Your data matches 6 different statistics following compositions of up to 3 maps.
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Matching statistic: St000356
(load all 4 compositions to match this statistic)
Mapping
St000356: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 1
[1,4,2,3] => 2
[1,4,3,2] => 2
[2,1,3,4] => 0
[2,1,4,3] => 1
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 0
[3,1,4,2] => 1
[3,2,1,4] => 0
[3,2,4,1] => 0
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,3,2] => 1
[4,2,1,3] => 0
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 0
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 1
[1,2,5,3,4] => 2
[1,2,5,4,3] => 2
[1,3,2,4,5] => 1
[1,3,2,5,4] => 2
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 2
[1,3,5,4,2] => 2
[1,4,2,3,5] => 2
[1,4,2,5,3] => 3
[1,4,3,2,5] => 2
[1,4,3,5,2] => 2
[1,4,5,2,3] => 2
Description
The number of occurrences of the pattern 13-2. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $13\!\!-\!\!2$.
Matching statistic: St000223
(load all 3 compositions to match this statistic)
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000223: Permutations ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 81%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [2,1] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [2,3,1] => [3,1,2] => 0
[1,3,2] => [2,3,1] => [3,2,1] => [3,2,1] => 1
[2,1,3] => [3,1,2] => [3,1,2] => [2,3,1] => 0
[2,3,1] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[3,1,2] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [2,3,4,1] => [4,1,2,3] => 0
[1,2,4,3] => [3,4,2,1] => [2,4,3,1] => [4,1,3,2] => 1
[1,3,2,4] => [4,2,3,1] => [3,4,2,1] => [4,3,1,2] => 1
[1,3,4,2] => [2,4,3,1] => [3,2,4,1] => [4,2,1,3] => 1
[1,4,2,3] => [3,2,4,1] => [4,3,2,1] => [4,3,2,1] => 2
[1,4,3,2] => [2,3,4,1] => [4,2,3,1] => [4,2,3,1] => 2
[2,1,3,4] => [4,3,1,2] => [3,1,4,2] => [2,4,1,3] => 0
[2,1,4,3] => [3,4,1,2] => [4,1,3,2] => [2,4,3,1] => 1
[2,3,1,4] => [4,1,3,2] => [3,4,1,2] => [3,4,1,2] => 0
[2,3,4,1] => [1,4,3,2] => [1,3,4,2] => [1,4,2,3] => 0
[2,4,1,3] => [3,1,4,2] => [4,3,1,2] => [3,4,2,1] => 1
[2,4,3,1] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1
[3,1,2,4] => [4,2,1,3] => [2,4,1,3] => [3,1,4,2] => 0
[3,1,4,2] => [2,4,1,3] => [4,2,1,3] => [3,2,4,1] => 1
[3,2,1,4] => [4,1,2,3] => [4,1,2,3] => [2,3,4,1] => 0
[3,2,4,1] => [1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 0
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[4,1,2,3] => [3,2,1,4] => [2,3,1,4] => [3,1,2,4] => 0
[4,1,3,2] => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[4,2,1,3] => [3,1,2,4] => [3,1,2,4] => [2,3,1,4] => 0
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[4,3,1,2] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [5,4,3,2,1] => [2,3,4,5,1] => [5,1,2,3,4] => 0
[1,2,3,5,4] => [4,5,3,2,1] => [2,3,5,4,1] => [5,1,2,4,3] => 1
[1,2,4,3,5] => [5,3,4,2,1] => [2,4,5,3,1] => [5,1,4,2,3] => 1
[1,2,4,5,3] => [3,5,4,2,1] => [2,4,3,5,1] => [5,1,3,2,4] => 1
[1,2,5,3,4] => [4,3,5,2,1] => [2,5,4,3,1] => [5,1,4,3,2] => 2
[1,2,5,4,3] => [3,4,5,2,1] => [2,5,3,4,1] => [5,1,3,4,2] => 2
[1,3,2,4,5] => [5,4,2,3,1] => [3,4,2,5,1] => [5,3,1,2,4] => 1
[1,3,2,5,4] => [4,5,2,3,1] => [3,5,2,4,1] => [5,3,1,4,2] => 2
[1,3,4,2,5] => [5,2,4,3,1] => [3,4,5,2,1] => [5,4,1,2,3] => 1
[1,3,4,5,2] => [2,5,4,3,1] => [3,2,4,5,1] => [5,2,1,3,4] => 1
[1,3,5,2,4] => [4,2,5,3,1] => [3,5,4,2,1] => [5,4,1,3,2] => 2
[1,3,5,4,2] => [2,4,5,3,1] => [3,2,5,4,1] => [5,2,1,4,3] => 2
[1,4,2,3,5] => [5,3,2,4,1] => [4,3,5,2,1] => [5,4,2,1,3] => 2
[1,4,2,5,3] => [3,5,2,4,1] => [4,5,3,2,1] => [5,4,3,1,2] => 3
[1,4,3,2,5] => [5,2,3,4,1] => [4,5,2,3,1] => [5,3,4,1,2] => 2
[1,4,3,5,2] => [2,5,3,4,1] => [4,2,5,3,1] => [5,2,4,1,3] => 2
[1,4,5,2,3] => [3,2,5,4,1] => [4,3,2,5,1] => [5,3,2,1,4] => 2
[1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [2,3,4,5,7,6,1] => [7,1,2,3,4,6,5] => ? = 1
[1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [2,3,4,6,7,5,1] => [7,1,2,3,6,4,5] => ? = 1
[1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => [2,3,4,6,5,7,1] => [7,1,2,3,5,4,6] => ? = 1
[1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => [2,3,4,7,6,5,1] => [7,1,2,3,6,5,4] => ? = 2
[1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [2,3,4,7,5,6,1] => [7,1,2,3,5,6,4] => ? = 2
