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Your data matches 84 different statistics following compositions of up to 3 maps.
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Matching statistic: St000069
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 1
[1,2] => [2,1] => ([],2)
=> 2
[2,1] => [1,2] => ([(0,1)],2)
=> 1
[1,2,3] => [3,2,1] => ([],3)
=> 3
[1,3,2] => [3,1,2] => ([(1,2)],3)
=> 2
[2,1,3] => [2,3,1] => ([(1,2)],3)
=> 2
[2,3,1] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[3,1,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,2,3,4] => [4,3,2,1] => ([],4)
=> 4
[1,2,4,3] => [4,3,1,2] => ([(2,3)],4)
=> 3
[1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> 3
[1,3,4,2] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 2
[1,4,2,3] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> 3
[1,4,3,2] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2
[2,1,3,4] => [3,4,2,1] => ([(2,3)],4)
=> 3
[2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[2,3,1,4] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 2
[2,3,4,1] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 2
[2,4,3,1] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[3,1,2,4] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> 3
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 2
[3,2,1,4] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2
[3,2,4,1] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[3,4,2,1] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
[4,1,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 3
[4,1,3,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[4,2,1,3] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
[4,2,3,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 5
[1,2,3,5,4] => [5,4,3,1,2] => ([(3,4)],5)
=> 4
[1,2,4,3,5] => [5,4,2,3,1] => ([(3,4)],5)
=> 4
[1,2,4,5,3] => [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> 3
[1,2,5,3,4] => [5,4,1,3,2] => ([(2,3),(2,4)],5)
=> 4
[1,2,5,4,3] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> 3
[1,3,2,4,5] => [5,3,4,2,1] => ([(3,4)],5)
=> 4
[1,3,2,5,4] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> 3
[1,3,4,2,5] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> 3
[1,3,4,5,2] => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,3,5,2,4] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
=> 3
[1,3,5,4,2] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 2
[1,4,2,3,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> 4
[1,4,2,5,3] => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
=> 3
[1,4,3,2,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> 3
[1,4,3,5,2] => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> 2
[1,4,5,2,3] => [5,2,1,4,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
Description
The number of maximal elements of a poset.
Matching statistic: St000010
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00066: Permutations —inverse⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1]
=> 1
[1,2] => [1,2] => [1,2] => [1,1]
=> 2
[2,1] => [2,1] => [2,1] => [2]
=> 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,1,1]
=> 3
[1,3,2] => [1,3,2] => [1,3,2] => [2,1]
=> 2
[2,1,3] => [2,1,3] => [2,1,3] => [2,1]
=> 2
[2,3,1] => [3,1,2] => [2,3,1] => [3]
=> 1
[3,1,2] => [2,3,1] => [3,2,1] => [2,1]
=> 2
[3,2,1] => [3,2,1] => [3,1,2] => [3]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 3
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 3
[1,3,4,2] => [1,4,2,3] => [1,3,4,2] => [3,1]
=> 2
[1,4,2,3] => [1,3,4,2] => [1,4,3,2] => [2,1,1]
=> 3
[1,4,3,2] => [1,4,3,2] => [1,4,2,3] => [3,1]
=> 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [2,1,1]
=> 3
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,2]
=> 2
[2,3,1,4] => [3,1,2,4] => [2,3,1,4] => [3,1]
=> 2
[2,3,4,1] => [4,1,2,3] => [2,3,4,1] => [4]
=> 1
[2,4,1,3] => [3,1,4,2] => [3,4,1,2] => [2,2]
=> 2
[2,4,3,1] => [4,1,3,2] => [3,4,2,1] => [4]
=> 1
[3,1,2,4] => [2,3,1,4] => [3,2,1,4] => [2,1,1]
=> 3
[3,1,4,2] => [2,4,1,3] => [3,2,4,1] => [3,1]
=> 2
[3,2,1,4] => [3,2,1,4] => [3,1,2,4] => [3,1]
=> 2
[3,2,4,1] => [4,2,1,3] => [3,1,4,2] => [4]
=> 1
[3,4,1,2] => [3,4,1,2] => [2,4,3,1] => [3,1]
=> 2
[3,4,2,1] => [4,3,1,2] => [2,4,1,3] => [4]
=> 1
[4,1,2,3] => [2,3,4,1] => [4,2,3,1] => [2,1,1]
=> 3
[4,1,3,2] => [2,4,3,1] => [4,2,1,3] => [3,1]
=> 2
[4,2,1,3] => [3,2,4,1] => [4,3,2,1] => [2,2]
=> 2
[4,2,3,1] => [4,2,3,1] => [4,3,1,2] => [4]
=> 1
[4,3,1,2] => [3,4,2,1] => [4,1,3,2] => [3,1]
=> 2
[4,3,2,1] => [4,3,2,1] => [4,1,2,3] => [4]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 4
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 4
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,4,5,3] => [3,1,1]
=> 3
[1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,4,3] => [2,1,1,1]
=> 4
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,3,4] => [3,1,1]
=> 3
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> 4
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> 3
[1,3,4,2,5] => [1,4,2,3,5] => [1,3,4,2,5] => [3,1,1]
=> 3
[1,3,4,5,2] => [1,5,2,3,4] => [1,3,4,5,2] => [4,1]
=> 2
[1,3,5,2,4] => [1,4,2,5,3] => [1,4,5,2,3] => [2,2,1]
=> 3
[1,3,5,4,2] => [1,5,2,4,3] => [1,4,5,3,2] => [4,1]
=> 2
[1,4,2,3,5] => [1,3,4,2,5] => [1,4,3,2,5] => [2,1,1,1]
=> 4
[1,4,2,5,3] => [1,3,5,2,4] => [1,4,3,5,2] => [3,1,1]
=> 3
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,2,3,5] => [3,1,1]
=> 3
[1,4,3,5,2] => [1,5,3,2,4] => [1,4,2,5,3] => [4,1]
=> 2
[1,4,5,2,3] => [1,4,5,2,3] => [1,3,5,4,2] => [3,1,1]
=> 3
[8,6,7,2,3,4,5,1] => [8,4,5,6,7,2,3,1] => ? => ?
=> ? = 1
[8,4,5,3,6,2,7,1] => [8,6,4,2,3,5,7,1] => [8,3,5,2,7,4,1,6] => ?
=> ? = 1
[8,3,4,2,5,6,7,1] => [8,4,2,3,5,6,7,1] => [8,3,5,2,6,7,1,4] => ?
=> ? = 1
[5,6,3,4,7,8,1,2] => [7,8,3,4,1,2,5,6] => [2,5,4,1,6,8,7,3] => ?
=> ? = 2
[2,3,4,6,7,5,8,1] => [8,1,2,3,6,4,5,7] => [2,3,6,5,7,4,8,1] => ?
=> ? = 1
[2,4,5,6,3,7,8,1] => [8,1,5,2,3,4,6,7] => [5,3,4,6,2,7,8,1] => ?
=> ? = 1
[2,3,5,6,4,7,8,1] => [8,1,2,5,3,4,6,7] => [2,5,4,6,3,7,8,1] => ?
=> ? = 1
[2,3,1,7,6,5,4,8] => [3,1,2,7,6,5,4,8] => [2,3,1,7,4,5,6,8] => ?
=> ? = 3
[2,1,3,7,6,5,4,8] => [2,1,3,7,6,5,4,8] => [2,1,3,7,4,5,6,8] => ?
=> ? = 4
[2,1,6,5,4,3,7,8] => [2,1,6,5,4,3,7,8] => ? => ?
=> ? = 4
[3,2,1,6,5,4,7,8] => [3,2,1,6,5,4,7,8] => [3,1,2,6,4,5,7,8] => ?
=> ? = 4
[2,6,3,4,5,7,8,1] => [8,1,3,4,5,2,6,7] => [3,6,4,5,2,7,8,1] => ?
=> ? = 1
[8,2,3,4,6,7,5,1] => [8,2,3,4,7,5,6,1] => [8,3,4,7,6,1,5,2] => ?
