Your data matches 11 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000358
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [2,1] => [1,2] => [1,2] => 0
[1,0,1,0]
 => [3,1,2] => [1,3,2] => [1,3,2] => 0
[1,1,0,0]
 => [2,3,1] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0]
 => [4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 0
[1,0,1,1,0,0]
 => [3,1,4,2] => [1,3,4,2] => [1,4,2,3] => 1
[1,1,0,0,1,0]
 => [2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 0
[1,1,0,1,0,0]
 => [4,3,1,2] => [1,4,2,3] => [1,3,4,2] => 0
[1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [1,4,5,3,2] => [1,5,3,2,4] => 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [1,3,5,4,2] => [1,5,4,2,3] => 1
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [1,5,3,4,2] => [1,4,2,5,3] => 1
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [1,3,4,5,2] => [1,5,2,3,4] => 2
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => 1
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,5,4,2,3] => [1,3,5,4,2] => 0
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [1,5,3,2,4] => [1,4,5,3,2] => 0
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [1,4,5,2,3] => [1,3,5,2,4] => 1
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [1,2,5,3,4] => [1,2,4,5,3] => 0
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [1,5,2,3,4] => [1,3,4,5,2] => 0
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [1,5,6,4,3,2] => [1,6,4,3,2,5] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [1,4,6,5,3,2] => [1,6,5,3,2,4] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [1,6,4,5,3,2] => [1,5,3,2,6,4] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [1,4,5,6,3,2] => [1,6,3,2,4,5] => 2
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [1,3,6,5,4,2] => [1,6,5,4,2,3] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [1,3,5,6,4,2] => [1,6,4,2,3,5] => 2
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [1,6,5,3,4,2] => [1,4,2,6,5,3] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [1,6,4,2,3,5] => [1,3,5,6,4,2] => 0
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [1,5,6,3,4,2] => [1,4,2,6,3,5] => 2
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [1,3,4,6,5,2] => [1,6,5,2,3,4] => 2
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [1,3,6,4,5,2] => [1,5,2,3,6,4] => 2
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [1,6,3,4,5,2] => [1,5,2,4,6,3] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [1,3,4,5,6,2] => [1,6,2,3,4,5] => 3
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [1,2,5,6,4,3] => [1,2,6,4,3,5] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [1,2,4,6,5,3] => [1,2,6,5,3,4] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [1,2,6,4,5,3] => [1,2,5,3,6,4] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [1,2,4,5,6,3] => [1,2,6,3,4,5] => 2
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [1,6,5,4,2,3] => [1,3,6,5,4,2] => 0
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [1,5,6,4,2,3] => [1,3,6,4,2,5] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [1,6,5,3,2,4] => [1,4,6,5,3,2] => 0
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [1,5,3,2,6,4] => [1,6,4,5,3,2] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [1,5,6,3,2,4] => [1,4,6,3,2,5] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [1,4,6,5,2,3] => [1,3,6,5,2,4] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [1,6,4,5,2,3] => [1,3,5,2,6,4] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [1,6,3,2,4,5] => [1,4,5,6,3,2] => 0
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [1,4,5,6,2,3] => [1,3,6,2,4,5] => 2
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: St000371
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [2,1] => [1,2] => [1,2] => 0
[1,0,1,0]
 => [3,1,2] => [1,3,2] => [1,3,2] => 0
[1,1,0,0]
 => [2,3,1] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0]
 => [4,1,2,3] => [1,4,3,2] => [1,3,4,2] => 0
[1,0,1,1,0,0]
 => [3,1,4,2] => [1,3,4,2] => [1,4,3,2] => 1
[1,1,0,0,1,0]
 => [2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 0
[1,1,0,1,0,0]
 => [4,3,1,2] => [1,4,2,3] => [1,4,2,3] => 0
[1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [1,5,4,3,2] => [1,3,4,5,2] => 0
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [1,4,5,3,2] => [1,3,5,4,2] => 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [1,3,5,4,2] => [1,4,3,5,2] => 1
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [1,5,3,4,2] => [1,4,5,3,2] => 1
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [1,3,4,5,2] => [1,5,3,4,2] => 2
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => 1
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,5,4,2,3] => [1,4,2,5,3] => 0
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [1,5,3,2,4] => [1,3,5,2,4] => 0
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [1,4,5,2,3] => [1,5,2,4,3] => 1
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [1,6,5,4,3,2] => [1,3,4,5,6,2] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [1,5,6,4,3,2] => [1,3,4,6,5,2] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [1,4,6,5,3,2] => [1,3,5,4,6,2] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [1,6,4,5,3,2] => [1,3,5,6,4,2] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [1,4,5,6,3,2] => [1,3,6,4,5,2] => 2
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [1,3,6,5,4,2] => [1,4,3,5,6,2] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [1,3,5,6,4,2] => [1,4,3,6,5,2] => 2
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [1,6,5,3,4,2] => [1,4,5,3,6,2] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [1,6,4,2,3,5] => [1,4,2,6,3,5] => 0
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [1,5,6,3,4,2] => [1,4,6,3,5,2] => 2
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [1,3,4,6,5,2] => [1,5,3,4,6,2] => 2
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [1,3,6,4,5,2] => [1,5,3,6,4,2] => 2
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [1,6,3,4,5,2] => [1,5,6,3,4,2] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [1,3,4,5,6,2] => [1,6,3,4,5,2] => 3
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [1,2,6,5,4,3] => [1,2,4,5,6,3] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [1,2,5,6,4,3] => [1,2,4,6,5,3] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [1,2,4,6,5,3] => [1,2,5,4,6,3] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [1,2,6,4,5,3] => [1,2,5,6,4,3] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [1,2,4,5,6,3] => [1,2,6,4,5,3] => 2
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [1,6,5,4,2,3] => [1,4,2,5,6,3] => 0
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [1,5,6,4,2,3] => [1,4,2,6,5,3] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [1,6,5,3,2,4] => [1,3,5,2,6,4] => 0
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [1,5,3,2,6,4] => [1,3,6,5,2,4] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [1,5,6,3,2,4] => [1,3,6,2,5,4] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [1,4,6,5,2,3] => [1,5,2,4,6,3] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [1,6,4,5,2,3] => [1,5,2,6,4,3] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [1,6,3,2,4,5] => [1,3,6,2,4,5] => 0
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [1,4,5,6,2,3] => [1,6,2,4,5,3] => 2
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [5,1,4,2,6,7,3] => [1,5,6,7,3,4,2] => [1,4,7,3,5,6,2] => ? = 3
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => [1,7,5,2,3,4,6] => [1,5,2,3,7,4,6] => ? = 0
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [7,1,6,5,2,3,4] => [1,7,4,5,2,3,6] => [1,5,2,7,4,3,6] => ? = 1
[1,1,0,1,1,0,0,0,1,0,1,0]
 => [4,3,1,7,2,5,6] => [1,4,7,6,5,2,3] => [1,5,2,4,6,7,3] => ? = 1
[1,1,0,1,1,0,0,0,1,1,0,0]
 => [4,3,1,6,2,7,5] => [1,4,6,7,5,2,3] => [1,5,2,4,7,6,3] => ? = 2
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [7,3,1,5,2,4,6] => [1,7,6,4,5,2,3] => [1,5,2,6,4,7,3] => ? = 1
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [7,3,1,6,2,4,5] => [1,7,5,2,3,4,6] => [1,5,2,3,7,4,6] => ? = 0
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [6,3,1,5,2,7,4] => [1,6,7,4,5,2,3] => [1,5,2,7,4,6,3] => ? = 2
[1,1,0,1,1,1,0,0,0,0,1,0]
 => [4,3,1,5,7,2,6] => [1,4,5,7,6,2,3] => [1,6,2,4,5,7,3] => ? = 2
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [4,3,1,7,6,2,5] => [1,4,7,5,6,2,3] => [1,6,2,4,7,5,3] => ? = 2
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [7,3,1,5,6,2,4] => [1,7,4,5,6,2,3] => [1,6,2,7,4,5,3] => ? = 2
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [4,3,1,5,6,7,2] => [1,4,5,6,7,2,3] => [1,7,2,4,5,6,3] => ? = 3
[1,1,1,0,1,0,0,0,1,0,1,0]
 => [7,3,4,1,2,5,6] => [1,7,6,5,2,3,4] => [1,5,2,3,6,7,4] => ? = 0
[1,1,1,0,1,0,0,0,1,1,0,0]
 => [6,3,4,1,2,7,5] => [1,6,7,5,2,3,4] => [1,5,2,3,7,6,4] => ? = 1
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [6,7,4,1,2,3,5] => [1,6,3,4,2,7,5] => [1,4,7,3,6,2,5] => ? = 2
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [6,5,4,1,2,7,3] => [1,6,7,3,4,2,5] => [1,4,7,3,2,6,5] => ? = 2
[1,1,1,0,1,1,0,0,0,0,1,0]
 => [5,3,4,1,7,2,6] => [1,5,7,6,2,3,4] => [1,6,2,3,5,7,4] => ? = 1
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [7,3,4,1,6,2,5] => [1,7,5,6,2,3,4] => [1,6,2,3,7,5,4] => ? = 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [7,5,4,1,6,2,3] => [1,7,3,4,2,5,6] => [1,4,7,3,2,5,6] => ? = 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [5,3,4,1,6,7,2] => [1,5,6,7,2,3,4] => [1,7,2,3,5,6,4] => ? = 2
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [7,3,4,6,1,2,5] => [1,7,5,2,3,4,6] => [1,5,2,3,7,4,6] => ? = 0
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [7,3,6,5,1,2,4] => [1,7,4,5,2,3,6] => [1,5,2,7,4,3,6] => ? = 1
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [6,3,4,5,1,7,2] => [1,6,7,2,3,4,5] => [1,7,2,3,4,6,5] => ? = 1
Description
The number of mid points of decreasing subsequences of length 3 in a permutation. For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) > \pi(j) > \pi(k)$. In other words, this is the number of indices that are neither left-to-right maxima nor right-to-left minima. This statistic can also be expressed as the number of occurrences of the mesh pattern ([3,2,1], {(0,2),(0,3),(2,0),(3,0)}): the shading fixes the first and the last element of the decreasing subsequence. See also [[St000119]].