[1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [2,3,5,6,4,7,1] => [7,1,2,5,3,4,6] => ? = 1
[1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [2,3,5,7,4,6,1] => [7,1,2,5,3,6,4] => ? = 2
[1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => [2,3,5,6,7,4,1] => [7,1,2,6,3,4,5] => ? = 1
[1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => [2,3,5,4,6,7,1] => [7,1,2,4,3,5,6] => ? = 1
[1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [2,3,5,7,6,4,1] => [7,1,2,6,3,5,4] => ? = 2
[1,2,3,5,7,6,4] => [4,6,7,5,3,2,1] => [2,3,5,4,7,6,1] => [7,1,2,4,3,6,5] => ? = 2
[1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => [2,3,6,5,7,4,1] => [7,1,2,6,4,3,5] => ? = 2
[1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [2,3,6,7,5,4,1] => [7,1,2,6,5,3,4] => ? = 3
[1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => [2,3,6,7,4,5,1] => [7,1,2,5,6,3,4] => ? = 2
[1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => [2,3,6,4,7,5,1] => [7,1,2,4,6,3,5] => ? = 2
[1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => [2,3,6,5,4,7,1] => [7,1,2,5,4,3,6] => ? = 2
[1,2,3,6,7,5,4] => [4,5,7,6,3,2,1] => [2,3,6,4,5,7,1] => [7,1,2,4,5,3,6] => ? = 2
[1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [2,3,7,5,6,4,1] => [7,1,2,6,4,5,3] => ? = 3
[1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => [2,3,7,6,5,4,1] => [7,1,2,6,5,4,3] => ? = 4
[1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => [2,3,7,6,4,5,1] => [7,1,2,5,6,4,3] => ? = 3
[1,2,3,7,5,6,4] => [4,6,5,7,3,2,1] => [2,3,7,4,6,5,1] => [7,1,2,4,6,5,3] => ? = 3
[1,2,3,7,6,4,5] => [5,4,6,7,3,2,1] => [2,3,7,5,4,6,1] => [7,1,2,5,4,6,3] => ? = 3
[1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => [2,3,7,4,5,6,1] => [7,1,2,4,5,6,3] => ? = 3
[1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [2,4,5,3,6,7,1] => [7,1,4,2,3,5,6] => ? = 1
[1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [2,4,5,3,7,6,1] => [7,1,4,2,3,6,5] => ? = 2
[1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => [2,4,6,3,7,5,1] => [7,1,4,2,6,3,5] => ? = 2
[1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => [2,4,6,3,5,7,1] => [7,1,4,2,5,3,6] => ? = 2
[1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => [2,4,7,3,6,5,1] => [7,1,4,2,6,5,3] => ? = 3
[1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => [2,4,7,3,5,6,1] => [7,1,4,2,5,6,3] => ? = 3
[1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => [2,4,5,6,3,7,1] => [7,1,5,2,3,4,6] => ? = 1
[1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => [2,4,5,7,3,6,1] => [7,1,5,2,3,6,4] => ? = 2
[1,2,4,5,6,3,7] => [7,3,6,5,4,2,1] => [2,4,5,6,7,3,1] => [7,1,6,2,3,4,5] => ? = 1
[1,2,4,5,6,7,3] => [3,7,6,5,4,2,1] => [2,4,3,5,6,7,1] => [7,1,3,2,4,5,6] => ? = 1
[1,2,4,5,7,3,6] => [6,3,7,5,4,2,1] => [2,4,5,7,6,3,1] => [7,1,6,2,3,5,4] => ? = 2
[1,2,4,5,7,6,3] => [3,6,7,5,4,2,1] => [2,4,3,5,7,6,1] => [7,1,3,2,4,6,5] => ? = 2
[1,2,4,6,3,5,7] => [7,5,3,6,4,2,1] => [2,4,6,5,7,3,1] => [7,1,6,2,4,3,5] => ? = 2
[1,2,4,6,3,7,5] => [5,7,3,6,4,2,1] => [2,4,6,7,5,3,1] => [7,1,6,2,5,3,4] => ? = 3
[1,2,4,6,5,3,7] => [7,3,5,6,4,2,1] => [2,4,6,7,3,5,1] => [7,1,5,2,6,3,4] => ? = 2
[1,2,4,6,5,7,3] => [3,7,5,6,4,2,1] => [2,4,3,6,7,5,1] => [7,1,3,2,6,4,5] => ? = 2
[1,2,4,6,7,3,5] => [5,3,7,6,4,2,1] => [2,4,6,5,3,7,1] => [7,1,5,2,4,3,6] => ? = 2
[1,2,4,6,7,5,3] => [3,5,7,6,4,2,1] => [2,4,3,6,5,7,1] => [7,1,3,2,5,4,6] => ? = 2
[1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => [2,4,7,5,6,3,1] => [7,1,6,2,4,5,3] => ? = 3
[1,2,4,7,3,6,5] => [5,6,3,7,4,2,1] => [2,4,7,6,5,3,1] => [7,1,6,2,5,4,3] => ? = 4
[1,2,4,7,5,3,6] => [6,3,5,7,4,2,1] => [2,4,7,6,3,5,1] => [7,1,5,2,6,4,3] => ? = 3
[1,2,4,7,5,6,3] => [3,6,5,7,4,2,1] => [2,4,3,7,6,5,1] => [7,1,3,2,6,5,4] => ? = 3
[1,2,4,7,6,3,5] => [5,3,6,7,4,2,1] => [2,4,7,5,3,6,1] => [7,1,5,2,4,6,3] => ? = 3
[1,2,4,7,6,5,3] => [3,5,6,7,4,2,1] => [2,4,3,7,5,6,1] => [7,1,3,2,5,6,4] => ? = 3
[1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => [2,5,4,6,3,7,1] => [7,1,5,3,2,4,6] => ? = 2
[1,2,5,3,4,7,6] => [6,7,4,3,5,2,1] => [2,5,4,7,3,6,1] => [7,1,5,3,2,6,4] => ? = 3
[1,2,5,3,6,4,7] => [7,4,6,3,5,2,1] => [2,5,6,7,4,3,1] => [7,1,6,5,2,3,4] => ? = 3
Description
The number of nestings in the permutation.