=> ? = 1
[8,2,4,5,6,7,3,1] => [8,2,7,3,4,5,6,1] => [8,7,4,5,6,1,3,2] => ?
=> ? = 1
[1,4,5,2,3,6,7,8] => [1,4,5,2,3,6,7,8] => [1,3,5,4,2,6,7,8] => ?
=> ? = 6
[6,8,4,7,1,2,3,5] => [5,6,7,3,8,1,4,2] => [4,8,7,2,5,6,3,1] => ?
=> ? = 4
[6,8,4,7,2,5,1,3] => [7,5,8,3,6,1,4,2] => [4,8,6,2,7,1,5,3] => ?
=> ? = 2
[8,1,2,5,3,4,6,7] => [2,3,5,6,4,7,8,1] => [8,2,3,6,5,4,7,1] => ?
=> ? = 6
[9,1,2,3,4,5,8,6,7] => [2,3,4,5,6,8,9,7,1] => [9,2,3,4,5,6,1,8,7] => ?
=> ? = 7
[9,1,2,5,3,4,6,7,8] => [2,3,5,6,4,7,8,9,1] => [9,2,3,6,5,4,7,8,1] => ?
=> ? = 7
[4,1,7,2,6,5,3,8] => [2,4,7,1,6,5,3,8] => [6,2,7,4,3,5,1,8] => ?
=> ? = 4
[1,6,4,3,8,2,5,7] => [1,6,4,3,7,2,8,5] => [1,7,6,3,8,4,2,5] => ?
=> ? = 4
[6,8,4,7,1,5,2,3] => [5,7,8,3,6,1,4,2] => [4,8,6,2,5,1,7,3] => ?
=> ? = 3
[9,8,10,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,2,1,3] => [3,1,10,2,4,5,6,7,8,9] => ?
=> ? = 1
[1,6,4,3,7,2,8,5] => [1,6,4,3,8,2,5,7] => [1,5,6,3,7,4,8,2] => ?
=> ? = 3
[3,4,7,8,5,6,1,2] => [7,8,1,2,5,6,3,4] => [2,5,4,8,6,3,7,1] => ?
=> ? = 2
[7,8,1,2,5,6,3,4] => [3,4,7,8,5,6,1,2] => [2,8,3,4,6,1,7,5] => ?
=> ? = 4
[2,4,7,1,6,5,3,8] => [4,1,7,2,6,5,3,8] => [4,6,7,1,3,5,2,8] => ?
=> ? = 3
[7,8,5,6,1,3,2,4] => [5,7,6,8,3,4,1,2] => [2,8,4,1,5,7,6,3] => ?
=> ? = 3
[3,7,5,8,4,6,1,2] => [7,8,1,5,3,6,2,4] => [5,4,6,8,3,2,7,1] => ?
=> ? = 2
[8,1,2,3,5,4,6,7] => [2,3,4,6,5,7,8,1] => [8,2,3,4,6,5,7,1] => ?
=> ? = 6
[7,9,8,6,5,4,3,2,1] => [9,8,7,6,5,4,1,3,2] => [3,9,2,1,4,5,6,7,8] => ?
=> ? = 1
[8,10,9,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,1,3,2] => [3,10,2,1,4,5,6,7,8,9] => ?
=> ? = 1
[8,5,7,6,4,2,3,1] => [8,6,7,5,2,4,3,1] => [8,4,1,3,2,7,5,6] => ?
=> ? = 1
[7,8,1,3,2,4,6,5] => [3,5,4,6,8,7,1,2] => [2,8,3,5,4,6,1,7] => ?
=> ? = 4
[8,5,4,3,2,1,7,6] => [6,5,4,3,2,8,7,1] => [8,6,2,3,4,5,1,7] => ?
=> ? = 2
[9,6,5,4,3,2,1,8,7] => [7,6,5,4,3,2,9,8,1] => [9,7,2,3,4,5,6,1,8] => ?
=> ? = 2
[8,10,6,9,4,7,2,5,1,3] => [9,7,10,5,8,3,6,1,4,2] => [4,10,6,2,8,1,9,3,7,5] => ?
=> ? = 2
[7,9,5,8,3,6,1,4,2] => [7,9,5,8,3,6,1,4,2] => [4,9,6,2,8,1,7,3,5] => ?
=> ? = 2
[8,1,3,4,5,2,6,7] => [2,6,3,4,5,7,8,1] => [8,2,4,5,6,3,7,1] => ?
=> ? = 4
[8,1,3,2,4,5,6,7] => [2,4,3,5,6,7,8,1] => [8,2,4,3,5,6,7,1] => ?
=> ? = 6
[6,8,3,4,5,7,2,1] => [8,7,3,4,5,1,6,2] => [6,8,4,5,1,2,3,7] => ?
=> ? = 1
[8,1,2,3,5,7,4,6] => [2,3,4,7,5,8,6,1] => [8,2,3,4,7,1,5,6] => ?
=> ? = 5
[8,7,5,3,6,4,2,1] => [8,7,4,6,3,5,2,1] => [8,1,5,6,2,3,4,7] => ?
=> ? = 1
[10,7,6,5,4,3,2,1,9,8] => [8,7,6,5,4,3,2,10,9,1] => [10,8,2,3,4,5,6,7,1,9] => ?
=> ? = 2
[8,6,4,2,1,7,5,3] => [5,4,8,3,7,2,6,1] => [8,6,7,5,4,1,2,3] => ?
=> ? = 2
[9,1,2,3,5,4,6,7,8] => [2,3,4,6,5,7,8,9,1] => [9,2,3,4,6,5,7,8,1] => ?
=> ? = 7
[8,1,2,4,5,3,6,7] => [2,3,6,4,5,7,8,1] => [8,2,3,5,6,4,7,1] => ?
=> ? = 5
[2,3,4,6,5,7,8,9,1] => [9,1,2,3,5,4,6,7,8] => [2,3,5,6,4,7,8,9,1] => ?
=> ? = 1
[2,3,4,5,7,6,8,9,10,1] => [10,1,2,3,4,6,5,7,8,9] => [2,3,4,6,7,5,8,9,10,1] => ?
=> ? = 1
Description
The length of the partition.
Matching statistic: St000340
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 94%●distinct values known / distinct values provided: 73%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 94%●distinct values known / distinct values provided: 73%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 0 = 1 - 1
[1,2] => [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1 = 2 - 1
[2,1] => [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[2,4,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[3,4,2,1] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[4,1,3,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,2,5,3,4] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 4 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,3,5,2,4] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,3,5,4,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,4,2,3,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,4,2,5,3] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,4,3,5,2] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[5,6,3,4,1,2,7,8] => [5,6,3,4,1,2,7,8] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 4 - 1
[3,4,5,6,1,2,7,8] => [5,6,1,2,3,4,7,8] => ?
=> ?