Matching statistic: St001744
(load all 2 compositions to match this statistic)
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001744: Permutations ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [2,1] => [1,2] => [1,2] => 0
[1,0,1,0]
 => [3,1,2] => [1,3,2] => [1,3,2] => 0
[1,1,0,0]
 => [2,3,1] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0]
 => [4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 0
[1,0,1,1,0,0]
 => [3,1,4,2] => [1,3,4,2] => [1,4,2,3] => 1
[1,1,0,0,1,0]
 => [2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 0
[1,1,0,1,0,0]
 => [4,3,1,2] => [1,4,2,3] => [1,3,4,2] => 0
[1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [1,4,5,3,2] => [1,5,4,2,3] => 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [1,3,5,4,2] => [1,5,2,4,3] => 1
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [1,5,3,4,2] => [1,5,3,4,2] => 1
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [1,3,4,5,2] => [1,5,2,3,4] => 2
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => 1
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,5,4,2,3] => [1,4,5,3,2] => 0
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [1,5,3,2,4] => [1,4,3,5,2] => 0
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [1,4,5,2,3] => [1,4,5,2,3] => 1
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [1,2,5,3,4] => [1,2,4,5,3] => 0
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [1,5,2,3,4] => [1,3,4,5,2] => 0
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [1,5,6,4,3,2] => [1,6,5,4,2,3] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [1,4,6,5,3,2] => [1,6,5,2,4,3] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [1,6,4,5,3,2] => [1,6,5,3,4,2] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [1,4,5,6,3,2] => [1,6,5,2,3,4] => 2
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [1,3,6,5,4,2] => [1,6,2,5,4,3] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [1,3,5,6,4,2] => [1,6,2,5,3,4] => 2
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [1,6,5,3,4,2] => [1,6,4,5,3,2] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [1,6,4,2,3,5] => [1,4,5,3,6,2] => 0
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [1,5,6,3,4,2] => [1,6,4,5,2,3] => 2
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [1,3,4,6,5,2] => [1,6,2,3,5,4] => 2
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [1,3,6,4,5,2] => [1,6,2,4,5,3] => 2
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [1,6,3,4,5,2] => [1,6,3,4,5,2] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [1,3,4,5,6,2] => [1,6,2,3,4,5] => 3
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [1,2,5,6,4,3] => [1,2,6,5,3,4] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [1,2,4,6,5,3] => [1,2,6,3,5,4] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [1,2,4,5,6,3] => [1,2,6,3,4,5] => 2
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [1,6,5,4,2,3] => [1,5,6,4,3,2] => 0
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [1,5,6,4,2,3] => [1,5,6,4,2,3] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [1,6,5,3,2,4] => [1,5,4,6,3,2] => 0
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [1,5,3,2,6,4] => [1,4,3,6,2,5] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [1,5,6,3,2,4] => [1,5,4,6,2,3] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [1,4,6,5,2,3] => [1,5,6,2,4,3] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [1,6,4,5,2,3] => [1,5,6,3,4,2] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [1,6,3,2,4,5] => [1,4,3,5,6,2] => 0
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [1,4,5,6,2,3] => [1,5,6,2,3,4] => 2
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [7,1,4,2,3,5,6] => [1,7,6,5,3,4,2] => [1,7,5,6,4,3,2] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [6,1,4,2,3,7,5] => [1,6,7,5,3,4,2] => [1,7,5,6,4,2,3] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [7,1,5,2,3,4,6] => [1,7,6,4,2,3,5] => [1,5,6,4,7,3,2] => ? = 0
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [6,1,5,2,3,7,4] => [1,6,7,4,2,3,5] => [1,5,6,4,7,2,3] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,1,4,2,7,3,6] => [1,5,7,6,3,4,2] => [1,7,5,6,2,4,3] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [7,1,4,2,6,3,5] => [1,7,5,6,3,4,2] => [1,7,5,6,3,4,2] => ? = 2
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [5,1,4,2,6,7,3] => [1,5,6,7,3,4,2] => [1,7,5,6,2,3,4] => ? = 3
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [7,1,6,5,2,3,4] => [1,7,4,5,2,3,6] => [1,5,6,3,4,7,2] => ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [7,3,1,2,4,5,6] => [1,7,6,5,4,2,3] => [1,6,7,5,4,3,2] => ? = 0
[1,1,0,1,0,0,1,0,1,1,0,0]
 => [6,3,1,2,4,7,5] => [1,6,7,5,4,2,3] => [1,6,7,5,4,2,3] => ? = 1
[1,1,0,1,0,0,1,1,0,0,1,0]
 => [5,3,1,2,7,4,6] => [1,5,7,6,4,2,3] => [1,6,7,5,2,4,3] => ? = 1
[1,1,0,1,0,0,1,1,0,1,0,0]
 => [7,3,1,2,6,4,5] => [1,7,5,6,4,2,3] => [1,6,7,5,3,4,2] => ? = 1
[1,1,0,1,0,0,1,1,1,0,0,0]
 => [5,3,1,2,6,7,4] => [1,5,6,7,4,2,3] => [1,6,7,5,2,3,4] => ? = 2
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [7,4,1,2,6,3,5] => [1,7,5,6,3,2,4] => [1,6,5,7,3,4,2] => ? = 1
[1,1,0,1,1,0,0,0,1,0,1,0]
 => [4,3,1,7,2,5,6] => [1,4,7,6,5,2,3] => [1,6,7,2,5,4,3] => ? = 1
[1,1,0,1,1,0,0,0,1,1,0,0]
 => [4,3,1,6,2,7,5] => [1,4,6,7,5,2,3] => [1,6,7,2,5,3,4] => ? = 2
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [7,3,1,5,2,4,6] => [1,7,6,4,5,2,3] => [1,6,7,4,5,3,2] => ? = 1
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [6,3,1,5,2,7,4] => [1,6,7,4,5,2,3] => [1,6,7,4,5,2,3] => ? = 2
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [7,4,1,5,2,3,6] => [1,7,6,3,2,4,5] => [1,5,4,6,7,3,2] => ? = 0
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [6,4,1,5,2,7,3] => [1,6,7,3,2,4,5] => [1,5,4,6,7,2,3] => ? = 1
[1,1,0,1,1,1,0,0,0,0,1,0]
 => [4,3,1,5,7,2,6] => [1,4,5,7,6,2,3] => [1,6,7,2,3,5,4] => ? = 2
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [4,3,1,7,6,2,5] => [1,4,7,5,6,2,3] => [1,6,7,2,4,5,3] => ? = 2
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [7,3,1,5,6,2,4] => [1,7,4,5,6,2,3] => [1,6,7,3,4,5,2] => ? = 2
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [4,3,1,5,6,7,2] => [1,4,5,6,7,2,3] => [1,6,7,2,3,4,5] => ? = 3
[1,1,1,0,1,0,0,0,1,0,1,0]
 => [7,3,4,1,2,5,6] => [1,7,6,5,2,3,4] => [1,5,6,7,4,3,2] => ? = 0
[1,1,1,0,1,0,0,0,1,1,0,0]
 => [6,3,4,1,2,7,5] => [1,6,7,5,2,3,4] => [1,5,6,7,4,2,3] => ? = 1
[1,1,1,0,1,0,0,1,0,0,1,0]
 => [7,3,5,1,2,4,6] => [1,7,6,4,2,3,5] => [1,5,6,4,7,3,2] => ? = 0
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [6,3,5,1,2,7,4] => [1,6,7,4,2,3,5] => [1,5,6,4,7,2,3] => ? = 1
[1,1,1,0,1,0,1,0,0,0,1,0]
 => [7,5,4,1,2,3,6] => [1,7,6,3,4,2,5] => [1,6,4,5,7,3,2] => ? = 1
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [6,7,4,1,2,3,5] => [1,6,3,4,2,7,5] => [1,5,3,4,7,2,6] => ? = 2
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [6,5,4,1,2,7,3] => [1,6,7,3,4,2,5] => [1,6,4,5,7,2,3] => ? = 2
[1,1,1,0,1,1,0,0,0,0,1,0]
 => [5,3,4,1,7,2,6] => [1,5,7,6,2,3,4] => [1,5,6,7,2,4,3] => ? = 1
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [7,3,4,1,6,2,5] => [1,7,5,6,2,3,4] => [1,5,6,7,3,4,2] => ? = 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [7,5,4,1,6,2,3] => [1,7,3,4,2,5,6] => [1,5,3,4,6,7,2] => ? = 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [5,3,4,1,6,7,2] => [1,5,6,7,2,3,4] => [1,5,6,7,2,3,4] => ? = 2