Matching statistic: St000358
(load all 2 compositions to match this statistic)
Mapping
Mp00069: Permutations —complement⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 44%
Values
[1] => [1] => 0
[1,2] => [2,1] => 0
[2,1] => [1,2] => 0
[1,2,3] => [3,2,1] => 0
[1,3,2] => [3,1,2] => 1
[2,1,3] => [2,3,1] => 0
[2,3,1] => [2,1,3] => 0
[3,1,2] => [1,3,2] => 0
[3,2,1] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [4,3,1,2] => 1
[1,3,2,4] => [4,2,3,1] => 1
[1,3,4,2] => [4,2,1,3] => 1
[1,4,2,3] => [4,1,3,2] => 2
[1,4,3,2] => [4,1,2,3] => 2
[2,1,3,4] => [3,4,2,1] => 0
[2,1,4,3] => [3,4,1,2] => 1
[2,3,1,4] => [3,2,4,1] => 0
[2,3,4,1] => [3,2,1,4] => 0
[2,4,1,3] => [3,1,4,2] => 1
[2,4,3,1] => [3,1,2,4] => 1
[3,1,2,4] => [2,4,3,1] => 0
[3,1,4,2] => [2,4,1,3] => 1
[3,2,1,4] => [2,3,4,1] => 0
[3,2,4,1] => [2,3,1,4] => 0
[3,4,1,2] => [2,1,4,3] => 0
[3,4,2,1] => [2,1,3,4] => 0
[4,1,2,3] => [1,4,3,2] => 0
[4,1,3,2] => [1,4,2,3] => 1
[4,2,1,3] => [1,3,4,2] => 0
[4,2,3,1] => [1,3,2,4] => 0
[4,3,1,2] => [1,2,4,3] => 0
[4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [5,4,3,1,2] => 1
[1,2,4,3,5] => [5,4,2,3,1] => 1
[1,2,4,5,3] => [5,4,2,1,3] => 1
[1,2,5,3,4] => [5,4,1,3,2] => 2
[1,2,5,4,3] => [5,4,1,2,3] => 2
[1,3,2,4,5] => [5,3,4,2,1] => 1
[1,3,2,5,4] => [5,3,4,1,2] => 2
[1,3,4,2,5] => [5,3,2,4,1] => 1
[1,3,4,5,2] => [5,3,2,1,4] => 1
[1,3,5,2,4] => [5,3,1,4,2] => 2
[1,3,5,4,2] => [5,3,1,2,4] => 2
[1,4,2,3,5] => [5,2,4,3,1] => 2
[1,4,2,5,3] => [5,2,4,1,3] => 3
[1,4,3,2,5] => [5,2,3,4,1] => 2
[1,4,3,5,2] => [5,2,3,1,4] => 2
[1,4,5,2,3] => [5,2,1,4,3] => 2
[1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0
[1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => ? = 1
[1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 1
[1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 1
[1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 2
[1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => ? = 2
[1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 1
[1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => ? = 2
[1,2,3,5,6,4,7] => [7,6,5,3,2,4,1] => ? = 1
[1,2,3,5,6,7,4] => [7,6,5,3,2,1,4] => ? = 1
[1,2,3,5,7,4,6] => [7,6,5,3,1,4,2] => ? = 2
[1,2,3,5,7,6,4] => [7,6,5,3,1,2,4] => ? = 2
[1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 2
[1,2,3,6,4,7,5] => [7,6,5,2,4,1,3] => ? = 3
[1,2,3,6,5,4,7] => [7,6,5,2,3,4,1] => ? = 2
[1,2,3,6,5,7,4] => [7,6,5,2,3,1,4] => ? = 2
[1,2,3,6,7,4,5] => [7,6,5,2,1,4,3] => ? = 2
[1,2,3,6,7,5,4] => [7,6,5,2,1,3,4] => ? = 2
[1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => ? = 3
[1,2,3,7,4,6,5] => [7,6,5,1,4,2,3] => ? = 4
[1,2,3,7,5,4,6] => [7,6,5,1,3,4,2] => ? = 3
[1,2,3,7,5,6,4] => [7,6,5,1,3,2,4] => ? = 3
[1,2,3,7,6,4,5] => [7,6,5,1,2,4,3] => ? = 3
[1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => ? = 3
[1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => ? = 1
[1,2,4,3,5,7,6] => [7,6,4,5,3,1,2] => ? = 2
[1,2,4,3,6,5,7] => [7,6,4,5,2,3,1] => ? = 2
[1,2,4,3,6,7,5] => [7,6,4,5,2,1,3] => ? = 2
[1,2,4,3,7,5,6] => [7,6,4,5,1,3,2] => ? = 3
[1,2,4,3,7,6,5] => [7,6,4,5,1,2,3] => ? = 3
[1,2,4,5,3,6,7] => [7,6,4,3,5,2,1] => ? = 1
[1,2,4,5,3,7,6] => [7,6,4,3,5,1,2] => ? = 2
[1,2,4,5,6,3,7] => [7,6,4,3,2,5,1] => ? = 1
[1,2,4,5,6,7,3] => [7,6,4,3,2,1,5] => ? = 1
[1,2,4,5,7,3,6] => [7,6,4,3,1,5,2] => ? = 2
[1,2,4,5,7,6,3] => [7,6,4,3,1,2,5] => ? = 2
[1,2,4,6,3,5,7] => [7,6,4,2,5,3,1] => ? = 2
[1,2,4,6,3,7,5] => [7,6,4,2,5,1,3] => ? = 3
[1,2,4,6,5,3,7] => [7,6,4,2,3,5,1] => ? = 2
[1,2,4,6,5,7,3] => [7,6,4,2,3,1,5] => ? = 2
[1,2,4,6,7,3,5] => [7,6,4,2,1,5,3] => ? = 2
[1,2,4,6,7,5,3] => [7,6,4,2,1,3,5] => ? = 2
[1,2,4,7,3,5,6] => [7,6,4,1,5,3,2] => ? = 3
[1,2,4,7,3,6,5] => [7,6,4,1,5,2,3] => ? = 4
[1,2,4,7,5,3,6] => [7,6,4,1,3,5,2] => ? = 3
[1,2,4,7,5,6,3] => [7,6,4,1,3,2,5] => ? = 3
[1,2,4,7,6,3,5] => [7,6,4,1,2,5,3] => ? = 3
[1,2,4,7,6,5,3] => [7,6,4,1,2,3,5] => ? = 3
[1,2,5,3,4,6,7] => [7,6,3,5,4,2,1] => ? = 2
[1,2,5,3,4,7,6] => [7,6,3,5,4,1,2] => ? = 3
Description
The number of occurrences of the pattern 31-2. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $31\!\!-\!\!2$.