=> ? = 4 - 1
[5,6,1,2,3,4,7,8] => [3,4,5,6,1,2,7,8] => [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]
=> ? = 6 - 1
[4,3,2,1,5,6,7,8] => [4,3,2,1,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 5 - 1
[3,4,1,2,5,6,7,8] => [3,4,1,2,5,6,7,8] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,1,0,0]
=> ? = 6 - 1
[4,3,2,1,8,7,6,5] => [4,3,2,1,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[4,3,2,1,7,8,6,5] => [4,3,2,1,8,7,5,6] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[4,3,2,1,6,8,7,5] => [4,3,2,1,8,5,7,6] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[4,3,2,1,7,6,8,5] => [4,3,2,1,8,6,5,7] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[4,3,2,1,6,7,8,5] => [4,3,2,1,8,5,6,7] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[3,4,2,1,8,7,6,5] => [4,3,1,2,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[3,4,2,1,7,8,6,5] => [4,3,1,2,8,7,5,6] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[2,4,3,1,8,7,6,5] => [4,1,3,2,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[2,4,3,1,6,8,7,5] => [4,1,3,2,8,5,7,6] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[2,4,3,1,7,6,8,5] => [4,1,3,2,8,6,5,7] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[3,2,4,1,8,7,6,5] => [4,2,1,3,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[3,2,4,1,6,8,7,5] => [4,2,1,3,8,5,7,6] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[3,2,4,1,7,6,8,5] => [4,2,1,3,8,6,5,7] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[2,3,4,1,8,7,6,5] => [4,1,2,3,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[2,3,4,1,6,7,8,5] => [4,1,2,3,8,5,6,7] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[1,4,3,2,8,7,6,5] => [1,4,3,2,8,7,6,5] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 3 - 1
[1,3,4,2,8,7,6,5] => [1,4,2,3,8,7,6,5] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 3 - 1
[2,1,4,3,8,7,6,5] => [2,1,4,3,8,7,6,5] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 3 - 1
[1,2,4,3,8,7,6,5] => [1,2,4,3,8,7,6,5] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 4 - 1
[3,2,1,4,8,7,6,5] => [3,2,1,4,8,7,6,5] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
=> ? = 3 - 1
[2,3,1,4,8,7,6,5] => [3,1,2,4,8,7,6,5] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
=> ? = 3 - 1
[1,3,2,4,8,7,6,5] => [1,3,2,4,8,7,6,5] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 4 - 1
[2,1,3,4,8,7,6,5] => [2,1,3,4,8,7,6,5] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 4 - 1
[1,2,3,4,8,7,6,5] => [1,2,3,4,8,7,6,5] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 5 - 1
[1,5,4,3,2,8,7,6] => [1,5,4,3,2,8,7,6] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 3 - 1
[1,5,4,3,2,7,8,6] => [1,5,4,3,2,8,6,7] => ?
=> ?
=> ? = 3 - 1
[1,2,5,4,3,8,7,6] => [1,2,5,4,3,8,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 4 - 1
[4,3,2,1,5,8,7,6] => [4,3,2,1,5,8,7,6] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 3 - 1
[4,3,2,1,5,7,8,6] => [4,3,2,1,5,8,6,7] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 3 - 1
[1,4,3,2,5,8,7,6] => [1,4,3,2,5,8,7,6] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 4 - 1
[3,2,1,4,5,8,7,6] => [3,2,1,4,5,8,7,6] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 4 - 1
[2,3,1,4,5,7,8,6] => [3,1,2,4,5,8,6,7] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 4 - 1
[2,1,6,5,4,3,8,7] => [2,1,6,5,4,3,8,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 3 - 1
[1,2,6,5,4,3,8,7] => [1,2,6,5,4,3,8,7] => ?
=> ?
=> ? = 4 - 1
[4,3,2,1,6,5,8,7] => [4,3,2,1,6,5,8,7] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
[2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 1
[1,2,3,4,6,5,8,7] => [1,2,3,4,6,5,8,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> ? = 6 - 1
[1,2,3,5,4,6,8,7] => [1,2,3,5,4,6,8,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 6 - 1
[4,3,2,1,5,6,8,7] => [4,3,2,1,5,6,8,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> ? = 4 - 1
[1,2,4,3,5,6,8,7] => [1,2,4,3,5,6,8,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> ? = 6 - 1
[1,3,2,4,5,6,8,7] => [1,3,2,4,5,6,8,7] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 6 - 1
[2,1,3,4,5,6,8,7] => [2,1,3,4,5,6,8,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 6 - 1
[3,2,1,7,6,5,4,8] => [3,2,1,7,6,5,4,8] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
[2,3,1,7,6,5,4,8] => [3,1,2,7,6,5,4,8] => ?
=> ?
=> ? = 3 - 1
[2,1,3,7,6,5,4,8] => [2,1,3,7,6,5,4,8] => ?
=> ?
=> ? = 4 - 1
Description
The number of non-final maximal constant sub-paths of length greater than one.
This is the total number of occurrences of the patterns $110$ and $001$.
Matching statistic: St000011
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 91%●distinct values known / distinct values provided: 82%
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 91%●distinct values known / distinct values provided: 82%
Values
[1] => [.,.]
=> [1,0]
=> [1,0]
=> 1
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[8,5,6,7,4,2,3,1] => [[[[[.,.],[.,[.,.]]],.],[.,.]],.]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 1
[8,6,7,4,5,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 1
[8,4,5,3,6,2,7,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 1
[6,5,7,8,3,4,1,2] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 2
[5,6,7,8,3,4,1,2] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 2
[7,8,4,3,5,6,1,2] => [[[[.,[.,.]],.],[.,[.,.]]],[.,.]]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2
[7,8,3,4,5,6,1,2] => [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2
[5,6,3,4,7,8,1,2] => [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2
[5,4,3,6,7,8,1,2] => [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2
[7,8,5,6,1,2,3,4] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 4
[7,8,3,4,1,2,5,6] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 4
[8,2,3,4,1,5,6,7] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> ? = 4
[5,6,3,4,1,2,7,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 4
[5,6,1,2,3,4,7,8] => ?
=> ?
=> ?
=> ? = 6
[2,3,5,6,7,4,8,1] => [[.,[.,[[.,[.,[.,.]]],[.,.]]]],.]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 1
[2,4,5,3,6,7,8,1] => [[.,[[.,[.,.]],[.,[.,[.,.]]]]],.]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> ? = 1
[3,4,2,1,7,8,6,5] => [[[.,[.,.]],.],[[[.,[.,.]],.],.]]
=> [1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2
[2,4,3,1,6,8,7,5] => [[.,[[.,.],.]],[[.,[[.,.],.]],.]]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 2
[2,4,3,1,7,6,8,5] => [[.,[[.,.],.]],[[[.,.],[.,.]],.]]
=> [1,1,0,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0]
=> ? = 2
[3,2,4,1,7,6,8,5] => [[[.,.],[.,.]],[[[.,.],[.,.]],.]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 2
[2,3,4,1,6,7,8,5] => [[.,[.,[.,.]]],[[.,[.,[.,.]]],.]]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> ? = 2
[1,4,3,2,8,7,6,5] => [.,[[[.,.],.],[[[[.,.],.],.],.]]]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 3
[1,2,4,3,8,7,6,5] => [.,[.,[[.,.],[[[[.,.],.],.],.]]]]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[1,3,2,4,8,7,6,5] => [.,[[.,.],[.,[[[[.,.],.],.],.]]]]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 4
[2,1,3,4,8,7,6,5] => [[.,.],[.,[.,[[[[.,.],.],.],.]]]]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 4
[1,2,3,4,8,7,6,5] => [.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 5
[1,5,4,3,2,8,7,6] => [.,[[[[.,.],.],.],[[[.,.],.],.]]]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 3
[1,5,4,3,2,7,8,6] => [.,[[[[.,.],.],.],[[.,[.,.]],.]]]
=> [1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0,1,0]
=> ? = 3
[1,2,5,4,3,8,7,6] => [.,[.,[[[.,.],.],[[[.,.],.],.]]]]
=> [1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 4
[4,3,2,1,5,7,8,6] => [[[[.,.],.],.],[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 3
[1,4,3,2,5,8,7,6] => [.,[[[.,.],.],[.,[[[.,.],.],.]]]]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 4
[3,2,1,4,5,8,7,6] => [[[.,.],.],[.,[.,[[[.,.],.],.]]]]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 4
[2,1,6,5,4,3,8,7] => [[.,.],[[[[.,.],.],.],[[.,.],.]]]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3
[1,2,6,5,4,3,8,7] => ?
=> ?
=> ?
=> ? = 4
[2,1,4,3,6,5,8,7] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4
[1,2,3,4,6,5,8,7] => [.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 6
[2,3,1,7,6,5,4,8] => ?
=> ?
=> ?
=> ? = 3
[1,3,2,7,6,5,4,8] => [.,[[.,.],[[[[.,.],.],.],[.,.]]]]
=> [1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 4
[2,1,3,7,6,5,4,8] => ?
=> ?
=> ?
=> ? = 4
[4,3,2,1,6,7,5,8] => [[[[.,.],.],.],[[.,[.,.]],[.,.]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 3
[1,4,3,2,7,6,5,8] => [.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 4
[1,3,4,2,6,7,5,8] => [.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> [1,1,1,0,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 4
[3,2,1,4,7,6,5,8] => [[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 4
[2,3,1,4,6,7,5,8] => [[.,[.,.]],[.,[[.,[.,.]],[.,.]]]]