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [7,3,6,5,1,2,4] => [1,7,4,5,2,3,6] => [1,5,6,3,4,7,2] => ? = 1
Description
The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. Let $\nu$ be a (partial) permutation of $[k]$ with $m$ letters together with dashes between some of its letters. An occurrence of $\nu$ in a permutation $\tau$ is a subsequence $\tau_{a_1},\dots,\tau_{a_m}$ such that $a_i + 1 = a_{i+1}$ whenever there is a dash between the $i$-th and the $(i+1)$-st letter of $\nu$, which is order isomorphic to $\nu$. Thus, $\nu$ is a vincular pattern, except that it is not required to be a permutation. An arrow pattern of size $k$ consists of such a generalized vincular pattern $\nu$ and arrows $b_1\to c_1, b_2\to c_2,\dots$, such that precisely the numbers $1,\dots,k$ appear in the vincular pattern and the arrows. Let $\Phi$ be the map [[Mp00087]]. Let $\tau$ be a permutation and $\sigma = \Phi(\tau)$. Then a subsequence $w = (x_{a_1},\dots,x_{a_m})$ of $\tau$ is an occurrence of the arrow pattern if $w$ is an occurrence of $\nu$, for each arrow $b\to c$ we have $\sigma(x_b) = x_c$ and $x_1 < x_2 < \dots < x_k$.
Matching statistic: St000223
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000223: Permutations ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [2,1] => [1,2] => [1,2] => 0
[1,0,1,0]
 => [3,1,2] => [1,3,2] => [1,3,2] => 0
[1,1,0,0]
 => [2,3,1] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0]
 => [4,1,2,3] => [1,4,3,2] => [1,3,4,2] => 0
[1,0,1,1,0,0]
 => [3,1,4,2] => [1,3,4,2] => [1,4,3,2] => 1
[1,1,0,0,1,0]
 => [2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 0
[1,1,0,1,0,0]
 => [4,3,1,2] => [1,4,2,3] => [1,4,2,3] => 0
[1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [1,5,4,3,2] => [1,3,4,5,2] => 0
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [1,4,5,3,2] => [1,3,5,4,2] => 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [1,3,5,4,2] => [1,4,3,5,2] => 1
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [1,5,3,4,2] => [1,4,5,3,2] => 1
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [1,3,4,5,2] => [1,5,3,4,2] => 2
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => 1
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,5,4,2,3] => [1,4,2,5,3] => 0
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [1,5,3,2,4] => [1,3,5,2,4] => 0
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [1,4,5,2,3] => [1,5,2,4,3] => 1
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [1,6,5,4,3,2] => [1,3,4,5,6,2] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [1,5,6,4,3,2] => [1,3,4,6,5,2] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [1,4,6,5,3,2] => [1,3,5,4,6,2] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [1,6,4,5,3,2] => [1,3,5,6,4,2] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [1,4,5,6,3,2] => [1,3,6,4,5,2] => 2
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [1,3,6,5,4,2] => [1,4,3,5,6,2] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [1,3,5,6,4,2] => [1,4,3,6,5,2] => 2
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [1,6,5,3,4,2] => [1,4,5,3,6,2] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [1,6,4,2,3,5] => [1,4,2,6,3,5] => 0
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [1,5,6,3,4,2] => [1,4,6,3,5,2] => 2
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [1,3,4,6,5,2] => [1,5,3,4,6,2] => 2
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [1,3,6,4,5,2] => [1,5,3,6,4,2] => 2
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [1,6,3,4,5,2] => [1,5,6,3,4,2] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [1,3,4,5,6,2] => [1,6,3,4,5,2] => 3
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [1,2,6,5,4,3] => [1,2,4,5,6,3] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [1,2,5,6,4,3] => [1,2,4,6,5,3] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [1,2,4,6,5,3] => [1,2,5,4,6,3] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [1,2,6,4,5,3] => [1,2,5,6,4,3] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [1,2,4,5,6,3] => [1,2,6,4,5,3] => 2
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [1,6,5,4,2,3] => [1,4,2,5,6,3] => 0
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [1,5,6,4,2,3] => [1,4,2,6,5,3] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [1,6,5,3,2,4] => [1,3,5,2,6,4] => 0
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [1,5,3,2,6,4] => [1,3,6,5,2,4] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [1,5,6,3,2,4] => [1,3,6,2,5,4] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [1,4,6,5,2,3] => [1,5,2,4,6,3] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [1,6,4,5,2,3] => [1,5,2,6,4,3] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [1,6,3,2,4,5] => [1,3,6,2,4,5] => 0
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [1,4,5,6,2,3] => [1,6,2,4,5,3] => 2
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [7,1,4,2,3,5,6] => [1,7,6,5,3,4,2] => [1,4,5,3,6,7,2] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [6,1,4,2,3,7,5] => [1,6,7,5,3,4,2] => [1,4,5,3,7,6,2] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [7,1,5,2,3,4,6] => [1,7,6,4,2,3,5] => [1,4,2,6,3,7,5] => ? = 0
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [6,1,7,2,3,4,5] => [1,6,4,2,3,7,5] => [1,4,2,7,6,3,5] => ? = 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [6,1,5,2,3,7,4] => [1,6,7,4,2,3,5] => [1,4,2,7,3,6,5] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,1,4,2,7,3,6] => [1,5,7,6,3,4,2] => [1,4,6,3,5,7,2] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [7,1,4,2,6,3,5] => [1,7,5,6,3,4,2] => [1,4,6,3,7,5,2] => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [7,1,5,2,6,3,4] => [1,7,4,2,3,5,6] => [1,4,2,7,3,5,6] => ? = 0
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [5,1,4,2,6,7,3] => [1,5,6,7,3,4,2] => [1,4,7,3,5,6,2] => ? = 3
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => [1,7,5,2,3,4,6] => [1,5,2,3,7,4,6] => ? = 0
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [7,1,6,5,2,3,4] => [1,7,4,5,2,3,6] => [1,5,2,7,4,3,6] => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,7,1,3,4,5,6] => [1,2,7,6,5,4,3] => [1,2,4,5,6,7,3] => ? = 0
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,6,1,3,4,7,5] => [1,2,6,7,5,4,3] => [1,2,4,5,7,6,3] => ? = 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,5,1,3,7,4,6] => [1,2,5,7,6,4,3] => [1,2,4,6,5,7,3] => ? = 1
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [2,7,1,3,6,4,5] => [1,2,7,5,6,4,3] => [1,2,4,6,7,5,3] => ? = 1
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,5,1,3,6,7,4] => [1,2,5,6,7,4,3] => [1,2,4,7,5,6,3] => ? = 2
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,4,1,7,3,5,6] => [1,2,4,7,6,5,3] => [1,2,5,4,6,7,3] => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => [1,2,4,6,7,5,3] => [1,2,5,4,7,6,3] => ? = 2
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,7,1,5,3,4,6] => [1,2,7,6,4,5,3] => [1,2,5,6,4,7,3] => ? = 1
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,7,1,6,3,4,5] => [1,2,7,5,3,4,6] => [1,2,5,3,7,4,6] => ? = 0
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [2,6,1,5,3,7,4] => [1,2,6,7,4,5,3] => [1,2,5,7,4,6,3] => ? = 2