Matching statistic: St000039
(load all 2 compositions to match this statistic)
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000039: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 44%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [2,1] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [2,3,1] => [3,2,1] => 0
[1,3,2] => [2,3,1] => [3,2,1] => [2,3,1] => 1
[2,1,3] => [3,1,2] => [3,1,2] => [3,1,2] => 0
[2,3,1] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[3,1,2] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [2,3,4,1] => [4,2,3,1] => 0
[1,2,4,3] => [3,4,2,1] => [2,4,3,1] => [3,2,4,1] => 1
[1,3,2,4] => [4,2,3,1] => [3,4,2,1] => [4,3,1,2] => 1
[1,3,4,2] => [2,4,3,1] => [3,2,4,1] => [2,4,3,1] => 1
[1,4,2,3] => [3,2,4,1] => [4,3,2,1] => [3,4,1,2] => 2
[1,4,3,2] => [2,3,4,1] => [4,2,3,1] => [2,3,4,1] => 2
[2,1,3,4] => [4,3,1,2] => [3,1,4,2] => [4,1,3,2] => 0
[2,1,4,3] => [3,4,1,2] => [4,1,3,2] => [3,1,4,2] => 1
[2,3,1,4] => [4,1,3,2] => [3,4,1,2] => [4,3,2,1] => 0
[2,3,4,1] => [1,4,3,2] => [1,3,4,2] => [1,4,3,2] => 0
[2,4,1,3] => [3,1,4,2] => [4,3,1,2] => [3,4,2,1] => 1
[2,4,3,1] => [1,3,4,2] => [1,4,3,2] => [1,3,4,2] => 1
[3,1,2,4] => [4,2,1,3] => [2,4,1,3] => [4,2,1,3] => 0
[3,1,4,2] => [2,4,1,3] => [4,2,1,3] => [2,4,1,3] => 1
[3,2,1,4] => [4,1,2,3] => [4,1,2,3] => [4,1,2,3] => 0
[3,2,4,1] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 0
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[4,1,2,3] => [3,2,1,4] => [2,3,1,4] => [3,2,1,4] => 0
[4,1,3,2] => [2,3,1,4] => [3,2,1,4] => [2,3,1,4] => 1
[4,2,1,3] => [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 0
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[4,3,1,2] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [5,4,3,2,1] => [2,3,4,5,1] => [5,2,3,4,1] => 0
[1,2,3,5,4] => [4,5,3,2,1] => [2,3,5,4,1] => [4,2,3,5,1] => 1
[1,2,4,3,5] => [5,3,4,2,1] => [2,4,5,3,1] => [5,2,4,1,3] => 1
[1,2,4,5,3] => [3,5,4,2,1] => [2,4,3,5,1] => [3,2,5,4,1] => 1
[1,2,5,3,4] => [4,3,5,2,1] => [2,5,4,3,1] => [4,2,5,1,3] => 2
[1,2,5,4,3] => [3,4,5,2,1] => [2,5,3,4,1] => [3,2,4,5,1] => 2
[1,3,2,4,5] => [5,4,2,3,1] => [3,4,2,5,1] => [5,3,1,4,2] => 1
[1,3,2,5,4] => [4,5,2,3,1] => [3,5,2,4,1] => [4,3,1,5,2] => 2
[1,3,4,2,5] => [5,2,4,3,1] => [3,4,5,2,1] => [5,4,3,1,2] => 1
[1,3,4,5,2] => [2,5,4,3,1] => [3,2,4,5,1] => [2,5,3,4,1] => 1
[1,3,5,2,4] => [4,2,5,3,1] => [3,5,4,2,1] => [5,3,4,1,2] => 2
[1,3,5,4,2] => [2,4,5,3,1] => [3,2,5,4,1] => [2,4,3,5,1] => 2
[1,4,2,3,5] => [5,3,2,4,1] => [4,3,5,2,1] => [4,5,3,1,2] => 2
[1,4,2,5,3] => [3,5,2,4,1] => [4,5,3,2,1] => [4,3,5,1,2] => 3
[1,4,3,2,5] => [5,2,3,4,1] => [4,5,2,3,1] => [5,4,1,2,3] => 2
[1,4,3,5,2] => [2,5,3,4,1] => [4,2,5,3,1] => [2,5,4,1,3] => 2
[1,4,5,2,3] => [3,2,5,4,1] => [4,3,2,5,1] => [3,5,1,4,2] => 2
[1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [2,3,4,5,6,7,1] => [7,2,3,4,5,6,1] => ? = 0
[1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [2,3,4,5,7,6,1] => [6,2,3,4,5,7,1] => ? = 1
[1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [2,3,4,6,7,5,1] => [7,2,3,4,6,1,5] => ? = 1
[1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => [2,3,4,6,5,7,1] => [5,2,3,4,7,6,1] => ? = 1
[1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => [2,3,4,7,6,5,1] => [6,2,3,4,7,1,5] => ? = 2
[1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [2,3,4,7,5,6,1] => [5,2,3,4,6,7,1] => ? = 2
[1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [2,3,5,6,4,7,1] => [7,2,3,5,1,6,4] => ? = 1
[1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [2,3,5,7,4,6,1] => [6,2,3,5,1,7,4] => ? = 2
[1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => [2,3,5,6,7,4,1] => [7,2,3,6,5,1,4] => ? = 1
[1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => [2,3,5,4,6,7,1] => [4,2,3,7,5,6,1] => ? = 1
[1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [2,3,5,7,6,4,1] => [7,2,3,5,6,1,4] => ? = 2
[1,2,3,5,7,6,4] => [4,6,7,5,3,2,1] => [2,3,5,4,7,6,1] => [4,2,3,6,5,7,1] => ? = 2
[1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => [2,3,6,5,7,4,1] => [6,2,3,7,5,1,4] => ? = 2