=> [1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 4
[1,5,4,3,2,7,6,8] => [.,[[[[.,.],.],.],[[.,.],[.,.]]]]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 4
[1,2,3,5,4,7,6,8] => [.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 6
[1,2,4,3,5,7,6,8] => [.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 6
[2,1,6,5,4,3,7,8] => ?
=> ?
=> ?
=> ? = 4
[3,2,1,6,5,4,7,8] => [[[.,.],.],[[[.,.],.],[.,[.,.]]]]
=> [1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[2,3,1,5,6,4,7,8] => [[.,[.,.]],[[.,[.,.]],[.,[.,.]]]]
=> [1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> ? = 4
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000678
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 91%●distinct values known / distinct values provided: 73%
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 91%●distinct values known / distinct values provided: 73%
Values
[1] => [.,.]
=> [1,0]
=> [1,0]
=> ? = 1
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 4
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 4
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[8,7,5,6,4,3,2,1] => [[[[[[[.,.],.],[.,.]],.],.],.],.]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[8,7,6,4,5,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[8,7,6,5,3,4,2,1] => [[[[[[[.,.],.],.],.],[.,.]],.],.]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
[8,7,6,5,4,2,3,1] => [[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1
[8,7,5,6,4,2,3,1] => [[[[[[.,.],.],[.,.]],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
[8,6,5,7,4,2,3,1] => [[[[[[.,.],.],[.,.]],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
[5,4,3,6,7,8,1,2] => [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[8,6,5,4,3,2,1,7] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 2
[8,4,1,2,3,5,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[8,3,1,2,4,5,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[8,2,1,3,4,5,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[5,6,1,2,3,4,7,8] => ?
=> ?
=> ?
=> ? = 6
[4,3,2,1,5,6,7,8] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[6,8,7,5,4,3,2,1] => [[[[[[.,[[.,.],.]],.],.],.],.],.]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[5,6,8,7,4,3,2,1] => [[[[[.,[.,[[.,.],.]]],.],.],.],.]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[5,7,6,8,4,3,2,1] => [[[[[.,[[.,.],[.,.]]],.],.],.],.]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[2,3,4,5,6,8,7,1] => [[.,[.,[.,[.,[.,[[.,.],.]]]]]],.]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1
[2,3,4,5,7,6,8,1] => [[.,[.,[.,[.,[[.,.],[.,.]]]]]],.]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 1
[2,3,4,6,5,7,8,1] => [[.,[.,[.,[[.,.],[.,[.,.]]]]]],.]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> ? = 1
[2,3,5,4,6,7,8,1] => [[.,[.,[[.,.],[.,[.,[.,.]]]]]],.]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 1
[4,3,2,1,8,7,6,5] => [[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 2
[4,3,2,1,7,8,6,5] => [[[[.,.],.],.],[[[.,[.,.]],.],.]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
=> ? = 2
[4,3,2,1,6,8,7,5] => [[[[.,.],.],.],[[.,[[.,.],.]],.]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,1,0,0,0]
=> ? = 2
[4,3,2,1,7,6,8,5] => [[[[.,.],.],.],[[[.,.],[.,.]],.]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> ? = 2
[4,3,2,1,6,7,8,5] => [[[[.,.],.],.],[[.,[.,[.,.]]],.]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,1,0,0]
=> ? = 2
[2,4,3,1,8,7,6,5] => [[.,[[.,.],.]],[[[[.,.],.],.],.]]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 2
[2,4,3,1,6,8,7,5] => [[.,[[.,.],.]],[[.,[[.,.],.]],.]]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> ? = 2
[2,4,3,1,7,6,8,5] => [[.,[[.,.],.]],[[[.,.],[.,.]],.]]
=> [1,1,0,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 2
[1,4,3,2,8,7,6,5] => [.,[[[.,.],.],[[[[.,.],.],.],.]]]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> ? = 3
[2,1,4,3,8,7,6,5] => [[.,.],[[.,.],[[[[.,.],.],.],.]]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> ? = 3
[1,2,4,3,8,7,6,5] => [.,[.,[[.,.],[[[[.,.],.],.],.]]]]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> ? = 4
[3,2,1,4,8,7,6,5] => [[[.,.],.],[.,[[[[.,.],.],.],.]]]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> ? = 3
[1,3,2,4,8,7,6,5] => [.,[[.,.],[.,[[[[.,.],.],.],.]]]]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> ? = 4
[2,1,3,4,8,7,6,5] => [[.,.],[.,[.,[[[[.,.],.],.],.]]]]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> ? = 4
[1,2,3,4,8,7,6,5] => [.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 5
[1,5,4,3,2,8,7,6] => [.,[[[[.,.],.],.],[[[.,.],.],.]]]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
=> ? = 3
[1,5,4,3,2,7,8,6] => [.,[[[[.,.],.],.],[[.,[.,.]],.]]]
=> [1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 3
[1,2,5,4,3,8,7,6] => [.,[.,[[[.,.],.],[[[.,.],.],.]]]]
=> [1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0]
=> ? = 4
[4,3,2,1,5,8,7,6] => [[[[.,.],.],.],[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> ? = 3
[4,3,2,1,5,7,8,6] => [[[[.,.],.],.],[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,1,0,0]
=> ? = 3
[1,4,3,2,5,8,7,6] => [.,[[[.,.],.],[.,[[[.,.],.],.]]]]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0]
=> ? = 4
[3,2,1,4,5,8,7,6] => [[[.,.],.],[.,[.,[[[.,.],.],.]]]]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
=> ? = 4
[2,1,6,5,4,3,8,7] => [[.,.],[[[[.,.],.],.],[[.,.],.]]]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> ? = 3
[1,2,6,5,4,3,8,7] => ?
=> ?
=> ?
=> ? = 4
[4,3,2,1,6,5,8,7] => [[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 3
[2,1,4,3,6,5,8,7] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 4
[1,2,3,4,6,5,8,7] => [.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 6
[1,2,3,5,4,6,8,7] => [.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 6
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000439
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 90%●distinct values known / distinct values provided: 82%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 90%●distinct values known / distinct values provided: 82%
Values
[1] => [1] => [.,.]
=> [1,0]
=> 2 = 1 + 1
[1,2] => [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 3 = 2 + 1
[2,1] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,2,3] => [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,3,2] => [2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[2,1,3] => [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
[2,3,1] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[3,1,2] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[3,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,2,4,3] => [3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,3,2,4] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,3,4,2] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,4,2,3] => [3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,4,3,2] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[2,1,3,4] => [4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[2,1,4,3] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[2,3,1,4] => [4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[2,3,4,1] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,4,1,3] => [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[2,4,3,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[3,1,2,4] => [4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[3,1,4,2] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[3,2,1,4] => [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[3,2,4,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[3,4,1,2] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[3,4,2,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[4,1,2,3] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[4,1,3,2] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[4,2,1,3] => [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[4,2,3,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[4,3,1,2] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,3,4,5] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,2,3,5,4] => [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[1,2,4,3,5] => [5,3,4,2,1] => [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5 = 4 + 1
[1,2,4,5,3] => [3,5,4,2,1] => [[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,2,5,3,4] => [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[1,2,5,4,3] => [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,3,2,4,5] => [5,4,2,3,1] => [[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5 = 4 + 1
[1,3,2,5,4] => [4,5,2,3,1] => [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,3,4,2,5] => [5,2,4,3,1] => [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4 = 3 + 1
[1,3,4,5,2] => [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,5,2,4] => [4,2,5,3,1] => [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,3,5,4,2] => [2,4,5,3,1] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,4,2,3,5] => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5 = 4 + 1
[1,4,2,5,3] => [3,5,2,4,1] => [[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,4,3,2,5] => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4 = 3 + 1
[1,4,3,5,2] => [2,5,3,4,1] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,4,5,2,3] => [3,2,5,4,1] => [[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[8,6,5,7,4,2,3,1] => [1,3,2,4,7,5,6,8] => ?
