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Your data matches 6 different statistics following compositions of up to 3 maps.
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Matching statistic: St001315
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001315: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001315: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> 2
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 4
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,5,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,4,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,6,4,5,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,5,4,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,3,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,3,6,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 5
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,5,3,4,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,6,3,4,5,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,6,3,5,4,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,5,4,3,6,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,6,4,3,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,6,4,5,3,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,2,4,6,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [3,2,5,4,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,6,4,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,2,6,5,4,1] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,2,3,5,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,2,3,6,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,2,3,4,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,2,3,5,4,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,4,3,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [6,2,4,3,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [6,2,4,5,3,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
Description
The dissociation number of a graph.
Matching statistic: St000836
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000836: Permutations ⟶ ℤResult quality: 83% ●values known / values provided: 99%●distinct values known / distinct values provided: 83%
St000836: Permutations ⟶ ℤResult quality: 83% ●values known / values provided: 99%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1] => ? = 2 - 2
[1,0,1,0]
=> [2,1] => 0 = 2 - 2
[1,1,0,0]
=> [1,2] => 0 = 2 - 2
[1,0,1,0,1,0]
=> [3,2,1] => 1 = 3 - 2
[1,0,1,1,0,0]
=> [2,3,1] => 1 = 3 - 2
[1,1,0,0,1,0]
=> [3,1,2] => 1 = 3 - 2
[1,1,0,1,0,0]
=> [2,1,3] => 0 = 2 - 2
[1,1,1,0,0,0]
=> [1,2,3] => 0 = 2 - 2
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 2 = 4 - 2
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2 = 4 - 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2 = 4 - 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1 = 3 - 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 3 - 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 2 = 4 - 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2 = 4 - 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 1 = 3 - 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1 = 3 - 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1 = 3 - 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1 = 3 - 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1 = 3 - 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 0 = 2 - 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 2 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 3 = 5 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 3 = 5 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 3 = 5 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 2 = 4 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 2 = 4 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 3 = 5 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 3 = 5 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 2 = 4 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 2 = 4 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2 = 4 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 2 = 4 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 2 = 4 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1 = 3 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 3 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 3 = 5 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 3 = 5 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 3 = 5 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 2 = 4 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 2 = 4 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 2 = 4 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 2 = 4 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 2 = 4 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 2 = 4 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => 2 = 4 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 2 = 4 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => 2 = 4 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 1 = 3 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 2 = 4 - 2
[]
=> [] => ? = 1 - 2
Description
The number of descents of distance 2 of a permutation.
This is, $\operatorname{des}_2(\pi) = | \{ i : \pi(i) > \pi(i+2) \} |$.
Matching statistic: St000837
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000837: Permutations ⟶ ℤResult quality: 83% ●values known / values provided: 99%●distinct values known / distinct values provided: 83%
St000837: Permutations ⟶ ℤResult quality: 83% ●values known / values provided: 99%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1] => ? = 2 - 2
[1,0,1,0]
=> [1,2] => 0 = 2 - 2
[1,1,0,0]
=> [2,1] => 0 = 2 - 2
[1,0,1,0,1,0]
=> [1,2,3] => 1 = 3 - 2
[1,0,1,1,0,0]
=> [1,3,2] => 1 = 3 - 2
[1,1,0,0,1,0]
=> [2,1,3] => 1 = 3 - 2
[1,1,0,1,0,0]
=> [2,3,1] => 0 = 2 - 2
[1,1,1,0,0,0]
=> [3,2,1] => 0 = 2 - 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 2 = 4 - 2
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2 = 4 - 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2 = 4 - 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1 = 3 - 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1 = 3 - 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2 = 4 - 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 4 - 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1 = 3 - 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1 = 3 - 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1 = 3 - 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1 = 3 - 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1 = 3 - 2
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 0 = 2 - 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0 = 2 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 3 = 5 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 3 = 5 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 3 = 5 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2 = 4 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2 = 4 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 3 = 5 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 3 = 5 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 2 = 4 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 2 = 4 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 2 = 4 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2 = 4 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 2 = 4 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 1 = 3 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1 = 3 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 3 = 5 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 3 = 5 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 3 = 5 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 2 = 4 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 2 = 4 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 2 = 4 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 2 = 4 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 2 = 4 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 2 = 4 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 2 = 4 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 2 = 4 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 2 = 4 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 1 = 3 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 2 = 4 - 2
[]
=> [] => ? = 1 - 2
Description
The number of ascents of distance 2 of a permutation.