[1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [2,3,6,7,5,4,1] => [6,2,3,5,7,1,4] => ? = 3
[1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => [2,3,6,7,4,5,1] => [7,2,3,6,1,4,5] => ? = 2
[1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => [2,3,6,4,7,5,1] => [4,2,3,7,6,1,5] => ? = 2
[1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => [2,3,6,5,4,7,1] => [5,2,3,7,1,6,4] => ? = 2
[1,2,3,6,7,5,4] => [4,5,7,6,3,2,1] => [2,3,6,4,5,7,1] => [4,2,3,5,7,6,1] => ? = 2
[1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [2,3,7,5,6,4,1] => [5,2,3,7,6,1,4] => ? = 3
[1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => [2,3,7,6,5,4,1] => [5,2,3,6,7,1,4] => ? = 4
[1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => [2,3,7,6,4,5,1] => [6,2,3,7,1,4,5] => ? = 3
[1,2,3,7,5,6,4] => [4,6,5,7,3,2,1] => [2,3,7,4,6,5,1] => [4,2,3,6,7,1,5] => ? = 3
[1,2,3,7,6,4,5] => [5,4,6,7,3,2,1] => [2,3,7,5,4,6,1] => [5,2,3,6,1,7,4] => ? = 3
[1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => [2,3,7,4,5,6,1] => [4,2,3,5,6,7,1] => ? = 3
[1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [2,4,5,3,6,7,1] => [7,2,4,1,5,6,3] => ? = 1
[1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [2,4,5,3,7,6,1] => [6,2,4,1,5,7,3] => ? = 2
[1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => [2,4,6,3,7,5,1] => [7,2,4,1,6,3,5] => ? = 2
[1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => [2,4,6,3,5,7,1] => [5,2,4,1,7,6,3] => ? = 2
[1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => [2,4,7,3,6,5,1] => [6,2,4,1,7,3,5] => ? = 3
[1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => [2,4,7,3,5,6,1] => [5,2,4,1,6,7,3] => ? = 3
[1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => [2,4,5,6,3,7,1] => [7,2,5,4,1,6,3] => ? = 1
[1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => [2,4,5,7,3,6,1] => [6,2,5,4,1,7,3] => ? = 2
[1,2,4,5,6,3,7] => [7,3,6,5,4,2,1] => [2,4,5,6,7,3,1] => [7,2,6,4,5,1,3] => ? = 1
[1,2,4,5,6,7,3] => [3,7,6,5,4,2,1] => [2,4,3,5,6,7,1] => [3,2,7,4,5,6,1] => ? = 1
[1,2,4,5,7,3,6] => [6,3,7,5,4,2,1] => [2,4,5,7,6,3,1] => [7,2,5,4,6,1,3] => ? = 2
[1,2,4,5,7,6,3] => [3,6,7,5,4,2,1] => [2,4,3,5,7,6,1] => [3,2,6,4,5,7,1] => ? = 2
[1,2,4,6,3,5,7] => [7,5,3,6,4,2,1] => [2,4,6,5,7,3,1] => [7,2,4,6,5,1,3] => ? = 2
[1,2,4,6,3,7,5] => [5,7,3,6,4,2,1] => [2,4,6,7,5,3,1] => [6,2,5,4,7,1,3] => ? = 3
[1,2,4,6,5,3,7] => [7,3,5,6,4,2,1] => [2,4,6,7,3,5,1] => [7,2,6,4,1,3,5] => ? = 2
[1,2,4,6,5,7,3] => [3,7,5,6,4,2,1] => [2,4,3,6,7,5,1] => [3,2,7,4,6,1,5] => ? = 2
[1,2,4,6,7,3,5] => [5,3,7,6,4,2,1] => [2,4,6,5,3,7,1] => [7,2,4,5,1,6,3] => ? = 2
[1,2,4,6,7,5,3] => [3,5,7,6,4,2,1] => [2,4,3,6,5,7,1] => [3,2,5,4,7,6,1] => ? = 2
[1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => [2,4,7,5,6,3,1] => [7,2,4,5,6,1,3] => ? = 3
[1,2,4,7,3,6,5] => [5,6,3,7,4,2,1] => [2,4,7,6,5,3,1] => [6,2,4,5,7,1,3] => ? = 4
[1,2,4,7,5,3,6] => [6,3,5,7,4,2,1] => [2,4,7,6,3,5,1] => [7,2,4,6,1,3,5] => ? = 3
[1,2,4,7,5,6,3] => [3,6,5,7,4,2,1] => [2,4,3,7,6,5,1] => [3,2,6,4,7,1,5] => ? = 3
[1,2,4,7,6,3,5] => [5,3,6,7,4,2,1] => [2,4,7,5,3,6,1] => [6,2,4,5,1,7,3] => ? = 3
[1,2,4,7,6,5,3] => [3,5,6,7,4,2,1] => [2,4,3,7,5,6,1] => [3,2,5,4,6,7,1] => ? = 3
[1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => [2,5,4,6,3,7,1] => [5,2,7,4,1,6,3] => ? = 2
[1,2,5,3,4,7,6] => [6,7,4,3,5,2,1] => [2,5,4,7,3,6,1] => [5,2,6,4,1,7,3] => ? = 3
Description
The number of crossings of a permutation. A crossing of a permutation $\pi$ is given by a pair $(i,j)$ such that either $i < j \leq \pi(i) \leq \pi(j)$ or $\pi(i) < \pi(j) < i < j$. Pictorially, the diagram of a permutation is obtained by writing the numbers from $1$ to $n$ in this order on a line, and connecting $i$ and $\pi(i)$ with an arc above the line if $i\leq\pi(i)$ and with an arc below the line if $i > \pi(i)$. Then the number of crossings is the number of pairs of arcs above the line that cross or touch, plus the number of arcs below the line that cross.