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => [1,2,4,5,7,6,3] => [1,2,6,4,5,7,3] => ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,4,1,7,6,3,5] => [1,2,4,7,5,6,3] => [1,2,6,4,7,5,3] => ? = 2
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [2,7,1,5,6,3,4] => [1,2,7,4,5,6,3] => [1,2,6,7,4,5,3] => ? = 2
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,4,1,5,6,7,3] => [1,2,4,5,6,7,3] => [1,2,7,4,5,6,3] => ? = 3
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [7,3,1,2,4,5,6] => [1,7,6,5,4,2,3] => [1,4,2,5,6,7,3] => ? = 0
[1,1,0,1,0,0,1,0,1,1,0,0]
 => [6,3,1,2,4,7,5] => [1,6,7,5,4,2,3] => [1,4,2,5,7,6,3] => ? = 1
[1,1,0,1,0,0,1,1,0,0,1,0]
 => [5,3,1,2,7,4,6] => [1,5,7,6,4,2,3] => [1,4,2,6,5,7,3] => ? = 1
[1,1,0,1,0,0,1,1,0,1,0,0]
 => [7,3,1,2,6,4,5] => [1,7,5,6,4,2,3] => [1,4,2,6,7,5,3] => ? = 1
[1,1,0,1,0,0,1,1,1,0,0,0]
 => [5,3,1,2,6,7,4] => [1,5,6,7,4,2,3] => [1,4,2,7,5,6,3] => ? = 2
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [7,4,1,2,6,3,5] => [1,7,5,6,3,2,4] => [1,3,6,2,7,5,4] => ? = 1
[1,1,0,1,1,0,0,0,1,0,1,0]
 => [4,3,1,7,2,5,6] => [1,4,7,6,5,2,3] => [1,5,2,4,6,7,3] => ? = 1
[1,1,0,1,1,0,0,0,1,1,0,0]
 => [4,3,1,6,2,7,5] => [1,4,6,7,5,2,3] => [1,5,2,4,7,6,3] => ? = 2
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [7,3,1,5,2,4,6] => [1,7,6,4,5,2,3] => [1,5,2,6,4,7,3] => ? = 1
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [7,3,1,6,2,4,5] => [1,7,5,2,3,4,6] => [1,5,2,3,7,4,6] => ? = 0
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [6,3,1,5,2,7,4] => [1,6,7,4,5,2,3] => [1,5,2,7,4,6,3] => ? = 2
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [7,4,1,5,2,3,6] => [1,7,6,3,2,4,5] => [1,3,6,2,4,7,5] => ? = 0
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [6,4,1,5,2,7,3] => [1,6,7,3,2,4,5] => [1,3,7,2,4,6,5] => ? = 1
[1,1,0,1,1,1,0,0,0,0,1,0]
 => [4,3,1,5,7,2,6] => [1,4,5,7,6,2,3] => [1,6,2,4,5,7,3] => ? = 2
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [4,3,1,7,6,2,5] => [1,4,7,5,6,2,3] => [1,6,2,4,7,5,3] => ? = 2
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [7,3,1,5,6,2,4] => [1,7,4,5,6,2,3] => [1,6,2,7,4,5,3] => ? = 2
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [4,3,1,5,6,7,2] => [1,4,5,6,7,2,3] => [1,7,2,4,5,6,3] => ? = 3
[1,1,1,0,0,0,1,0,1,0,1,0]
 => [2,3,7,1,4,5,6] => [1,2,3,7,6,5,4] => [1,2,3,5,6,7,4] => ? = 0
[1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => [1,2,3,6,7,5,4] => [1,2,3,5,7,6,4] => ? = 1
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => [1,2,3,5,7,6,4] => [1,2,3,6,5,7,4] => ? = 1
[1,1,1,0,0,0,1,1,0,1,0,0]
 => [2,3,7,1,6,4,5] => [1,2,3,7,5,6,4] => [1,2,3,6,7,5,4] => ? = 1
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => [1,2,3,5,6,7,4] => [1,2,3,7,5,6,4] => ? = 2
[1,1,1,0,0,1,0,0,1,0,1,0]
 => [2,7,4,1,3,5,6] => [1,2,7,6,5,3,4] => [1,2,5,3,6,7,4] => ? = 0
[1,1,1,0,0,1,0,0,1,1,0,0]
 => [2,6,4,1,3,7,5] => [1,2,6,7,5,3,4] => [1,2,5,3,7,6,4] => ? = 1
[1,1,1,0,0,1,0,1,0,0,1,0]
 => [2,7,5,1,3,4,6] => [1,2,7,6,4,3,5] => [1,2,4,6,3,7,5] => ? = 0
Description
The number of nestings in the permutation.
Matching statistic: St000356
(load all 2 compositions to match this statistic)
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [2,1] => [1,2] => [2,1] => 0
[1,0,1,0]
 => [3,1,2] => [1,3,2] => [2,3,1] => 0
[1,1,0,0]
 => [2,3,1] => [1,2,3] => [3,2,1] => 0
[1,0,1,0,1,0]
 => [4,1,2,3] => [1,4,3,2] => [2,3,4,1] => 0
[1,0,1,1,0,0]
 => [3,1,4,2] => [1,3,4,2] => [2,4,3,1] => 1
[1,1,0,0,1,0]
 => [2,4,1,3] => [1,2,4,3] => [3,4,2,1] => 0
[1,1,0,1,0,0]
 => [4,3,1,2] => [1,4,2,3] => [3,2,4,1] => 0
[1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [1,4,5,3,2] => [2,3,5,4,1] => 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [1,3,5,4,2] => [2,4,5,3,1] => 1
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [1,5,3,4,2] => [2,4,3,5,1] => 1
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [1,3,4,5,2] => [2,5,4,3,1] => 2
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [1,2,5,4,3] => [3,4,5,2,1] => 0
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => 1
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,5,4,2,3] => [3,2,4,5,1] => 0
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [1,5,3,2,4] => [4,2,3,5,1] => 0
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [1,4,5,2,3] => [3,2,5,4,1] => 1
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [1,2,3,5,4] => [4,5,3,2,1] => 0
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [1,2,5,3,4] => [4,3,5,2,1] => 0
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [1,5,2,3,4] => [4,3,2,5,1] => 0
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [1,5,6,4,3,2] => [2,3,4,6,5,1] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [1,4,6,5,3,2] => [2,3,5,6,4,1] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [1,6,4,5,3,2] => [2,3,5,4,6,1] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [1,4,5,6,3,2] => [2,3,6,5,4,1] => 2
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [1,3,6,5,4,2] => [2,4,5,6,3,1] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [1,3,5,6,4,2] => [2,4,6,5,3,1] => 2
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [1,6,5,3,4,2] => [2,4,3,5,6,1] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [1,6,4,2,3,5] => [5,3,2,4,6,1] => 0
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [1,5,6,3,4,2] => [2,4,3,6,5,1] => 2
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [1,3,4,6,5,2] => [2,5,6,4,3,1] => 2
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [1,3,6,4,5,2] => [2,5,4,6,3,1] => 2
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [1,6,3,4,5,2] => [2,5,4,3,6,1] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [1,3,4,5,6,2] => [2,6,5,4,3,1] => 3
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [1,2,6,5,4,3] => [3,4,5,6,2,1] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [1,2,5,6,4,3] => [3,4,6,5,2,1] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [1,2,4,6,5,3] => [3,5,6,4,2,1] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [1,2,6,4,5,3] => [3,5,4,6,2,1] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [1,2,4,5,6,3] => [3,6,5,4,2,1] => 2
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [1,6,5,4,2,3] => [3,2,4,5,6,1] => 0
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [1,5,6,4,2,3] => [3,2,4,6,5,1] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [1,6,5,3,2,4] => [4,2,3,5,6,1] => 0
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [1,5,3,2,6,4] => [4,6,2,3,5,1] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [1,5,6,3,2,4] => [4,2,3,6,5,1] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [1,4,6,5,2,3] => [3,2,5,6,4,1] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [1,6,4,5,2,3] => [3,2,5,4,6,1] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [1,6,3,2,4,5] => [5,4,2,3,6,1] => 0
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [1,4,5,6,2,3] => [3,2,6,5,4,1] => 2
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [7,1,4,2,3,5,6] => [1,7,6,5,3,4,2] => [2,4,3,5,6,7,1] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [6,1,4,2,3,7,5] => [1,6,7,5,3,4,2] => [2,4,3,5,7,6,1] => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [6,1,7,2,3,4,5] => [1,6,4,2,3,7,5] => [5,7,3,2,4,6,1] => ? = 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [6,1,5,2,3,7,4] => [1,6,7,4,2,3,5] => [5,3,2,4,7,6,1] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,1,4,2,7,3,6] => [1,5,7,6,3,4,2] => [2,4,3,6,7,5,1] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [7,1,4,2,6,3,5] => [1,7,5,6,3,4,2] => [2,4,3,6,5,7,1] => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [7,1,5,2,6,3,4] => [1,7,4,2,3,5,6] => [6,5,3,2,4,7,1] => ? = 0
