be indexed by standard tableaux of shape $\lambda$; this statistic gives all the eigenvalues of the operator acting on the module.
This statistic bears different names, such as the type in [2] or eig in [3].
Similarly, the eigenvalues of the random-to-random operator acting on a simple module is [[St000508]].
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