Matching statistic: St001866
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001866: Signed permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 19%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [2,1] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [2,3,1] => [2,3,1] => 0
[1,3,2] => [2,3,1] => [3,2,1] => [3,2,1] => 1
[2,1,3] => [3,1,2] => [3,1,2] => [3,1,2] => 0
[2,3,1] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[3,1,2] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [2,3,4,1] => [2,3,4,1] => 0
[1,2,4,3] => [3,4,2,1] => [2,4,3,1] => [2,4,3,1] => 1
[1,3,2,4] => [4,2,3,1] => [3,4,2,1] => [3,4,2,1] => 1
[1,3,4,2] => [2,4,3,1] => [3,2,4,1] => [3,2,4,1] => 1
[1,4,2,3] => [3,2,4,1] => [4,3,2,1] => [4,3,2,1] => 2
[1,4,3,2] => [2,3,4,1] => [4,2,3,1] => [4,2,3,1] => 2
[2,1,3,4] => [4,3,1,2] => [3,1,4,2] => [3,1,4,2] => 0
[2,1,4,3] => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[2,3,1,4] => [4,1,3,2] => [3,4,1,2] => [3,4,1,2] => 0
[2,3,4,1] => [1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 0
[2,4,1,3] => [3,1,4,2] => [4,3,1,2] => [4,3,1,2] => 1
[2,4,3,1] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1
[3,1,2,4] => [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 0
[3,1,4,2] => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[3,2,1,4] => [4,1,2,3] => [4,1,2,3] => [4,1,2,3] => 0
[3,2,4,1] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 0
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[4,1,2,3] => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 0
[4,1,3,2] => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[4,2,1,3] => [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 0
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[4,3,1,2] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [5,4,3,2,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,2,3,5,4] => [4,5,3,2,1] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 1
[1,2,4,3,5] => [5,3,4,2,1] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 1
[1,2,4,5,3] => [3,5,4,2,1] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 1
[1,2,5,3,4] => [4,3,5,2,1] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 2
[1,2,5,4,3] => [3,4,5,2,1] => [2,5,3,4,1] => [2,5,3,4,1] => ? = 2
[1,3,2,4,5] => [5,4,2,3,1] => [3,4,2,5,1] => [3,4,2,5,1] => ? = 1
[1,3,2,5,4] => [4,5,2,3,1] => [3,5,2,4,1] => [3,5,2,4,1] => ? = 2
[1,3,4,2,5] => [5,2,4,3,1] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 1
[1,3,4,5,2] => [2,5,4,3,1] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 1
[1,3,5,2,4] => [4,2,5,3,1] => [3,5,4,2,1] => [3,5,4,2,1] => ? = 2
[1,3,5,4,2] => [2,4,5,3,1] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 2
[1,4,2,3,5] => [5,3,2,4,1] => [4,3,5,2,1] => [4,3,5,2,1] => ? = 2
[1,4,2,5,3] => [3,5,2,4,1] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 3
[1,4,3,2,5] => [5,2,3,4,1] => [4,5,2,3,1] => [4,5,2,3,1] => ? = 2
[1,4,3,5,2] => [2,5,3,4,1] => [4,2,5,3,1] => [4,2,5,3,1] => ? = 2
[1,4,5,2,3] => [3,2,5,4,1] => [4,3,2,5,1] => [4,3,2,5,1] => ? = 2
[1,4,5,3,2] => [2,3,5,4,1] => [4,2,3,5,1] => [4,2,3,5,1] => ? = 2
[1,5,2,3,4] => [4,3,2,5,1] => [5,3,4,2,1] => [5,3,4,2,1] => ? = 3
[1,5,2,4,3] => [3,4,2,5,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 4
[1,5,3,2,4] => [4,2,3,5,1] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 3
[1,5,3,4,2] => [2,4,3,5,1] => [5,2,4,3,1] => [5,2,4,3,1] => ? = 3
[1,5,4,2,3] => [3,2,4,5,1] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 3
[1,5,4,3,2] => [2,3,4,5,1] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 3
[2,1,3,4,5] => [5,4,3,1,2] => [3,1,4,5,2] => [3,1,4,5,2] => ? = 0
[2,1,3,5,4] => [4,5,3,1,2] => [3,1,5,4,2] => [3,1,5,4,2] => ? = 1
[2,1,4,3,5] => [5,3,4,1,2] => [4,1,5,3,2] => [4,1,5,3,2] => ? = 1
[2,1,4,5,3] => [3,5,4,1,2] => [4,1,3,5,2] => [4,1,3,5,2] => ? = 1
[2,1,5,3,4] => [4,3,5,1,2] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 2
[2,1,5,4,3] => [3,4,5,1,2] => [5,1,3,4,2] => [5,1,3,4,2] => ? = 2
[2,3,1,4,5] => [5,4,1,3,2] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 0
[2,3,1,5,4] => [4,5,1,3,2] => [3,5,1,4,2] => [3,5,1,4,2] => ? = 1
[2,3,4,1,5] => [5,1,4,3,2] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 0
[2,3,4,5,1] => [1,5,4,3,2] => [1,3,4,5,2] => [1,3,4,5,2] => 0
[2,3,5,1,4] => [4,1,5,3,2] => [3,5,4,1,2] => [3,5,4,1,2] => ? = 1
[2,3,5,4,1] => [1,4,5,3,2] => [1,3,5,4,2] => [1,3,5,4,2] => 1
[2,4,1,3,5] => [5,3,1,4,2] => [4,3,5,1,2] => [4,3,5,1,2] => ? = 1