=> ?
=> ? = 1 + 1
[7,8,3,4,1,2,5,6] => [6,5,2,1,4,3,8,7] => [[[[.,.],[[.,.],.]],.],[[.,.],.]]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,1,0,0]
=> ? = 4 + 1
[5,6,3,4,1,2,7,8] => [8,7,2,1,4,3,6,5] => [[[[.,.],[[.,.],[[.,.],.]]],.],.]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 4 + 1
[3,4,5,6,1,2,7,8] => [8,7,2,1,6,5,4,3] => [[[[.,.],[[[[.,.],.],.],.]],.],.]
=> [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 4 + 1
[5,6,1,2,3,4,7,8] => [8,7,4,3,2,1,6,5] => [[[[[[.,.],.],.],[[.,.],.]],.],.]
=> [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> ? = 6 + 1
[4,3,2,1,6,8,7,5] => [5,7,8,6,1,2,3,4] => [[.,[.,[.,[.,.]]]],[[.,.],[.,.]]]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2 + 1
[4,3,2,1,7,6,8,5] => [5,8,6,7,1,2,3,4] => [[.,[.,[.,[.,.]]]],[[.,[.,.]],.]]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 2 + 1
[4,3,2,1,6,7,8,5] => [5,8,7,6,1,2,3,4] => [[.,[.,[.,[.,.]]]],[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 2 + 1
[3,4,2,1,7,8,6,5] => [5,6,8,7,1,2,4,3] => [[.,[.,[[.,.],.]]],[.,[[.,.],.]]]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 2 + 1
[2,4,3,1,8,7,6,5] => [5,6,7,8,1,3,4,2] => [[.,[[.,.],[.,.]]],[.,[.,[.,.]]]]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[2,4,3,1,6,8,7,5] => [5,7,8,6,1,3,4,2] => [[.,[[.,.],[.,.]]],[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 2 + 1
[2,4,3,1,7,6,8,5] => [5,8,6,7,1,3,4,2] => [[.,[[.,.],[.,.]]],[[.,[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> ? = 2 + 1
[3,2,4,1,8,7,6,5] => [5,6,7,8,1,4,2,3] => [[.,[[.,[.,.]],.]],[.,[.,[.,.]]]]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[3,2,4,1,6,8,7,5] => [5,7,8,6,1,4,2,3] => [[.,[[.,[.,.]],.]],[[.,.],[.,.]]]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> ? = 2 + 1
[2,3,4,1,8,7,6,5] => [5,6,7,8,1,4,3,2] => [[.,[[[.,.],.],.]],[.,[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 2 + 1
[2,3,4,1,6,7,8,5] => [5,8,7,6,1,4,3,2] => [[.,[[[.,.],.],.]],[[[.,.],.],.]]
=> [1,1,0,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 2 + 1
[1,4,3,2,8,7,6,5] => [5,6,7,8,2,3,4,1] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[1,3,4,2,8,7,6,5] => [5,6,7,8,2,4,3,1] => [[[.,.],[[.,.],.]],[.,[.,[.,.]]]]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[2,1,4,3,8,7,6,5] => [5,6,7,8,3,4,1,2] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[1,2,4,3,8,7,6,5] => [5,6,7,8,3,4,2,1] => [[[[.,.],.],[.,.]],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 4 + 1
[2,3,1,4,8,7,6,5] => [5,6,7,8,4,1,3,2] => [[[.,[[.,.],.]],.],[.,[.,[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[1,3,2,4,8,7,6,5] => [5,6,7,8,4,2,3,1] => [[[[.,.],[.,.]],.],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 4 + 1
[2,1,3,4,8,7,6,5] => [5,6,7,8,4,3,1,2] => [[[[.,[.,.]],.],.],[.,[.,[.,.]]]]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 4 + 1
[1,2,3,4,8,7,6,5] => [5,6,7,8,4,3,2,1] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 5 + 1
[1,5,4,3,2,7,8,6] => [6,8,7,2,3,4,5,1] => [[[.,.],[.,[.,[.,.]]]],[[.,.],.]]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 3 + 1
[1,2,5,4,3,8,7,6] => [6,7,8,3,4,5,2,1] => [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 4 + 1
[1,4,3,2,5,8,7,6] => [6,7,8,5,2,3,4,1] => [[[[.,.],[.,[.,.]]],.],[.,[.,.]]]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 4 + 1
[3,2,1,4,5,8,7,6] => [6,7,8,5,4,1,2,3] => [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 4 + 1
[2,1,6,5,4,3,8,7] => [7,8,3,4,5,6,1,2] => [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 3 + 1
[1,2,6,5,4,3,8,7] => [7,8,3,4,5,6,2,1] => [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 4 + 1
[4,3,2,1,6,5,8,7] => [7,8,5,6,1,2,3,4] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 3 + 1
[1,2,3,4,6,5,8,7] => [7,8,5,6,4,3,2,1] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ? = 6 + 1
[1,2,3,5,4,6,8,7] => [7,8,6,4,5,3,2,1] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> ? = 6 + 1
[4,3,2,1,5,6,8,7] => [7,8,6,5,1,2,3,4] => [[[[.,[.,[.,[.,.]]]],.],.],[.,.]]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 4 + 1
[2,3,1,7,6,5,4,8] => [8,4,5,6,7,1,3,2] => ?
=> ?
=> ? = 3 + 1
[1,3,2,7,6,5,4,8] => [8,4,5,6,7,2,3,1] => [[[[.,.],[.,.]],[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
[2,1,3,7,6,5,4,8] => [8,4,5,6,7,3,1,2] => ?
=> ?
=> ? = 4 + 1
[1,2,3,7,6,5,4,8] => [8,4,5,6,7,3,2,1] => [[[[[.,.],.],.],[.,[.,[.,.]]]],.]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 5 + 1
[4,3,2,1,7,6,5,8] => [8,5,6,7,1,2,3,4] => [[[.,[.,[.,[.,.]]]],[.,[.,.]]],.]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 + 1
[4,3,2,1,6,7,5,8] => [8,5,7,6,1,2,3,4] => [[[.,[.,[.,[.,.]]]],[[.,.],.]],.]
=> [1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> ? = 3 + 1
[1,4,3,2,7,6,5,8] => [8,5,6,7,2,3,4,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 4 + 1
[1,3,4,2,6,7,5,8] => [8,5,7,6,2,4,3,1] => [[[[.,.],[[.,.],.]],[[.,.],.]],.]
=> [1,1,1,1,0,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 4 + 1
[3,2,1,4,7,6,5,8] => [8,5,6,7,4,1,2,3] => [[[[.,[.,[.,.]]],.],[.,[.,.]]],.]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 4 + 1
[2,3,1,4,6,7,5,8] => [8,5,7,6,4,1,3,2] => [[[[.,[[.,.],.]],.],[[.,.],.]],.]
=> [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 4 + 1
[1,5,4,3,2,7,6,8] => [8,6,7,2,3,4,5,1] => [[[[.,.],[.,[.,[.,.]]]],[.,.]],.]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 4 + 1
[1,2,3,5,4,7,6,8] => [8,6,7,4,5,3,2,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 6 + 1
[4,3,2,1,5,7,6,8] => [8,6,7,5,1,2,3,4] => [[[[.,[.,[.,[.,.]]]],.],[.,.]],.]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 4 + 1
[2,1,6,5,4,3,7,8] => [8,7,3,4,5,6,1,2] => [[[[.,[.,.]],[.,[.,[.,.]]]],.],.]
=> [1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 4 + 1
[1,2,6,5,4,3,7,8] => [8,7,3,4,5,6,2,1] => [[[[[.,.],.],[.,[.,[.,.]]]],.],.]
=> [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
=> ? = 5 + 1
[3,2,1,6,5,4,7,8] => [8,7,4,5,6,1,2,3] => [[[[.,[.,[.,.]]],[.,[.,.]]],.],.]
