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Matching statistic: St000837
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Mp00252: Permutations —restriction⟶ Permutations
St000837: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000837: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,2] => 0
[1,3,2] => [1,2] => 0
[2,1,3] => [2,1] => 0
[2,3,1] => [2,1] => 0
[3,1,2] => [1,2] => 0
[3,2,1] => [2,1] => 0
[1,2,3,4] => [1,2,3] => 1
[1,2,4,3] => [1,2,3] => 1
[1,3,2,4] => [1,3,2] => 1
[1,3,4,2] => [1,3,2] => 1
[1,4,2,3] => [1,2,3] => 1
[1,4,3,2] => [1,3,2] => 1
[2,1,3,4] => [2,1,3] => 1
[2,1,4,3] => [2,1,3] => 1
[2,3,1,4] => [2,3,1] => 0
[2,3,4,1] => [2,3,1] => 0
[2,4,1,3] => [2,1,3] => 1
[2,4,3,1] => [2,3,1] => 0
[3,1,2,4] => [3,1,2] => 0
[3,1,4,2] => [3,1,2] => 0
[3,2,1,4] => [3,2,1] => 0
[3,2,4,1] => [3,2,1] => 0
[3,4,1,2] => [3,1,2] => 0
[3,4,2,1] => [3,2,1] => 0
[4,1,2,3] => [1,2,3] => 1
[4,1,3,2] => [1,3,2] => 1
[4,2,1,3] => [2,1,3] => 1
[4,2,3,1] => [2,3,1] => 0
[4,3,1,2] => [3,1,2] => 0
[4,3,2,1] => [3,2,1] => 0
[1,2,3,4,5] => [1,2,3,4] => 2
[1,2,3,5,4] => [1,2,3,4] => 2
[1,2,4,3,5] => [1,2,4,3] => 2
[1,2,4,5,3] => [1,2,4,3] => 2
[1,2,5,3,4] => [1,2,3,4] => 2
[1,2,5,4,3] => [1,2,4,3] => 2
[1,3,2,4,5] => [1,3,2,4] => 2
[1,3,2,5,4] => [1,3,2,4] => 2
[1,3,4,2,5] => [1,3,4,2] => 1
[1,3,4,5,2] => [1,3,4,2] => 1
[1,3,5,2,4] => [1,3,2,4] => 2
[1,3,5,4,2] => [1,3,4,2] => 1
[1,4,2,3,5] => [1,4,2,3] => 1
[1,4,2,5,3] => [1,4,2,3] => 1
[1,4,3,2,5] => [1,4,3,2] => 1
[1,4,3,5,2] => [1,4,3,2] => 1
[1,4,5,2,3] => [1,4,2,3] => 1
[1,4,5,3,2] => [1,4,3,2] => 1
[1,5,2,3,4] => [1,2,3,4] => 2
[1,5,2,4,3] => [1,2,4,3] => 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: St000836
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000836: Permutations ⟶ ℤResult quality: 83% ●values known / values provided: 93%●distinct values known / distinct values provided: 83%
Mp00252: Permutations —restriction⟶ Permutations
St000836: Permutations ⟶ ℤResult quality: 83% ●values known / values provided: 93%●distinct values known / distinct values provided: 83%
Values
[1,2,3] => [3,2,1] => [2,1] => 0
[1,3,2] => [2,3,1] => [2,1] => 0
[2,1,3] => [3,1,2] => [1,2] => 0
[2,3,1] => [1,3,2] => [1,2] => 0
[3,1,2] => [2,1,3] => [2,1] => 0
[3,2,1] => [1,2,3] => [1,2] => 0
[1,2,3,4] => [4,3,2,1] => [3,2,1] => 1
[1,2,4,3] => [3,4,2,1] => [3,2,1] => 1
[1,3,2,4] => [4,2,3,1] => [2,3,1] => 1
[1,3,4,2] => [2,4,3,1] => [2,3,1] => 1