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [5,1,4,2,6,7,3] => [1,5,6,7,3,4,2] => [2,4,3,7,6,5,1] => ? = 3
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => [1,7,5,2,3,4,6] => [6,4,3,2,5,7,1] => ? = 0
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [7,1,6,5,2,3,4] => [1,7,4,5,2,3,6] => [6,3,2,5,4,7,1] => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,7,1,3,4,5,6] => [1,2,7,6,5,4,3] => [3,4,5,6,7,2,1] => ? = 0
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,6,1,3,4,7,5] => [1,2,6,7,5,4,3] => [3,4,5,7,6,2,1] => ? = 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,5,1,3,7,4,6] => [1,2,5,7,6,4,3] => [3,4,6,7,5,2,1] => ? = 1
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [2,7,1,3,6,4,5] => [1,2,7,5,6,4,3] => [3,4,6,5,7,2,1] => ? = 1
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,5,1,3,6,7,4] => [1,2,5,6,7,4,3] => [3,4,7,6,5,2,1] => ? = 2
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,4,1,7,3,5,6] => [1,2,4,7,6,5,3] => [3,5,6,7,4,2,1] => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => [1,2,4,6,7,5,3] => [3,5,7,6,4,2,1] => ? = 2
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,7,1,5,3,4,6] => [1,2,7,6,4,5,3] => [3,5,4,6,7,2,1] => ? = 1
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,7,1,6,3,4,5] => [1,2,7,5,3,4,6] => [6,4,3,5,7,2,1] => ? = 0
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [2,6,1,5,3,7,4] => [1,2,6,7,4,5,3] => [3,5,4,7,6,2,1] => ? = 2
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => [1,2,4,5,7,6,3] => [3,6,7,5,4,2,1] => ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,4,1,7,6,3,5] => [1,2,4,7,5,6,3] => [3,6,5,7,4,2,1] => ? = 2
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [2,7,1,5,6,3,4] => [1,2,7,4,5,6,3] => [3,6,5,4,7,2,1] => ? = 2
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,4,1,5,6,7,3] => [1,2,4,5,6,7,3] => [3,7,6,5,4,2,1] => ? = 3
[1,1,0,1,0,0,1,0,1,1,0,0]
 => [6,3,1,2,4,7,5] => [1,6,7,5,4,2,3] => [3,2,4,5,7,6,1] => ? = 1
[1,1,0,1,0,0,1,1,0,0,1,0]
 => [5,3,1,2,7,4,6] => [1,5,7,6,4,2,3] => [3,2,4,6,7,5,1] => ? = 1
[1,1,0,1,0,0,1,1,0,1,0,0]
 => [7,3,1,2,6,4,5] => [1,7,5,6,4,2,3] => [3,2,4,6,5,7,1] => ? = 1
[1,1,0,1,0,0,1,1,1,0,0,0]
 => [5,3,1,2,6,7,4] => [1,5,6,7,4,2,3] => [3,2,4,7,6,5,1] => ? = 2
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [7,4,1,2,6,3,5] => [1,7,5,6,3,2,4] => [4,2,3,6,5,7,1] => ? = 1
[1,1,0,1,1,0,0,0,1,0,1,0]
 => [4,3,1,7,2,5,6] => [1,4,7,6,5,2,3] => [3,2,5,6,7,4,1] => ? = 1
[1,1,0,1,1,0,0,0,1,1,0,0]
 => [4,3,1,6,2,7,5] => [1,4,6,7,5,2,3] => [3,2,5,7,6,4,1] => ? = 2
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [7,3,1,5,2,4,6] => [1,7,6,4,5,2,3] => [3,2,5,4,6,7,1] => ? = 1
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [7,3,1,6,2,4,5] => [1,7,5,2,3,4,6] => [6,4,3,2,5,7,1] => ? = 0
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [6,3,1,5,2,7,4] => [1,6,7,4,5,2,3] => [3,2,5,4,7,6,1] => ? = 2
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [7,4,1,5,2,3,6] => [1,7,6,3,2,4,5] => [5,4,2,3,6,7,1] => ? = 0
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [6,4,1,5,2,7,3] => [1,6,7,3,2,4,5] => [5,4,2,3,7,6,1] => ? = 1
[1,1,0,1,1,1,0,0,0,0,1,0]
 => [4,3,1,5,7,2,6] => [1,4,5,7,6,2,3] => [3,2,6,7,5,4,1] => ? = 2
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [4,3,1,7,6,2,5] => [1,4,7,5,6,2,3] => [3,2,6,5,7,4,1] => ? = 2
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [7,3,1,5,6,2,4] => [1,7,4,5,6,2,3] => [3,2,6,5,4,7,1] => ? = 2
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [7,4,1,5,6,2,3] => [1,7,3,2,4,5,6] => [6,5,4,2,3,7,1] => ? = 0
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [4,3,1,5,6,7,2] => [1,4,5,6,7,2,3] => [3,2,7,6,5,4,1] => ? = 3
[1,1,1,0,0,0,1,0,1,0,1,0]
 => [2,3,7,1,4,5,6] => [1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => ? = 0
[1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => [1,2,3,6,7,5,4] => [4,5,7,6,3,2,1] => ? = 1
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => [1,2,3,5,7,6,4] => [4,6,7,5,3,2,1] => ? = 1
[1,1,1,0,0,0,1,1,0,1,0,0]
 => [2,3,7,1,6,4,5] => [1,2,3,7,5,6,4] => [4,6,5,7,3,2,1] => ? = 1
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => [1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => ? = 2
[1,1,1,0,0,1,0,0,1,0,1,0]
 => [2,7,4,1,3,5,6] => [1,2,7,6,5,3,4] => [4,3,5,6,7,2,1] => ? = 0
[1,1,1,0,0,1,0,0,1,1,0,0]
 => [2,6,4,1,3,7,5] => [1,2,6,7,5,3,4] => [4,3,5,7,6,2,1] => ? = 1
[1,1,1,0,0,1,0,1,0,0,1,0]
 => [2,7,5,1,3,4,6] => [1,2,7,6,4,3,5] => [5,3,4,6,7,2,1] => ? = 0
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [2,6,7,1,3,4,5] => [1,2,6,4,3,7,5] => [5,7,3,4,6,2,1] => ? = 1
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: St001685
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St001685: Permutations ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [2,1] => [1,2] => [1,2] => 0
[1,0,1,0]
 => [3,1,2] => [1,3,2] => [3,1,2] => 0
[1,1,0,0]
 => [2,3,1] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0]
 => [4,1,2,3] => [1,4,3,2] => [4,3,1,2] => 0
[1,0,1,1,0,0]
 => [3,1,4,2] => [1,3,4,2] => [2,4,1,3] => 1
[1,1,0,0,1,0]
 => [2,4,1,3] => [1,2,4,3] => [4,1,2,3] => 0
[1,1,0,1,0,0]
 => [4,3,1,2] => [1,4,2,3] => [3,4,1,2] => 0
[1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [1,5,4,3,2] => [5,4,3,1,2] => 0
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [1,4,5,3,2] => [2,5,4,1,3] => 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [1,3,5,4,2] => [5,2,4,1,3] => 1
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [1,5,3,4,2] => [3,5,4,1,2] => 1
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [1,3,4,5,2] => [2,3,5,1,4] => 2
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => 0
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [1,2,4,5,3] => [3,5,1,2,4] => 1
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,5,4,2,3] => [4,5,3,1,2] => 0
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [1,5,3,2,4] => [4,3,5,1,2] => 0
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [1,4,5,2,3] => [4,2,5,1,3] => 1
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [1,2,3,5,4] => [5,1,2,3,4] => 0
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [1,2,5,3,4] => [4,5,1,2,3] => 0
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [1,5,2,3,4] => [3,4,5,1,2] => 0
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [1,6,5,4,3,2] => [6,5,4,3,1,2] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [1,5,6,4,3,2] => [2,6,5,4,1,3] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [1,4,6,5,3,2] => [6,2,5,4,1,3] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [1,6,4,5,3,2] => [3,6,5,4,1,2] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [1,4,5,6,3,2] => [2,3,6,5,1,4] => 2
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [1,3,6,5,4,2] => [6,5,2,4,1,3] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [1,3,5,6,4,2] => [3,6,2,5,1,4] => 2