=> [1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 4 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000675
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St000675: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 90%●distinct values known / distinct values provided: 64%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St000675: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 90%●distinct values known / distinct values provided: 64%
Values
[1] => [.,.]
=> [1,0]
=> [1,0]
=> ? = 1
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[7,8,6,5,4,3,2,1] => [[[[[[[.,[.,.]],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1
[8,6,7,5,4,3,2,1] => [[[[[[[.,.],[.,.]],.],.],.],.],.]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[7,6,8,5,4,3,2,1] => [[[[[[[.,.],[.,.]],.],.],.],.],.]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[6,7,8,5,4,3,2,1] => [[[[[[.,[.,[.,.]]],.],.],.],.],.]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
[8,7,5,6,4,3,2,1] => [[[[[[[.,.],.],[.,.]],.],.],.],.]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ? = 1
[7,5,6,8,4,3,2,1] => [[[[[[.,.],[.,[.,.]]],.],.],.],.]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 1
[6,5,7,8,4,3,2,1] => [[[[[[.,.],[.,[.,.]]],.],.],.],.]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 1
[5,6,7,8,4,3,2,1] => [[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[8,7,6,4,5,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> ? = 1
[7,4,5,6,8,3,2,1] => [[[[[.,.],[.,[.,[.,.]]]],.],.],.]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 1
[8,7,6,5,3,4,2,1] => [[[[[[[.,.],.],.],.],[.,.]],.],.]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 1
[8,6,7,5,3,4,2,1] => [[[[[[.,.],[.,.]],.],[.,.]],.],.]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> ? = 1
[8,7,6,5,4,2,3,1] => [[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 1
[8,6,7,5,4,2,3,1] => [[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> ? = 1
[8,7,5,6,4,2,3,1] => [[[[[[.,.],.],[.,.]],.],[.,.]],.]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 1
[8,6,5,7,4,2,3,1] => [[[[[[.,.],.],[.,.]],.],[.,.]],.]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 1
[8,5,6,7,4,2,3,1] => [[[[[.,.],[.,[.,.]]],.],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 1
[8,6,7,4,5,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[8,4,5,6,7,2,3,1] => [[[[.,.],[.,[.,[.,.]]]],[.,.]],.]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 1
[8,6,7,5,2,3,4,1] => [[[[[.,.],[.,.]],.],[.,[.,.]]],.]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0,1,0]
=> ? = 1
[8,6,7,2,3,4,5,1] => [[[[.,.],[.,.]],[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 1
[8,4,5,3,6,2,7,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[8,5,6,4,2,3,7,1] => [[[[[.,.],[.,.]],.],[.,[.,.]]],.]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0,1,0]
=> ? = 1
[8,3,4,2,5,6,7,1] => [[[[.,.],[.,.]],[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 1
[7,8,5,6,3,4,1,2] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2
[6,5,7,8,3,4,1,2] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[7,8,4,3,5,6,1,2] => [[[[.,[.,.]],.],[.,[.,.]]],[.,.]]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 2
[5,4,3,6,7,8,1,2] => [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2
[7,8,5,6,2,1,3,4] => [[[[.,[.,.]],[.,.]],.],[.,[.,.]]]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 3
[6,5,7,8,2,1,3,4] => [[[[.,.],[.,[.,.]]],.],[.,[.,.]]]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
[7,8,3,2,1,4,5,6] => [[[[.,[.,.]],.],.],[.,[.,[.,.]]]]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 4
[8,6,5,4,3,2,1,7] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 2
[8,4,1,2,3,5,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6
[8,3,1,2,4,5,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6
[8,2,1,3,4,5,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6
[5,6,1,2,3,4,7,8] => ?
=> ?
=> ?
=> ? = 6
[4,3,2,1,5,6,7,8] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 5
[1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[6,8,7,5,4,3,2,1] => [[[[[[.,[[.,.],.]],.],.],.],.],.]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> ? = 1
[5,6,8,7,4,3,2,1] => [[[[[.,[.,[[.,.],.]]],.],.],.],.]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 1
[5,7,6,8,4,3,2,1] => [[[[[.,[[.,.],[.,.]]],.],.],.],.]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,1,0,0]
=> ? = 1
[4,3,2,1,8,7,6,5] => [[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0,1,0]
=> ? = 2
[4,3,2,1,7,8,6,5] => [[[[.,.],.],.],[[[.,[.,.]],.],.]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0,1,0]
=> ? = 2
[4,3,2,1,6,8,7,5] => [[[[.,.],.],.],[[.,[[.,.],.]],.]]
=> [1,1,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0,1,0]
=> ? = 2
[4,3,2,1,7,6,8,5] => [[[[.,.],.],.],[[[.,.],[.,.]],.]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0,1,0]
=> ? = 2
[4,3,2,1,6,7,8,5] => [[[[.,.],.],.],[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0,1,0]
=> ? = 2
[1,4,3,2,8,7,6,5] => [.,[[[.,.],.],[[[[.,.],.],.],.]]]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 3
[1,3,4,2,8,7,6,5] => [.,[[.,[.,.]],[[[[.,.],.],.],.]]]
=> [1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 3
Description
The number of centered multitunnels of a Dyck path.
This is the number of factorisations $D = A B C$ of a Dyck path, such that $B$ is a Dyck path and $A$ and $B$ have the same length.
Matching statistic: St000925
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 64% ●values known / values provided: 89%●distinct values known / distinct values provided: 64%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 64% ●values known / values provided: 89%●distinct values known / distinct values provided: 64%
Values
[1] => [1] => [1,0]
=> {{1}}
=> ? = 1
[1,2] => [1,2] => [1,0,1,0]
=> {{1},{2}}
=> 2
[2,1] => [2,1] => [1,1,0,0]
=> {{1,2}}
=> 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 3
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 2
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
[2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 1
[3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 2
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 4
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 3
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 3
[1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
[1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 3
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 3
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2
[2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2
[2,4,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 3
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 2
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2
[3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 2
[3,4,2,1] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 3
[4,1,3,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 2
[4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 2
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 5
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 4
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 4
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 3
[1,2,5,3,4] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 4
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 3
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 4
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 3
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 3
[1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 2
[1,3,5,2,4] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 3
[1,3,5,4,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 2
[1,4,2,3,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 4
[1,4,2,5,3] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 3
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 3
[1,4,3,5,2] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 2
[1,4,5,2,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 3
[1,4,5,3,2] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 2
[7,8,5,6,2,1,3,4] => [6,5,7,8,3,4,1,2] => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,6},{7}}
=> ? = 3
[6,5,7,8,2,1,3,4] => [6,5,7,8,2,1,3,4] => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,6},{7}}
=> ? = 3
[7,8,5,6,1,2,3,4] => [5,6,7,8,3,4,1,2] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5},{6},{7}}
=> ? = 4
[5,6,7,8,1,2,3,4] => [5,6,7,8,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5},{6},{7}}
=> ? = 4
[7,8,3,4,1,2,5,6] => [5,6,3,4,7,8,1,2] => [1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0]
=> {{1,2,8},{3,4,6},{5},{7}}
=> ? = 4
[7,8,3,2,1,4,5,6] => [5,4,3,6,7,8,1,2] => [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
=> {{1,2,8},{3,4,5},{6},{7}}
=> ? = 4
[7,8,1,2,3,4,5,6] => [3,4,5,6,7,8,1,2] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> {{1,2,8},{3},{4},{5},{6},{7}}
=> ? = 6
[8,2,3,4,1,5,6,7] => [5,2,3,4,6,7,8,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> {{1,8},{2,3,4,5},{6},{7}}
=> ? = 4
[8,4,1,2,3,5,6,7] => [3,4,5,2,6,7,8,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> {{1,8},{2,5},{3},{4},{6},{7}}
=> ? = 6
[8,2,3,1,4,5,6,7] => [4,2,3,5,6,7,8,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2,3,4},{5},{6},{7}}
=> ? = 5
[8,3,1,2,4,5,6,7] => [3,4,2,5,6,7,8,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2,4},{3},{5},{6},{7}}
=> ? = 6
[8,2,1,3,4,5,6,7] => [3,2,4,5,6,7,8,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2,3},{4},{5},{6},{7}}
=> ? = 6
[8,1,2,3,4,5,6,7] => [2,3,4,5,6,7,8,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? = 7
[5,6,3,4,1,2,7,8] => [5,6,3,4,1,2,7,8] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,6},{5},{7},{8}}
=> ? = 4
[3,4,5,6,1,2,7,8] => [5,6,1,2,3,4,7,8] => ?