[1,4,2,3] => [3,2,4,1] => [3,2,1] => 1
[1,4,3,2] => [2,3,4,1] => [2,3,1] => 1
[2,1,3,4] => [4,3,1,2] => [3,1,2] => 1
[2,1,4,3] => [3,4,1,2] => [3,1,2] => 1
[2,3,1,4] => [4,1,3,2] => [1,3,2] => 0
[2,3,4,1] => [1,4,3,2] => [1,3,2] => 0
[2,4,1,3] => [3,1,4,2] => [3,1,2] => 1
[2,4,3,1] => [1,3,4,2] => [1,3,2] => 0
[3,1,2,4] => [4,2,1,3] => [2,1,3] => 0
[3,1,4,2] => [2,4,1,3] => [2,1,3] => 0
[3,2,1,4] => [4,1,2,3] => [1,2,3] => 0
[3,2,4,1] => [1,4,2,3] => [1,2,3] => 0
[3,4,1,2] => [2,1,4,3] => [2,1,3] => 0
[3,4,2,1] => [1,2,4,3] => [1,2,3] => 0
[4,1,2,3] => [3,2,1,4] => [3,2,1] => 1
[4,1,3,2] => [2,3,1,4] => [2,3,1] => 1
[4,2,1,3] => [3,1,2,4] => [3,1,2] => 1
[4,2,3,1] => [1,3,2,4] => [1,3,2] => 0
[4,3,1,2] => [2,1,3,4] => [2,1,3] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3] => 0
[1,2,3,4,5] => [5,4,3,2,1] => [4,3,2,1] => 2
[1,2,3,5,4] => [4,5,3,2,1] => [4,3,2,1] => 2
[1,2,4,3,5] => [5,3,4,2,1] => [3,4,2,1] => 2
[1,2,4,5,3] => [3,5,4,2,1] => [3,4,2,1] => 2
[1,2,5,3,4] => [4,3,5,2,1] => [4,3,2,1] => 2
[1,2,5,4,3] => [3,4,5,2,1] => [3,4,2,1] => 2
[1,3,2,4,5] => [5,4,2,3,1] => [4,2,3,1] => 2
[1,3,2,5,4] => [4,5,2,3,1] => [4,2,3,1] => 2
[1,3,4,2,5] => [5,2,4,3,1] => [2,4,3,1] => 1
[1,3,4,5,2] => [2,5,4,3,1] => [2,4,3,1] => 1
[1,3,5,2,4] => [4,2,5,3,1] => [4,2,3,1] => 2
[1,3,5,4,2] => [2,4,5,3,1] => [2,4,3,1] => 1
[1,4,2,3,5] => [5,3,2,4,1] => [3,2,4,1] => 1
[1,4,2,5,3] => [3,5,2,4,1] => [3,2,4,1] => 1
[1,4,3,2,5] => [5,2,3,4,1] => [2,3,4,1] => 1
[1,4,3,5,2] => [2,5,3,4,1] => [2,3,4,1] => 1
[1,4,5,2,3] => [3,2,5,4,1] => [3,2,4,1] => 1
[1,4,5,3,2] => [2,3,5,4,1] => [2,3,4,1] => 1
[1,5,2,3,4] => [4,3,2,5,1] => [4,3,2,1] => 2
[1,5,2,4,3] => [3,4,2,5,1] => [3,4,2,1] => 2
[8,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1] => ? = 5
[1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 5
[1,3,5,7,8,6,4,2] => [2,4,6,8,7,5,3,1] => [2,4,6,7,5,3,1] => ? = 3
[1,3,6,5,7,8,4,2] => [2,4,8,7,5,6,3,1] => ? => ? = 3
[1,3,5,6,7,8,4,2] => [2,4,8,7,6,5,3,1] => [2,4,7,6,5,3,1] => ? = 3
[1,4,3,8,7,6,5,2] => [2,5,6,7,8,3,4,1] => [2,5,6,7,3,4,1] => ? = 3
[1,4,3,7,8,6,5,2] => [2,5,6,8,7,3,4,1] => ? => ? = 3
[1,3,4,8,7,6,5,2] => [2,5,6,7,8,4,3,1] => [2,5,6,7,4,3,1] => ? = 3
[1,3,4,5,7,8,6,2] => [2,6,8,7,5,4,3,1] => [2,6,7,5,4,3,1] => ? = 4
[1,4,3,6,5,8,7,2] => [2,7,8,5,6,3,4,1] => [2,7,5,6,3,4,1] => ? = 4
[1,3,4,5,6,8,7,2] => [2,7,8,6,5,4,3,1] => [2,7,6,5,4,3,1] => ? = 4
[1,4,5,3,6,7,8,2] => [2,8,7,6,3,5,4,1] => ? => ? = 3