[2,4,1,5,3] => [3,5,1,4,2] => [4,5,3,1,2] => [4,5,3,1,2] => ? = 2
[2,4,3,1,5] => [5,1,3,4,2] => [4,5,1,3,2] => [4,5,1,3,2] => ? = 1
[2,4,3,5,1] => [1,5,3,4,2] => [1,4,5,3,2] => [1,4,5,3,2] => 1
[2,4,5,1,3] => [3,1,5,4,2] => [4,3,1,5,2] => [4,3,1,5,2] => ? = 1
[2,4,5,3,1] => [1,3,5,4,2] => [1,4,3,5,2] => [1,4,3,5,2] => 1
[2,5,1,3,4] => [4,3,1,5,2] => [5,3,4,1,2] => [5,3,4,1,2] => ? = 2
[2,5,1,4,3] => [3,4,1,5,2] => [5,4,3,1,2] => [5,4,3,1,2] => ? = 3
[2,5,3,1,4] => [4,1,3,5,2] => [5,4,1,3,2] => [5,4,1,3,2] => ? = 2
[2,5,3,4,1] => [1,4,3,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[2,5,4,1,3] => [3,1,4,5,2] => [5,3,1,4,2] => [5,3,1,4,2] => ? = 2
[2,5,4,3,1] => [1,3,4,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[3,1,2,4,5] => [5,4,2,1,3] => [2,4,1,5,3] => [2,4,1,5,3] => ? = 0
[3,1,2,5,4] => [4,5,2,1,3] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1
[3,1,4,2,5] => [5,2,4,1,3] => [4,5,2,1,3] => [4,5,2,1,3] => ? = 1
[3,1,4,5,2] => [2,5,4,1,3] => [4,2,1,5,3] => [4,2,1,5,3] => ? = 1
[3,1,5,2,4] => [4,2,5,1,3] => [5,4,2,1,3] => [5,4,2,1,3] => ? = 2
[3,1,5,4,2] => [2,4,5,1,3] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 2
[3,2,1,4,5] => [5,4,1,2,3] => [4,1,2,5,3] => [4,1,2,5,3] => ? = 0
[3,2,1,5,4] => [4,5,1,2,3] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 1
[3,2,4,5,1] => [1,5,4,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => 0
[3,2,5,4,1] => [1,4,5,2,3] => [1,5,2,4,3] => [1,5,2,4,3] => 1
[3,4,2,5,1] => [1,5,2,4,3] => [1,4,5,2,3] => [1,4,5,2,3] => 0
[3,4,5,2,1] => [1,2,5,4,3] => [1,2,4,5,3] => [1,2,4,5,3] => 0
[3,5,2,4,1] => [1,4,2,5,3] => [1,5,4,2,3] => [1,5,4,2,3] => 1
[3,5,4,2,1] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[4,2,3,5,1] => [1,5,3,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => 0
[4,2,5,3,1] => [1,3,5,2,4] => [1,5,3,2,4] => [1,5,3,2,4] => 1
[4,3,2,5,1] => [1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[4,3,5,2,1] => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[4,5,2,3,1] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
Description
The nesting alignments of a signed permutation. A nesting alignment of a signed permutation $\pi\in\mathfrak H_n$ is a pair $1\leq i, j \leq n$ such that * $-i < -j < -\pi(j) < -\pi(i)$, or * $-i < j \leq \pi(j) < -\pi(i)$, or * $i < j \leq \pi(j) < \pi(i)$.
Matching statistic: St001882
Mapping
Mp00066: Permutations —inverse⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001882: Signed permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 19%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [2,1] => 0
[2,1] => [2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[1,3,2] => [1,3,2] => [2,3,1] => [2,3,1] => 1
[2,1,3] => [2,1,3] => [3,1,2] => [3,1,2] => 0
[2,3,1] => [3,1,2] => [2,1,3] => [2,1,3] => 0
[3,1,2] => [2,3,1] => [1,3,2] => [1,3,2] => 0
[3,2,1] => [3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => [3,4,2,1] => 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 1
[1,3,4,2] => [1,4,2,3] => [3,2,4,1] => [3,2,4,1] => 1
[1,4,2,3] => [1,3,4,2] => [2,4,3,1] => [2,4,3,1] => 2
[1,4,3,2] => [1,4,3,2] => [2,3,4,1] => [2,3,4,1] => 2
[2,1,3,4] => [2,1,3,4] => [4,3,1,2] => [4,3,1,2] => 0
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [3,4,1,2] => 1
[2,3,1,4] => [3,1,2,4] => [4,2,1,3] => [4,2,1,3] => 0
[2,3,4,1] => [4,1,2,3] => [3,2,1,4] => [3,2,1,4] => 0
[2,4,1,3] => [3,1,4,2] => [2,4,1,3] => [2,4,1,3] => 1
[2,4,3,1] => [4,1,3,2] => [2,3,1,4] => [2,3,1,4] => 1
[3,1,2,4] => [2,3,1,4] => [4,1,3,2] => [4,1,3,2] => 0
[3,1,4,2] => [2,4,1,3] => [3,1,4,2] => [3,1,4,2] => 1
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 0
[3,2,4,1] => [4,2,1,3] => [3,1,2,4] => [3,1,2,4] => 0
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 0
[3,4,2,1] => [4,3,1,2] => [2,1,3,4] => [2,1,3,4] => 0
[4,1,2,3] => [2,3,4,1] => [1,4,3,2] => [1,4,3,2] => 0
[4,1,3,2] => [2,4,3,1] => [1,3,4,2] => [1,3,4,2] => 1
[4,2,1,3] => [3,2,4,1] => [1,4,2,3] => [1,4,2,3] => 0
[4,2,3,1] => [4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 0
[4,3,1,2] => [3,4,2,1] => [1,2,4,3] => [1,2,4,3] => 0
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [5,3,4,2,1] => ? = 1
[1,2,4,5,3] => [1,2,5,3,4] => [4,3,5,2,1] => [4,3,5,2,1] => ? = 1
[1,2,5,3,4] => [1,2,4,5,3] => [3,5,4,2,1] => [3,5,4,2,1] => ? = 2
[1,2,5,4,3] => [1,2,5,4,3] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 2
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => [4,5,2,3,1] => ? = 2
[1,3,4,2,5] => [1,4,2,3,5] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 1