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [1,6,5,3,4,2] => [4,6,5,3,1,2] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [1,6,4,2,3,5] => [4,5,3,6,1,2] => 0
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [1,5,6,3,4,2] => [4,2,6,5,1,3] => 2
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [1,3,4,6,5,2] => [6,2,3,5,1,4] => 2
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [1,3,6,4,5,2] => [4,6,2,5,1,3] => 2
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [1,6,3,4,5,2] => [3,4,6,5,1,2] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [1,3,4,5,6,2] => [2,3,4,6,1,5] => 3
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [1,2,6,5,4,3] => [6,5,4,1,2,3] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [1,2,5,6,4,3] => [3,6,5,1,2,4] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [1,2,4,6,5,3] => [6,3,5,1,2,4] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [1,2,6,4,5,3] => [4,6,5,1,2,3] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [1,2,4,5,6,3] => [3,4,6,1,2,5] => 2
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [1,6,5,4,2,3] => [5,6,4,3,1,2] => 0
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [1,5,6,4,2,3] => [5,2,6,4,1,3] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [1,6,5,3,2,4] => [5,4,6,3,1,2] => 0
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [1,5,3,2,6,4] => [3,6,4,1,2,5] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [1,5,6,3,2,4] => [5,4,2,6,1,3] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [1,4,6,5,2,3] => [2,5,6,4,1,3] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [1,6,4,5,2,3] => [5,3,6,4,1,2] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [1,6,3,2,4,5] => [4,3,5,6,1,2] => 0
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [1,4,5,6,2,3] => [2,5,3,6,1,4] => 2
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [7,1,4,2,3,5,6] => [1,7,6,5,3,4,2] => [5,7,6,4,3,1,2] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [6,1,4,2,3,7,5] => [1,6,7,5,3,4,2] => [5,2,7,6,4,1,3] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [7,1,5,2,3,4,6] => [1,7,6,4,2,3,5] => [5,6,4,7,3,1,2] => ? = 0
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [6,1,7,2,3,4,5] => [1,6,4,2,3,7,5] => [7,4,3,5,1,2,6] => ? = 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [6,1,5,2,3,7,4] => [1,6,7,4,2,3,5] => [5,6,4,2,7,1,3] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,1,4,2,7,3,6] => [1,5,7,6,3,4,2] => [2,5,7,6,4,1,3] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [7,1,4,2,6,3,5] => [1,7,5,6,3,4,2] => [5,3,7,6,4,1,2] => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [7,1,5,2,6,3,4] => [1,7,4,2,3,5,6] => [4,5,3,6,7,1,2] => ? = 0
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [5,1,4,2,6,7,3] => [1,5,6,7,3,4,2] => [2,5,3,7,6,1,4] => ? = 3
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => [1,7,5,2,3,4,6] => [4,5,6,3,7,1,2] => ? = 0
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [7,1,6,5,2,3,4] => [1,7,4,5,2,3,6] => [5,3,6,4,7,1,2] => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,7,1,3,4,5,6] => [1,2,7,6,5,4,3] => [7,6,5,4,1,2,3] => ? = 0
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,6,1,3,4,7,5] => [1,2,6,7,5,4,3] => [3,7,6,5,1,2,4] => ? = 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,5,1,3,7,4,6] => [1,2,5,7,6,4,3] => [7,3,6,5,1,2,4] => ? = 1
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [2,7,1,3,6,4,5] => [1,2,7,5,6,4,3] => [4,7,6,5,1,2,3] => ? = 1
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,5,1,3,6,7,4] => [1,2,5,6,7,4,3] => [3,4,7,6,1,2,5] => ? = 2
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,4,1,7,3,5,6] => [1,2,4,7,6,5,3] => [7,6,3,5,1,2,4] => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => [1,2,4,6,7,5,3] => [4,7,3,6,1,2,5] => ? = 2
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,7,1,5,3,4,6] => [1,2,7,6,4,5,3] => [5,7,6,4,1,2,3] => ? = 1
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,7,1,6,3,4,5] => [1,2,7,5,3,4,6] => [5,6,4,7,1,2,3] => ? = 0
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [2,6,1,5,3,7,4] => [1,2,6,7,4,5,3] => [5,3,7,6,1,2,4] => ? = 2
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => [1,2,4,5,7,6,3] => [7,3,4,6,1,2,5] => ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,4,1,7,6,3,5] => [1,2,4,7,5,6,3] => [5,7,3,6,1,2,4] => ? = 2
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [2,7,1,5,6,3,4] => [1,2,7,4,5,6,3] => [4,5,7,6,1,2,3] => ? = 2
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,4,1,5,6,7,3] => [1,2,4,5,6,7,3] => [3,4,5,7,1,2,6] => ? = 3
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [7,3,1,2,4,5,6] => [1,7,6,5,4,2,3] => [6,7,5,4,3,1,2] => ? = 0
[1,1,0,1,0,0,1,0,1,1,0,0]
 => [6,3,1,2,4,7,5] => [1,6,7,5,4,2,3] => [6,2,7,5,4,1,3] => ? = 1
[1,1,0,1,0,0,1,1,0,0,1,0]
 => [5,3,1,2,7,4,6] => [1,5,7,6,4,2,3] => [2,6,7,5,4,1,3] => ? = 1
[1,1,0,1,0,0,1,1,0,1,0,0]
 => [7,3,1,2,6,4,5] => [1,7,5,6,4,2,3] => [6,3,7,5,4,1,2] => ? = 1
[1,1,0,1,0,0,1,1,1,0,0,0]
 => [5,3,1,2,6,7,4] => [1,5,6,7,4,2,3] => [2,6,3,7,5,1,4] => ? = 2
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [7,4,1,2,6,3,5] => [1,7,5,6,3,2,4] => [6,5,3,7,4,1,2] => ? = 1
[1,1,0,1,1,0,0,0,1,0,1,0]
 => [4,3,1,7,2,5,6] => [1,4,7,6,5,2,3] => [6,7,2,5,4,1,3] => ? = 1
[1,1,0,1,1,0,0,0,1,1,0,0]
 => [4,3,1,6,2,7,5] => [1,4,6,7,5,2,3] => [6,2,3,7,5,1,4] => ? = 2
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [7,3,1,5,2,4,6] => [1,7,6,4,5,2,3] => [6,4,7,5,3,1,2] => ? = 1
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [7,3,1,6,2,4,5] => [1,7,5,2,3,4,6] => [4,5,6,3,7,1,2] => ? = 0
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [6,3,1,5,2,7,4] => [1,6,7,4,5,2,3] => [6,4,2,7,5,1,3] => ? = 2
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [7,4,1,5,2,3,6] => [1,7,6,3,2,4,5] => [5,4,6,7,3,1,2] => ? = 0
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [6,4,1,5,2,7,3] => [1,6,7,3,2,4,5] => [5,4,6,2,7,1,3] => ? = 1
[1,1,0,1,1,1,0,0,0,0,1,0]
 => [4,3,1,5,7,2,6] => [1,4,5,7,6,2,3] => [2,3,6,7,5,1,4] => ? = 2
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [4,3,1,7,6,2,5] => [1,4,7,5,6,2,3] => [6,2,4,7,5,1,3] => ? = 2
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [7,3,1,5,6,2,4] => [1,7,4,5,6,2,3] => [3,6,4,7,5,1,2] => ? = 2
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [7,4,1,5,6,2,3] => [1,7,3,2,4,5,6] => [4,3,5,6,7,1,2] => ? = 0
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [4,3,1,5,6,7,2] => [1,4,5,6,7,2,3] => [2,3,6,4,7,1,5] => ? = 3
[1,1,1,0,0,0,1,0,1,0,1,0]
 => [2,3,7,1,4,5,6] => [1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => ? = 0
[1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => [1,2,3,6,7,5,4] => [4,7,6,1,2,3,5] => ? = 1
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => [1,2,3,5,7,6,4] => [7,4,6,1,2,3,5] => ? = 1
[1,1,1,0,0,0,1,1,0,1,0,0]
 => [2,3,7,1,6,4,5] => [1,2,3,7,5,6,4] => [5,7,6,1,2,3,4] => ? = 1
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => [1,2,3,5,6,7,4] => [4,5,7,1,2,3,6] => ? = 2
[1,1,1,0,0,1,0,0,1,0,1,0]
 => [2,7,4,1,3,5,6] => [1,2,7,6,5,3,4] => [6,7,5,4,1,2,3] => ? = 0
[1,1,1,0,0,1,0,0,1,1,0,0]
 => [2,6,4,1,3,7,5] => [1,2,6,7,5,3,4] => [6,3,7,5,1,2,4] => ? = 1
Description
The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation.