This is, $\operatorname{asc}_2(\pi) = | \{ i : \pi(i) < \pi(i+2) \} |$.
Matching statistic: St001514
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001514: Dyck paths ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 67%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001514: Dyck paths ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> []
=> []
=> ? = 2 - 1
[1,0,1,0]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0]
=> []
=> []
=> ? = 2 - 1
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> []
=> []
=> ? = 2 - 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4 = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 6 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 6 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 5 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 5 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 6 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 6 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 5 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 5 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> ? = 5 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 5 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 5 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 4 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 6 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 6 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 6 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 5 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 5 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 5 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 5 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> ? = 5 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 5 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> ? = 5 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 4 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 4 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 5 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 5 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 5 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 5 - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 5 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 4 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 4 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 4 - 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 4 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 4 - 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 4 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 4 - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 3 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 6 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 6 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 6 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 5 - 1
Description
The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule.
Matching statistic: St000390
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 50%
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> 10 => 11 => 11 => 1 = 2 - 1
[1,0,1,0]
=> 1010 => 1101 => 1110 => 1 = 2 - 1
[1,1,0,0]
=> 1100 => 1101 => 1110 => 1 = 2 - 1
[1,0,1,0,1,0]
=> 101010 => 110101 => 111010 => 2 = 3 - 1
[1,0,1,1,0,0]
=> 101100 => 110101 => 111010 => 2 = 3 - 1
[1,1,0,0,1,0]
=> 110010 => 110101 => 111010 => 2 = 3 - 1
[1,1,0,1,0,0]
=> 110100 => 111001 => 111100 => 1 = 2 - 1
[1,1,1,0,0,0]
=> 111000 => 111001 => 111100 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> 10101010 => 11010101 => 11101010 => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> 10101100 => 11010101 => 11101010 => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> 10110010 => 11010101 => 11101010 => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> 10110100 => 11011001 => 11101100 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> 10111000 => 11011001 => 11101100 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> 11001010 => 11010101 => 11101010 => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> 11001100 => 11010101 => 11101010 => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> 11010010 => 11100101 => 11110010 => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> 11010100 => 11101001 => 11110100 => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> 11011000 => 11101001 => 11110100 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> 11100010 => 11100101 => 11110010 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> 11100100 => 11101001 => 11110100 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> 11101000 => 11110001 => 11111000 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> 11110000 => 11110001 => 11111000 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 1101010101 => 1110101010 => ? = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 1101010101 => 1110101010 => ? = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => 1101010101 => 1110101010 => ? = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => 1101011001 => 1110101100 => ? = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 1101011001 => 1110101100 => ? = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => 1101010101 => 1110101010 => ? = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 1101010101 => 1110101010 => ? = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => 1101100101 => 1110110010 => ? = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 1101101001 => 1110110100 => ? = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => 1101101001 => 1110110100 => ? = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 1101100101 => 1110110010 => ? = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => 1101101001 => 1110110100 => ? = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 1101110001 => 1110111000 => ? = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 1101110001 => 1110111000 => ? = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 1101010101 => 1110101010 => ? = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => 1101010101 => 1110101010 => ? = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => 1101010101 => 1110101010 => ? = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => 1101011001 => 1110101100 => ? = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 1101011001 => 1110101100 => ? = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => 1110010101 => 1111001010 => ? = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => 1110010101 => 1111001010 => ? = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => 1110100101 => 1111010010 => ? = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 1110101001 => 1111010100 => ? = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> 1101011000 => 1110101001 => 1111010100 => ? = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => 1110100101 => 1111010010 => ? = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> 1101100100 => 1110101001 => 1111010100 => ? = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => 1110110001 => 1111011000 => ? = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => 1110110001 => 1111011000 => ? = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => 1110010101 => 1111001010 => ? = 4 - 1
[1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => 1110010101 => 1111001010 => ? = 4 - 1
[1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => 1110100101 => 1111010010 => ? = 4 - 1
[1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => 1110101001 => 1111010100 => ? = 4 - 1
[1,1,1,0,0,1,1,0,0,0]
=> 1110011000 => 1110101001 => 1111010100 => ? = 4 - 1
[1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => 1111000101 => 1111100010 => ? = 3 - 1
[1,1,1,0,1,0,0,1,0,0]
=> 1110100100 => 1111001001 => 1111100100 => ? = 3 - 1
[1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => 1111010001 => 1111101000 => ? = 3 - 1
[1,1,1,0,1,1,0,0,0,0]
=> 1110110000 => 1111010001 => 1111101000 => ? = 3 - 1
[1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => 1111000101 => 1111100010 => ? = 3 - 1
[1,1,1,1,0,0,0,1,0,0]
=> 1111000100 => 1111001001 => 1111100100 => ? = 3 - 1
[1,1,1,1,0,0,1,0,0,0]
=> 1111001000 => 1111010001 => 1111101000 => ? = 3 - 1
[1,1,1,1,0,1,0,0,0,0]
=> 1111010000 => 1111100001 => 1111110000 => ? = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
=> 1111100000 => 1111100001 => 1111110000 => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => 110101010101 => 111010101010 => ? = 6 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => 110101010101 => 111010101010 => ? = 6 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> 101010110010 => 110101010101 => 111010101010 => ? = 6 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => 110101011001 => 111010101100 => ? = 5 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> 101010111000 => 110101011001 => 111010101100 => ? = 5 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> 101011001010 => 110101010101 => 111010101010 => ? = 6 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> 101011001100 => 110101010101 => 111010101010 => ? = 6 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> 101011010010 => 110101100101 => 111010110010 => ? = 5 - 1
Description
The number of runs of ones in a binary word.
Matching statistic: St000292
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 50%
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> 10 => 11 => 11 => 0 = 2 - 2
[1,0,1,0]
=> 1010 => 1101 => 1110 => 0 = 2 - 2
[1,1,0,0]
=> 1100 => 1101 => 1110 => 0 = 2 - 2
[1,0,1,0,1,0]
=> 101010 => 110101 => 111010 => 1 = 3 - 2
[1,0,1,1,0,0]
=> 101100 => 110101 => 111010 => 1 = 3 - 2
[1,1,0,0,1,0]
=> 110010 => 110101 => 111010 => 1 = 3 - 2
[1,1,0,1,0,0]
=> 110100 => 111001 => 111100 => 0 = 2 - 2
[1,1,1,0,0,0]
=> 111000 => 111001 => 111100 => 0 = 2 - 2
[1,0,1,0,1,0,1,0]
=> 10101010 => 11010101 => 11101010 => 2 = 4 - 2
[1,0,1,0,1,1,0,0]
=> 10101100 => 11010101 => 11101010 => 2 = 4 - 2
[1,0,1,1,0,0,1,0]
=> 10110010 => 11010101 => 11101010 => 2 = 4 - 2
[1,0,1,1,0,1,0,0]
=> 10110100 => 11011001 => 11101100 => 1 = 3 - 2
[1,0,1,1,1,0,0,0]
=> 10111000 => 11011001 => 11101100 => 1 = 3 - 2
[1,1,0,0,1,0,1,0]
=> 11001010 => 11010101 => 11101010 => 2 = 4 - 2
[1,1,0,0,1,1,0,0]
=> 11001100 => 11010101 => 11101010 => 2 = 4 - 2
[1,1,0,1,0,0,1,0]
=> 11010010 => 11100101 => 11110010 => 1 = 3 - 2