=> ?
=> ? = 4
[5,6,1,2,3,4,7,8] => [3,4,5,6,1,2,7,8] => [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> {{1,2,6},{3},{4},{5},{7},{8}}
=> ? = 6
[4,3,2,1,5,6,7,8] => [4,3,2,1,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 5
[3,4,1,2,5,6,7,8] => [3,4,1,2,5,6,7,8] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> {{1,2,4},{3},{5},{6},{7},{8}}
=> ? = 6
[1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 8
[4,3,2,1,8,7,6,5] => [4,3,2,1,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 2
[4,3,2,1,7,8,6,5] => [4,3,2,1,8,7,5,6] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 2
[4,3,2,1,6,8,7,5] => [4,3,2,1,8,5,7,6] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 2
[4,3,2,1,7,6,8,5] => [4,3,2,1,8,6,5,7] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 2
[4,3,2,1,6,7,8,5] => [4,3,2,1,8,5,6,7] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 2
[3,4,2,1,8,7,6,5] => [4,3,1,2,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 2
[3,4,2,1,7,8,6,5] => [4,3,1,2,8,7,5,6] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 2
[2,4,3,1,8,7,6,5] => [4,1,3,2,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 2
[2,4,3,1,6,8,7,5] => [4,1,3,2,8,5,7,6] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 2
[2,4,3,1,7,6,8,5] => [4,1,3,2,8,6,5,7] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 2
[3,2,4,1,8,7,6,5] => [4,2,1,3,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 2
[3,2,4,1,6,8,7,5] => [4,2,1,3,8,5,7,6] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 2
[3,2,4,1,7,6,8,5] => [4,2,1,3,8,6,5,7] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 2
[2,3,4,1,8,7,6,5] => [4,1,2,3,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 2
[2,3,4,1,6,7,8,5] => [4,1,2,3,8,5,6,7] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 2
[1,4,3,2,8,7,6,5] => [1,4,3,2,8,7,6,5] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4},{5,6,7,8}}
=> ? = 3
[1,3,4,2,8,7,6,5] => [1,4,2,3,8,7,6,5] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4},{5,6,7,8}}
=> ? = 3
[2,1,4,3,8,7,6,5] => [2,1,4,3,8,7,6,5] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> {{1,2},{3,4},{5,6,7,8}}
=> ? = 3
[1,2,4,3,8,7,6,5] => [1,2,4,3,8,7,6,5] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> {{1},{2},{3,4},{5,6,7,8}}
=> ? = 4
[3,2,1,4,8,7,6,5] => [3,2,1,4,8,7,6,5] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> {{1,2,3},{4},{5,6,7,8}}
=> ? = 3
[2,3,1,4,8,7,6,5] => [3,1,2,4,8,7,6,5] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> {{1,2,3},{4},{5,6,7,8}}
=> ? = 3
[1,3,2,4,8,7,6,5] => [1,3,2,4,8,7,6,5] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3},{4},{5,6,7,8}}
=> ? = 4
[2,1,3,4,8,7,6,5] => [2,1,3,4,8,7,6,5] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> {{1,2},{3},{4},{5,6,7,8}}
=> ? = 4
[1,2,3,4,8,7,6,5] => [1,2,3,4,8,7,6,5] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> {{1},{2},{3},{4},{5,6,7,8}}
=> ? = 5
[1,5,4,3,2,8,7,6] => [1,5,4,3,2,8,7,6] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> {{1},{2,3,4,5},{6,7,8}}
=> ? = 3
[1,5,4,3,2,7,8,6] => [1,5,4,3,2,8,6,7] => ?
=> ?
=> ? = 3
[1,2,5,4,3,8,7,6] => [1,2,5,4,3,8,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5},{6,7,8}}
=> ? = 4
[4,3,2,1,5,8,7,6] => [4,3,2,1,5,8,7,6] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> {{1,2,3,4},{5},{6,7,8}}
=> ? = 3
[4,3,2,1,5,7,8,6] => [4,3,2,1,5,8,6,7] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> {{1,2,3,4},{5},{6,7,8}}
=> ? = 3
[1,4,3,2,5,8,7,6] => [1,4,3,2,5,8,7,6] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> {{1},{2,3,4},{5},{6,7,8}}
=> ? = 4
Description
The number of topologically connected components of a set partition.
For example, the set partition $\{\{1,5\},\{2,3\},\{4,6\}\}$ has the two connected components $\{1,4,5,6\}$ and $\{2,3\}$.
The number of set partitions with only one block is [[oeis:A099947]].
Matching statistic: St001068
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 85%●distinct values known / distinct values provided: 64%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 85%●distinct values known / distinct values provided: 64%
Values
[1] => [1] => [1,0]
=> 1
[1,2] => [1,2] => [1,0,1,0]
=> 2
[2,1] => [2,1] => [1,1,0,0]
=> 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 3
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 3
[1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 3
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2
[2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2
[2,4,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 1
[3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[3,4,2,1] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3
[4,1,3,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2
[4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
[4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 4
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,2,5,3,4] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 4
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 4
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,3,5,2,4] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,3,5,4,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,4,2,3,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 4
[1,4,2,5,3] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 3
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,4,3,5,2] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,4,5,2,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 3
[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]
=> ? = 1
[7,8,6,5,4,3,2,1] => [8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[8,6,7,5,4,3,2,1] => [8,7,6,5,4,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[7,6,8,5,4,3,2,1] => [8,7,6,5,4,2,1,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[6,7,8,5,4,3,2,1] => [8,7,6,5,4,1,2,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[8,7,5,6,4,3,2,1] => [8,7,6,5,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[7,5,6,8,4,3,2,1] => [8,7,6,5,2,3,1,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[6,5,7,8,4,3,2,1] => [8,7,6,5,2,1,3,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[5,6,7,8,4,3,2,1] => [8,7,6,5,1,2,3,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[8,7,6,4,5,3,2,1] => [8,7,6,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[7,4,5,6,8,3,2,1] => [8,7,6,2,3,4,1,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[8,7,6,5,3,4,2,1] => [8,7,5,6,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[8,6,7,5,3,4,2,1] => [8,7,5,6,4,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[8,7,6,5,4,2,3,1] => [8,6,7,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[8,6,7,5,4,2,3,1] => [8,6,7,5,4,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[8,7,5,6,4,2,3,1] => [8,6,7,5,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[8,6,5,7,4,2,3,1] => [8,6,7,5,3,2,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[8,5,6,7,4,2,3,1] => [8,6,7,5,2,3,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[8,6,7,4,5,2,3,1] => [8,6,7,4,5,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[8,4,5,6,7,2,3,1] => [8,6,7,2,3,4,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[8,6,7,5,2,3,4,1] => [8,5,6,7,4,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[8,6,7,2,3,4,5,1] => [8,4,5,6,7,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[8,4,5,3,6,2,7,1] => [8,6,4,2,3,5,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[8,5,6,4,2,3,7,1] => [8,5,6,4,2,3,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[8,3,4,2,5,6,7,1] => [8,4,2,3,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[3,4,5,2,6,7,8,1] => [8,4,1,2,3,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[5,2,3,4,6,7,8,1] => [8,2,3,4,1,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[3,4,2,5,6,7,8,1] => [8,3,1,2,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[4,2,3,5,6,7,8,1] => [8,2,3,1,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[3,2,4,5,6,7,8,1] => [8,2,1,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[2,3,4,5,6,7,8,1] => [8,1,2,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[7,8,5,6,3,4,1,2] => [7,8,5,6,3,4,1,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[6,5,7,8,3,4,1,2] => [7,8,5,6,2,1,3,4] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[5,6,7,8,3,4,1,2] => [7,8,5,6,1,2,3,4] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[7,8,4,3,5,6,1,2] => [7,8,4,3,5,6,1,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[7,8,3,4,5,6,1,2] => [7,8,3,4,5,6,1,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[5,6,3,4,7,8,1,2] => [7,8,3,4,1,2,5,6] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[5,4,3,6,7,8,1,2] => [7,8,3,2,1,4,5,6] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[3,4,5,6,7,8,1,2] => [7,8,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[7,8,5,6,2,1,3,4] => [6,5,7,8,3,4,1,2] => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> ? = 3
[6,5,7,8,2,1,3,4] => [6,5,7,8,2,1,3,4] => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> ? = 3
[7,8,5,6,1,2,3,4] => [5,6,7,8,3,4,1,2] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 4
[5,6,7,8,1,2,3,4] => [5,6,7,8,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 4