Matching statistic: St001687
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001687: Permutations ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [2,1] => [1,2] => [2,1] => 0
[1,0,1,0]
 => [3,1,2] => [1,3,2] => [3,1,2] => 0
[1,1,0,0]
 => [2,3,1] => [1,2,3] => [3,2,1] => 0
[1,0,1,0,1,0]
 => [4,1,2,3] => [1,4,3,2] => [4,1,2,3] => 0
[1,0,1,1,0,0]
 => [3,1,4,2] => [1,3,4,2] => [4,2,1,3] => 1
[1,1,0,0,1,0]
 => [2,4,1,3] => [1,2,4,3] => [4,3,1,2] => 0
[1,1,0,1,0,0]
 => [4,3,1,2] => [1,4,2,3] => [4,1,3,2] => 0
[1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [1,4,5,3,2] => [5,2,1,3,4] => 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [1,3,5,4,2] => [5,3,1,2,4] => 1
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [1,5,3,4,2] => [5,1,3,2,4] => 1
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [1,3,4,5,2] => [5,3,2,1,4] => 2
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => 0
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => 1
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,5,4,2,3] => [5,1,2,4,3] => 0
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [1,5,3,2,4] => [5,1,3,4,2] => 0
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [1,4,5,2,3] => [5,2,1,4,3] => 1
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [1,2,3,5,4] => [5,4,3,1,2] => 0
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [1,2,5,3,4] => [5,4,1,3,2] => 0
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [1,5,2,3,4] => [5,1,4,3,2] => 0
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [1,6,5,4,3,2] => [6,1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [1,5,6,4,3,2] => [6,2,1,3,4,5] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [1,4,6,5,3,2] => [6,3,1,2,4,5] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [1,6,4,5,3,2] => [6,1,3,2,4,5] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [1,4,5,6,3,2] => [6,3,2,1,4,5] => 2
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [1,3,6,5,4,2] => [6,4,1,2,3,5] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [1,3,5,6,4,2] => [6,4,2,1,3,5] => 2
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [1,6,5,3,4,2] => [6,1,2,4,3,5] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [1,6,4,2,3,5] => [6,1,3,5,4,2] => 0
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [1,5,6,3,4,2] => [6,2,1,4,3,5] => 2
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [1,3,4,6,5,2] => [6,4,3,1,2,5] => 2
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [1,3,6,4,5,2] => [6,4,1,3,2,5] => 2
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [1,6,3,4,5,2] => [6,1,4,3,2,5] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [1,3,4,5,6,2] => [6,4,3,2,1,5] => 3
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [1,2,6,5,4,3] => [6,5,1,2,3,4] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [1,2,5,6,4,3] => [6,5,2,1,3,4] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [1,2,4,6,5,3] => [6,5,3,1,2,4] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [1,2,6,4,5,3] => [6,5,1,3,2,4] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [1,2,4,5,6,3] => [6,5,3,2,1,4] => 2
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [1,6,5,4,2,3] => [6,1,2,3,5,4] => 0
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [1,5,6,4,2,3] => [6,2,1,3,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [1,6,5,3,2,4] => [6,1,2,4,5,3] => 0
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [1,5,3,2,6,4] => [6,2,4,5,1,3] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [1,5,6,3,2,4] => [6,2,1,4,5,3] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [1,4,6,5,2,3] => [6,3,1,2,5,4] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [1,6,4,5,2,3] => [6,1,3,2,5,4] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [1,6,3,2,4,5] => [6,1,4,5,3,2] => 0
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [1,4,5,6,2,3] => [6,3,2,1,5,4] => 2
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [7,1,4,2,3,5,6] => [1,7,6,5,3,4,2] => [7,1,2,3,5,4,6] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [6,1,4,2,3,7,5] => [1,6,7,5,3,4,2] => [7,2,1,3,5,4,6] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [7,1,5,2,3,4,6] => [1,7,6,4,2,3,5] => [7,1,2,4,6,5,3] => ? = 0
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [6,1,7,2,3,4,5] => [1,6,4,2,3,7,5] => [7,2,4,6,5,1,3] => ? = 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [6,1,5,2,3,7,4] => [1,6,7,4,2,3,5] => [7,2,1,4,6,5,3] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,1,4,2,7,3,6] => [1,5,7,6,3,4,2] => [7,3,1,2,5,4,6] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [7,1,4,2,6,3,5] => [1,7,5,6,3,4,2] => [7,1,3,2,5,4,6] => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [7,1,5,2,6,3,4] => [1,7,4,2,3,5,6] => [7,1,4,6,5,3,2] => ? = 0
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [5,1,4,2,6,7,3] => [1,5,6,7,3,4,2] => [7,3,2,1,5,4,6] => ? = 3
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => [1,7,5,2,3,4,6] => [7,1,3,6,5,4,2] => ? = 0
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [7,1,6,5,2,3,4] => [1,7,4,5,2,3,6] => [7,1,4,3,6,5,2] => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,7,1,3,4,5,6] => [1,2,7,6,5,4,3] => [7,6,1,2,3,4,5] => ? = 0
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,6,1,3,4,7,5] => [1,2,6,7,5,4,3] => [7,6,2,1,3,4,5] => ? = 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,5,1,3,7,4,6] => [1,2,5,7,6,4,3] => [7,6,3,1,2,4,5] => ? = 1
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [2,7,1,3,6,4,5] => [1,2,7,5,6,4,3] => [7,6,1,3,2,4,5] => ? = 1
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,5,1,3,6,7,4] => [1,2,5,6,7,4,3] => [7,6,3,2,1,4,5] => ? = 2
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,4,1,7,3,5,6] => [1,2,4,7,6,5,3] => [7,6,4,1,2,3,5] => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => [1,2,4,6,7,5,3] => [7,6,4,2,1,3,5] => ? = 2
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,7,1,5,3,4,6] => [1,2,7,6,4,5,3] => [7,6,1,2,4,3,5] => ? = 1
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,7,1,6,3,4,5] => [1,2,7,5,3,4,6] => [7,6,1,3,5,4,2] => ? = 0
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [2,6,1,5,3,7,4] => [1,2,6,7,4,5,3] => [7,6,2,1,4,3,5] => ? = 2
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => [1,2,4,5,7,6,3] => [7,6,4,3,1,2,5] => ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,4,1,7,6,3,5] => [1,2,4,7,5,6,3] => [7,6,4,1,3,2,5] => ? = 2
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [2,7,1,5,6,3,4] => [1,2,7,4,5,6,3] => [7,6,1,4,3,2,5] => ? = 2
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,4,1,5,6,7,3] => [1,2,4,5,6,7,3] => [7,6,4,3,2,1,5] => ? = 3
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [7,3,1,2,4,5,6] => [1,7,6,5,4,2,3] => [7,1,2,3,4,6,5] => ? = 0
[1,1,0,1,0,0,1,0,1,1,0,0]
 => [6,3,1,2,4,7,5] => [1,6,7,5,4,2,3] => [7,2,1,3,4,6,5] => ? = 1
[1,1,0,1,0,0,1,1,0,0,1,0]
 => [5,3,1,2,7,4,6] => [1,5,7,6,4,2,3] => [7,3,1,2,4,6,5] => ? = 1
[1,1,0,1,0,0,1,1,0,1,0,0]
 => [7,3,1,2,6,4,5] => [1,7,5,6,4,2,3] => [7,1,3,2,4,6,5] => ? = 1
[1,1,0,1,0,0,1,1,1,0,0,0]
 => [5,3,1,2,6,7,4] => [1,5,6,7,4,2,3] => [7,3,2,1,4,6,5] => ? = 2
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [7,4,1,2,6,3,5] => [1,7,5,6,3,2,4] => [7,1,3,2,5,6,4] => ? = 1
[1,1,0,1,1,0,0,0,1,0,1,0]
 => [4,3,1,7,2,5,6] => [1,4,7,6,5,2,3] => [7,4,1,2,3,6,5] => ? = 1
[1,1,0,1,1,0,0,0,1,1,0,0]
 => [4,3,1,6,2,7,5] => [1,4,6,7,5,2,3] => [7,4,2,1,3,6,5] => ? = 2
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [7,3,1,5,2,4,6] => [1,7,6,4,5,2,3] => [7,1,2,4,3,6,5] => ? = 1