[1,4,3,5,6,7,8,2] => [2,8,7,6,5,3,4,1] => [2,7,6,5,3,4,1] => ? = 4
[1,3,4,5,6,7,8,2] => [2,8,7,6,5,4,3,1] => ? => ? = 4
[1,2,6,7,8,5,4,3] => [3,4,5,8,7,6,2,1] => [3,4,5,7,6,2,1] => ? = 2
[1,2,5,6,7,8,4,3] => [3,4,8,7,6,5,2,1] => [3,4,7,6,5,2,1] => ? = 3
[1,2,4,8,7,6,5,3] => [3,5,6,7,8,4,2,1] => [3,5,6,7,4,2,1] => ? = 3
[1,2,4,6,7,8,5,3] => [3,5,8,7,6,4,2,1] => [3,5,7,6,4,2,1] => ? = 3
[1,2,6,5,4,7,8,3] => [3,8,7,4,5,6,2,1] => ? => ? = 3
[1,2,5,6,4,7,8,3] => [3,8,7,4,6,5,2,1] => ? => ? = 3
[1,2,5,4,6,7,8,3] => [3,8,7,6,4,5,2,1] => [3,7,6,4,5,2,1] => ? = 4
[1,2,4,5,6,7,8,3] => [3,8,7,6,5,4,2,1] => ? => ? = 4
[1,3,2,8,7,6,5,4] => [4,5,6,7,8,2,3,1] => [4,5,6,7,2,3,1] => ? = 3
[1,2,3,8,7,6,5,4] => [4,5,6,7,8,3,2,1] => [4,5,6,7,3,2,1] => ? = 3
[1,2,3,7,8,6,5,4] => [4,5,6,8,7,3,2,1] => ? => ? = 3
[1,2,3,6,8,7,5,4] => [4,5,7,8,6,3,2,1] => ? => ? = 3
[1,2,3,7,6,8,5,4] => [4,5,8,6,7,3,2,1] => ? => ? = 3
[1,2,3,6,7,8,5,4] => [4,5,8,7,6,3,2,1] => ? => ? = 3
[1,2,3,5,8,7,6,4] => [4,6,7,8,5,3,2,1] => ? => ? = 4
[1,2,3,5,7,8,6,4] => [4,6,8,7,5,3,2,1] => ? => ? = 4
[1,2,3,6,5,8,7,4] => [4,7,8,5,6,3,2,1] => ? => ? = 4
[1,2,3,5,6,8,7,4] => [4,7,8,6,5,3,2,1] => ? => ? = 4
[1,2,3,7,6,5,8,4] => [4,8,5,6,7,3,2,1] => ? => ? = 3
[1,2,3,6,7,5,8,4] => [4,8,5,7,6,3,2,1] => ? => ? = 3
[1,2,3,5,7,6,8,4] => [4,8,6,7,5,3,2,1] => ? => ? = 4
[1,2,3,6,5,7,8,4] => [4,8,7,5,6,3,2,1] => [4,7,5,6,3,2,1] => ? = 4
[1,2,3,5,6,7,8,4] => [4,8,7,6,5,3,2,1] => [4,7,6,5,3,2,1] => ? = 4
[1,4,3,2,8,7,6,5] => [5,6,7,8,2,3,4,1] => [5,6,7,2,3,4,1] => ? = 3
[1,4,3,2,7,6,8,5] => [5,8,6,7,2,3,4,1] => ? => ? = 3
[1,4,3,2,6,7,8,5] => [5,8,7,6,2,3,4,1] => ? => ? = 3
[1,3,4,2,8,7,6,5] => [5,6,7,8,2,4,3,1] => [5,6,7,2,4,3,1] => ? = 3
[1,3,4,2,7,6,8,5] => [5,8,6,7,2,4,3,1] => ? => ? = 3
[1,3,4,2,6,7,8,5] => [5,8,7,6,2,4,3,1] => [5,7,6,2,4,3,1] => ? = 3
[1,2,4,3,8,7,6,5] => [5,6,7,8,3,4,2,1] => [5,6,7,3,4,2,1] => ? = 4
[1,2,4,3,7,6,8,5] => [5,8,6,7,3,4,2,1] => ? => ? = 4
[1,2,4,3,6,7,8,5] => [5,8,7,6,3,4,2,1] => [5,7,6,3,4,2,1] => ? = 4
[1,3,2,4,8,7,6,5] => [5,6,7,8,4,2,3,1] => [5,6,7,4,2,3,1] => ? = 4
[1,3,2,4,7,6,8,5] => [5,8,6,7,4,2,3,1] => ? => ? = 4
[1,3,2,4,6,7,8,5] => [5,8,7,6,4,2,3,1] => [5,7,6,4,2,3,1] => ? = 4
[1,2,3,4,8,7,6,5] => [5,6,7,8,4,3,2,1] => [5,6,7,4,3,2,1] => ? = 4
Description
The number of descents of distance 2 of a permutation.
This is, $\operatorname{des}_2(\pi) = | \{ i : \pi(i) > \pi(i+2) \} |$.
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