[1,3,4,5,2] => [1,5,2,3,4] => [4,3,2,5,1] => [4,3,2,5,1] => ? = 1
[1,3,5,2,4] => [1,4,2,5,3] => [3,5,2,4,1] => [3,5,2,4,1] => ? = 2
[1,3,5,4,2] => [1,5,2,4,3] => [3,4,2,5,1] => [3,4,2,5,1] => ? = 2
[1,4,2,3,5] => [1,3,4,2,5] => [5,2,4,3,1] => [5,2,4,3,1] => ? = 2
[1,4,2,5,3] => [1,3,5,2,4] => [4,2,5,3,1] => [4,2,5,3,1] => ? = 3
[1,4,3,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 2
[1,4,3,5,2] => [1,5,3,2,4] => [4,2,3,5,1] => [4,2,3,5,1] => ? = 2
[1,4,5,2,3] => [1,4,5,2,3] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 2
[1,4,5,3,2] => [1,5,4,2,3] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 2
[1,5,2,3,4] => [1,3,4,5,2] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 3
[1,5,2,4,3] => [1,3,5,4,2] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 4
[1,5,3,2,4] => [1,4,3,5,2] => [2,5,3,4,1] => [2,5,3,4,1] => ? = 3
[1,5,3,4,2] => [1,5,3,4,2] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 3
[1,5,4,2,3] => [1,4,5,3,2] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 3
[1,5,4,3,2] => [1,5,4,3,2] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 3
[2,1,3,4,5] => [2,1,3,4,5] => [5,4,3,1,2] => [5,4,3,1,2] => ? = 0
[2,1,3,5,4] => [2,1,3,5,4] => [4,5,3,1,2] => [4,5,3,1,2] => ? = 1
[2,1,4,3,5] => [2,1,4,3,5] => [5,3,4,1,2] => [5,3,4,1,2] => ? = 1
[2,1,4,5,3] => [2,1,5,3,4] => [4,3,5,1,2] => [4,3,5,1,2] => ? = 1
[2,1,5,3,4] => [2,1,4,5,3] => [3,5,4,1,2] => [3,5,4,1,2] => ? = 2
[2,1,5,4,3] => [2,1,5,4,3] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 2
[2,3,1,4,5] => [3,1,2,4,5] => [5,4,2,1,3] => [5,4,2,1,3] => ? = 0
[2,3,1,5,4] => [3,1,2,5,4] => [4,5,2,1,3] => [4,5,2,1,3] => ? = 1
[2,3,4,1,5] => [4,1,2,3,5] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 0
[2,3,4,5,1] => [5,1,2,3,4] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 0
[2,3,5,1,4] => [4,1,2,5,3] => [3,5,2,1,4] => [3,5,2,1,4] => ? = 1
[2,3,5,4,1] => [5,1,2,4,3] => [3,4,2,1,5] => [3,4,2,1,5] => ? = 1
[2,4,1,3,5] => [3,1,4,2,5] => [5,2,4,1,3] => [5,2,4,1,3] => ? = 1
[2,4,1,5,3] => [3,1,5,2,4] => [4,2,5,1,3] => [4,2,5,1,3] => ? = 2
[2,4,3,1,5] => [4,1,3,2,5] => [5,2,3,1,4] => [5,2,3,1,4] => ? = 1
[2,4,3,5,1] => [5,1,3,2,4] => [4,2,3,1,5] => [4,2,3,1,5] => ? = 1
[2,4,5,1,3] => [4,1,5,2,3] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 1
[2,4,5,3,1] => [5,1,4,2,3] => [3,2,4,1,5] => [3,2,4,1,5] => ? = 1
[2,5,1,3,4] => [3,1,4,5,2] => [2,5,4,1,3] => [2,5,4,1,3] => ? = 2
[2,5,1,4,3] => [3,1,5,4,2] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 3
[2,5,3,1,4] => [4,1,3,5,2] => [2,5,3,1,4] => [2,5,3,1,4] => ? = 2
[2,5,3,4,1] => [5,1,3,4,2] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 2
[2,5,4,1,3] => [4,1,5,3,2] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 2
[2,5,4,3,1] => [5,1,4,3,2] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 2
[3,1,2,4,5] => [2,3,1,4,5] => [5,4,1,3,2] => [5,4,1,3,2] => ? = 0
[3,1,2,5,4] => [2,3,1,5,4] => [4,5,1,3,2] => [4,5,1,3,2] => ? = 1
[5,1,2,3,4] => [2,3,4,5,1] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[5,1,2,4,3] => [2,3,5,4,1] => [1,4,5,3,2] => [1,4,5,3,2] => 1
[5,1,3,2,4] => [2,4,3,5,1] => [1,5,3,4,2] => [1,5,3,4,2] => 1
[5,1,3,4,2] => [2,5,3,4,1] => [1,4,3,5,2] => [1,4,3,5,2] => 1
[5,1,4,2,3] => [2,4,5,3,1] => [1,3,5,4,2] => [1,3,5,4,2] => 2
[5,1,4,3,2] => [2,5,4,3,1] => [1,3,4,5,2] => [1,3,4,5,2] => 2
[5,2,1,3,4] => [3,2,4,5,1] => [1,5,4,2,3] => [1,5,4,2,3] => 0
[5,2,1,4,3] => [3,2,5,4,1] => [1,4,5,2,3] => [1,4,5,2,3] => 1
[5,2,3,1,4] => [4,2,3,5,1] => [1,5,3,2,4] => [1,5,3,2,4] => 0
[5,2,3,4,1] => [5,2,3,4,1] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[5,2,4,1,3] => [4,2,5,3,1] => [1,3,5,2,4] => [1,3,5,2,4] => 1
[5,2,4,3,1] => [5,2,4,3,1] => [1,3,4,2,5] => [1,3,4,2,5] => 1
[5,3,1,2,4] => [3,4,2,5,1] => [1,5,2,4,3] => [1,5,2,4,3] => 0
[5,3,1,4,2] => [3,5,2,4,1] => [1,4,2,5,3] => [1,4,2,5,3] => 1
[5,3,2,1,4] => [4,3,2,5,1] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[5,3,2,4,1] => [5,3,2,4,1] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[5,3,4,1,2] => [4,5,2,3,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0
Description
The number of occurrences of a type-B 231 pattern in a signed permutation. For a signed permutation $\pi\in\mathfrak H_n$, a triple $-n \leq i < j < k\leq n$ is an occurrence of the type-B $231$ pattern, if $1 \leq j < k$, $\pi(i) < \pi(j)$ and $\pi(i)$ is one larger than $\pi(k)$, i.e., $\pi(i) = \pi(k) + 1$ if $\pi(k) \neq -1$ and $\pi(i) = 1$ otherwise.