[1,1,0,1,0,1,0,0]
=> 11010100 => 11101001 => 11110100 => 1 = 3 - 2
[1,1,0,1,1,0,0,0]
=> 11011000 => 11101001 => 11110100 => 1 = 3 - 2
[1,1,1,0,0,0,1,0]
=> 11100010 => 11100101 => 11110010 => 1 = 3 - 2
[1,1,1,0,0,1,0,0]
=> 11100100 => 11101001 => 11110100 => 1 = 3 - 2
[1,1,1,0,1,0,0,0]
=> 11101000 => 11110001 => 11111000 => 0 = 2 - 2
[1,1,1,1,0,0,0,0]
=> 11110000 => 11110001 => 11111000 => 0 = 2 - 2
[1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 1101010101 => 1110101010 => ? = 5 - 2
[1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 1101010101 => 1110101010 => ? = 5 - 2
[1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => 1101010101 => 1110101010 => ? = 5 - 2
[1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => 1101011001 => 1110101100 => ? = 4 - 2
[1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 1101011001 => 1110101100 => ? = 4 - 2
[1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => 1101010101 => 1110101010 => ? = 5 - 2
[1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 1101010101 => 1110101010 => ? = 5 - 2
[1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => 1101100101 => 1110110010 => ? = 4 - 2
[1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 1101101001 => 1110110100 => ? = 4 - 2
[1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => 1101101001 => 1110110100 => ? = 4 - 2
[1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 1101100101 => 1110110010 => ? = 4 - 2
[1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => 1101101001 => 1110110100 => ? = 4 - 2
[1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 1101110001 => 1110111000 => ? = 3 - 2
[1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 1101110001 => 1110111000 => ? = 3 - 2
[1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 1101010101 => 1110101010 => ? = 5 - 2
[1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => 1101010101 => 1110101010 => ? = 5 - 2
[1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => 1101010101 => 1110101010 => ? = 5 - 2
[1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => 1101011001 => 1110101100 => ? = 4 - 2
[1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 1101011001 => 1110101100 => ? = 4 - 2
[1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => 1110010101 => 1111001010 => ? = 4 - 2
[1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => 1110010101 => 1111001010 => ? = 4 - 2
[1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => 1110100101 => 1111010010 => ? = 4 - 2
[1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 1110101001 => 1111010100 => ? = 4 - 2
[1,1,0,1,0,1,1,0,0,0]
=> 1101011000 => 1110101001 => 1111010100 => ? = 4 - 2
[1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => 1110100101 => 1111010010 => ? = 4 - 2
[1,1,0,1,1,0,0,1,0,0]
=> 1101100100 => 1110101001 => 1111010100 => ? = 4 - 2
[1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => 1110110001 => 1111011000 => ? = 3 - 2
[1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => 1110110001 => 1111011000 => ? = 3 - 2
[1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => 1110010101 => 1111001010 => ? = 4 - 2
[1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => 1110010101 => 1111001010 => ? = 4 - 2
[1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => 1110100101 => 1111010010 => ? = 4 - 2
[1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => 1110101001 => 1111010100 => ? = 4 - 2
[1,1,1,0,0,1,1,0,0,0]
=> 1110011000 => 1110101001 => 1111010100 => ? = 4 - 2
[1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => 1111000101 => 1111100010 => ? = 3 - 2
[1,1,1,0,1,0,0,1,0,0]
=> 1110100100 => 1111001001 => 1111100100 => ? = 3 - 2
[1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => 1111010001 => 1111101000 => ? = 3 - 2
[1,1,1,0,1,1,0,0,0,0]
=> 1110110000 => 1111010001 => 1111101000 => ? = 3 - 2
[1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => 1111000101 => 1111100010 => ? = 3 - 2
[1,1,1,1,0,0,0,1,0,0]
=> 1111000100 => 1111001001 => 1111100100 => ? = 3 - 2
[1,1,1,1,0,0,1,0,0,0]
=> 1111001000 => 1111010001 => 1111101000 => ? = 3 - 2
[1,1,1,1,0,1,0,0,0,0]
=> 1111010000 => 1111100001 => 1111110000 => ? = 2 - 2
[1,1,1,1,1,0,0,0,0,0]
=> 1111100000 => 1111100001 => 1111110000 => ? = 2 - 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => 110101010101 => 111010101010 => ? = 6 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => 110101010101 => 111010101010 => ? = 6 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> 101010110010 => 110101010101 => 111010101010 => ? = 6 - 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => 110101011001 => 111010101100 => ? = 5 - 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> 101010111000 => 110101011001 => 111010101100 => ? = 5 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> 101011001010 => 110101010101 => 111010101010 => ? = 6 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> 101011001100 => 110101010101 => 111010101010 => ? = 6 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> 101011010010 => 110101100101 => 111010110010 => ? = 5 - 2
Description
The number of ascents of a binary word.
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