[7,8,3,4,1,2,5,6] => [5,6,3,4,7,8,1,2] => [1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0]
=> ? = 4
[7,8,3,2,1,4,5,6] => [5,4,3,6,7,8,1,2] => [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
=> ? = 4
[7,8,1,2,3,4,5,6] => [3,4,5,6,7,8,1,2] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 6
[8,6,5,4,3,2,1,7] => [7,6,5,4,3,2,8,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 2
[8,2,3,4,1,5,6,7] => [5,2,3,4,6,7,8,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 4
[8,4,1,2,3,5,6,7] => [3,4,5,2,6,7,8,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 6
[8,2,3,1,4,5,6,7] => [4,2,3,5,6,7,8,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 5
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
Matching statistic: St000053
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 85%●distinct values known / distinct values provided: 64%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 85%●distinct values known / distinct values provided: 64%
Values
[1] => [1] => [1,0]
=> 0 = 1 - 1
[1,2] => [1,2] => [1,0,1,0]
=> 1 = 2 - 1
[2,1] => [2,1] => [1,1,0,0]
=> 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> 1 = 2 - 1
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[2,4,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[3,4,2,1] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[4,1,3,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,2,5,3,4] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,3,5,2,4] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,3,5,4,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,4,2,3,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,4,2,5,3] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,4,3,5,2] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[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]
=> ? = 1 - 1
[7,8,6,5,4,3,2,1] => [8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[8,6,7,5,4,3,2,1] => [8,7,6,5,4,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[7,6,8,5,4,3,2,1] => [8,7,6,5,4,2,1,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[6,7,8,5,4,3,2,1] => [8,7,6,5,4,1,2,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[8,7,5,6,4,3,2,1] => [8,7,6,5,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[7,5,6,8,4,3,2,1] => [8,7,6,5,2,3,1,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[6,5,7,8,4,3,2,1] => [8,7,6,5,2,1,3,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[5,6,7,8,4,3,2,1] => [8,7,6,5,1,2,3,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[8,7,6,4,5,3,2,1] => [8,7,6,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[7,4,5,6,8,3,2,1] => [8,7,6,2,3,4,1,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[8,7,6,5,3,4,2,1] => [8,7,5,6,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[8,6,7,5,3,4,2,1] => [8,7,5,6,4,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[8,7,6,5,4,2,3,1] => [8,6,7,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[8,6,7,5,4,2,3,1] => [8,6,7,5,4,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[8,7,5,6,4,2,3,1] => [8,6,7,5,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[8,6,5,7,4,2,3,1] => [8,6,7,5,3,2,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[8,5,6,7,4,2,3,1] => [8,6,7,5,2,3,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[8,6,7,4,5,2,3,1] => [8,6,7,4,5,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[8,4,5,6,7,2,3,1] => [8,6,7,2,3,4,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[8,6,7,5,2,3,4,1] => [8,5,6,7,4,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[8,6,7,2,3,4,5,1] => [8,4,5,6,7,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[8,4,5,3,6,2,7,1] => [8,6,4,2,3,5,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[8,5,6,4,2,3,7,1] => [8,5,6,4,2,3,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[8,3,4,2,5,6,7,1] => [8,4,2,3,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[3,4,5,2,6,7,8,1] => [8,4,1,2,3,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[5,2,3,4,6,7,8,1] => [8,2,3,4,1,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[3,4,2,5,6,7,8,1] => [8,3,1,2,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[4,2,3,5,6,7,8,1] => [8,2,3,1,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[3,2,4,5,6,7,8,1] => [8,2,1,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[2,3,4,5,6,7,8,1] => [8,1,2,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[7,8,5,6,3,4,1,2] => [7,8,5,6,3,4,1,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2 - 1
[6,5,7,8,3,4,1,2] => [7,8,5,6,2,1,3,4] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2 - 1
[5,6,7,8,3,4,1,2] => [7,8,5,6,1,2,3,4] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2 - 1
[7,8,4,3,5,6,1,2] => [7,8,4,3,5,6,1,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2 - 1
[7,8,3,4,5,6,1,2] => [7,8,3,4,5,6,1,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2 - 1
[5,6,3,4,7,8,1,2] => [7,8,3,4,1,2,5,6] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2 - 1
[5,4,3,6,7,8,1,2] => [7,8,3,2,1,4,5,6] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2 - 1
[3,4,5,6,7,8,1,2] => [7,8,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2 - 1
[7,8,5,6,2,1,3,4] => [6,5,7,8,3,4,1,2] => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> ? = 3 - 1
[6,5,7,8,2,1,3,4] => [6,5,7,8,2,1,3,4] => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> ? = 3 - 1
[7,8,5,6,1,2,3,4] => [5,6,7,8,3,4,1,2] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 4 - 1
[5,6,7,8,1,2,3,4] => [5,6,7,8,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 4 - 1
[7,8,3,4,1,2,5,6] => [5,6,3,4,7,8,1,2] => [1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0]
=> ? = 4 - 1
[7,8,3,2,1,4,5,6] => [5,4,3,6,7,8,1,2] => [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
=> ? = 4 - 1
[7,8,1,2,3,4,5,6] => [3,4,5,6,7,8,1,2] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 6 - 1
[8,6,5,4,3,2,1,7] => [7,6,5,4,3,2,8,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 2 - 1
[8,2,3,4,1,5,6,7] => [5,2,3,4,6,7,8,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 4 - 1
[8,4,1,2,3,5,6,7] => [3,4,5,2,6,7,8,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
[8,2,3,1,4,5,6,7] => [4,2,3,5,6,7,8,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 1
Description
The number of valleys of the Dyck path.
The following 74 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000025The number of initial rises of a Dyck path. St000105The number of blocks in the set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000024The number of double up and double down steps of a Dyck path. St000167The number of leaves of an ordered tree. St000234The number of global ascents of a permutation. St000912The number of maximal antichains in a poset. St000068The number of minimal elements in a poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000989The number of final rises of a permutation. St000007The number of saliances of the permutation. St000546The number of global descents of a permutation. St000740The last entry of a permutation. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000354The number of recoils 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. St000702The number of weak deficiencies of a permutation. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000654The first descent of a permutation. St000292The number of ascents of a binary word. St001461The number of topologically connected components of the chord diagram of a permutation. St000054The first entry of the permutation. St000237The number of small exceedances. St000031The number of cycles in the cycle decomposition of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000990The first ascent of a permutation. St000843The decomposition number of a perfect matching. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000991The number of right-to-left minima of a permutation. St000314The number of left-to-right-maxima of a permutation. St000015The number of peaks of a Dyck path. St000084The number of subtrees. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000286The number of connected components of the complement of a graph. St000325The width of the tree associated to a permutation. 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. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000133The "bounce" of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000203The number of external nodes of a binary tree. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001180Number of indecomposable injective modules with projective dimension at most 1. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000061The number of nodes on the left branch of a binary tree. St000083The number of left oriented leafs of a binary tree except the first one. St000181The number of connected components of the Hasse diagram for the poset. St001863The number of weak excedances of a signed permutation. St000942The number of critical left to right maxima of the parking functions. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001712The number of natural descents of a standard Young tableau. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. 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 $eA$. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
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