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [7,3,1,6,2,4,5] => [1,7,5,2,3,4,6] => [7,1,3,6,5,4,2] => ? = 0
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [6,3,1,5,2,7,4] => [1,6,7,4,5,2,3] => [7,2,1,4,3,6,5] => ? = 2
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [7,4,1,5,2,3,6] => [1,7,6,3,2,4,5] => [7,1,2,5,6,4,3] => ? = 0
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [6,4,1,5,2,7,3] => [1,6,7,3,2,4,5] => [7,2,1,5,6,4,3] => ? = 1
[1,1,0,1,1,1,0,0,0,0,1,0]
 => [4,3,1,5,7,2,6] => [1,4,5,7,6,2,3] => [7,4,3,1,2,6,5] => ? = 2
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [4,3,1,7,6,2,5] => [1,4,7,5,6,2,3] => [7,4,1,3,2,6,5] => ? = 2
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [7,3,1,5,6,2,4] => [1,7,4,5,6,2,3] => [7,1,4,3,2,6,5] => ? = 2
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [7,4,1,5,6,2,3] => [1,7,3,2,4,5,6] => [7,1,5,6,4,3,2] => ? = 0
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [4,3,1,5,6,7,2] => [1,4,5,6,7,2,3] => [7,4,3,2,1,6,5] => ? = 3
[1,1,1,0,0,0,1,0,1,0,1,0]
 => [2,3,7,1,4,5,6] => [1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => ? = 0
[1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => [1,2,3,6,7,5,4] => [7,6,5,2,1,3,4] => ? = 1
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => [1,2,3,5,7,6,4] => [7,6,5,3,1,2,4] => ? = 1
[1,1,1,0,0,0,1,1,0,1,0,0]
 => [2,3,7,1,6,4,5] => [1,2,3,7,5,6,4] => [7,6,5,1,3,2,4] => ? = 1
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => [1,2,3,5,6,7,4] => [7,6,5,3,2,1,4] => ? = 2
[1,1,1,0,0,1,0,0,1,0,1,0]
 => [2,7,4,1,3,5,6] => [1,2,7,6,5,3,4] => [7,6,1,2,3,5,4] => ? = 0
[1,1,1,0,0,1,0,0,1,1,0,0]
 => [2,6,4,1,3,7,5] => [1,2,6,7,5,3,4] => [7,6,2,1,3,5,4] => ? = 1
Description
The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation.
Matching statistic: St001435
Mapping
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00187: Skew partitions —conjugate⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [[1],[]]
 => [[1],[]]
 => 0
[1,0,1,0]
 => [1,1,0,0]
 => [[2],[]]
 => [[1,1],[]]
 => 0
[1,1,0,0]
 => [1,0,1,0]
 => [[1,1],[]]
 => [[2],[]]
 => 0
[1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [[3],[]]
 => [[1,1,1],[]]
 => 0
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [[2,2],[1]]
 => [[2,2],[1]]
 => 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [[2,1],[]]
 => [[2,1],[]]
 => 0
[1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => [[2,2],[]]
 => [[2,2],[]]
 => 0
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [[1,1,1],[]]
 => [[3],[]]
 => 0
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [[4],[]]
 => [[1,1,1,1],[]]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => [[2,2,2],[1,1]]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [[2,2,1],[1]]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [[2,2,2],[1]]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => [[3,3],[2]]
 => 2
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [[3,1],[]]
 => [[2,1,1],[]]
 => 0
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => [[3,2],[1]]
 => 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => [[2,2,1],[]]
 => 0
[1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => [[2,2,2],[]]
 => ? = 0
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [[3,3],[1]]
 => 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [[2,1,1],[]]
 => [[3,1],[]]
 => 0
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [[2,2,1],[]]
 => [[3,2],[]]
 => 0
[1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => [[3,3],[]]
 => ? = 0
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [[1,1,1,1],[]]
 => [[4],[]]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => [[1,1,1,1,1],[]]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => [[2,2,2,2],[1,1,1]]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[4,3],[2]]
 => [[2,2,2,1],[1,1]]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [[4,4],[2]]
 => [[2,2,2,2],[1,1]]
 => ? = 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3],[2,2]]
 => [[3,3,3],[2,2]]
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[4,2],[1]]
 => [[2,2,1,1],[1]]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => [[3,3,2],[2,1]]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [[4,3],[1]]
 => [[2,2,2,1],[1]]
 => ? = 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [[4,4],[1]]
 => [[2,2,2,2],[1]]
 => ? = 0
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3],[2,1]]
 => [[3,3,3],[2,1]]
 => ? = 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2],[1,1]]
 => [[3,3,1],[2]]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [[3,3,2],[2]]
 => ? = 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => [[3,3,3],[2]]
 => ? = 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2],[1,1,1]]
 => [[4,4],[3]]
 => 3
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[4,1],[]]
 => [[2,1,1,1],[]]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1],[2]]
 => [[3,2,2],[1,1]]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [[3,2,1],[1]]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1],[1]]
 => [[3,2,2],[1]]
 => ? = 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => [[4,3],[2]]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[4,2],[]]
 => [[2,2,1,1],[]]
 => ? = 0
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => [[3,3,2],[1,1]]
 => ? = 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[4,3],[]]
 => [[2,2,2,1],[]]
 => ? = 0
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [[4,4],[]]
 => [[2,2,2,2],[]]
 => ? = 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => [[3,3,3],[1,1]]
 => ? = 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => [[3,3,1],[1]]
 => ? = 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2],[1]]
 => [[3,3,2],[1]]
 => ? = 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [[3,3,3],[1]]
 => [[3,3,3],[1]]
 => ? = 0
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [[4,4],[2]]
 => ? = 2
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [[3,1,1],[]]
 => [[3,1,1],[]]
 => 0
[1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1],[1]]
 => [[4,2],[1]]
 => 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[3,2,1],[]]
 => [[3,2,1],[]]
 => ? = 0
[1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1],[]]
 => [[3,2,2],[]]
 => ? = 0
[1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => [[4,3],[1]]
 => ? = 1
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[3,2,2],[]]
 => [[3,3,1],[]]
 => ? = 0
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [[3,3,2],[]]
 => [[3,3,2],[]]
 => ? = 0
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3],[]]
 => [[3,3,3],[]]
 => ? = 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[2,2,2,2],[1]]
 => [[4,4],[1]]
 => ? = 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [[2,1,1,1],[]]
 => [[4,1],[]]
 => 0
[1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[2,2,1,1],[]]
 => [[4,2],[]]
 => ? = 0
[1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[2,2,2,1],[]]
 => [[4,3],[]]
 => ? = 0
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [[2,2,2,2],[]]
 => [[4,4],[]]
 => ? = 0
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [[1,1,1,1,1],[]]
 => [[5],[]]
 => 0
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [[5,3],[1]]
 => [[2,2,2,1,1],[1]]
 => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [[4,4,3],[3,1]]
 => [[3,3,3,2],[2,1,1]]
 => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [[5,4],[1]]
 => [[2,2,2,2,1],[1]]
 => ? = 0
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [[5,5],[1]]
 => [[2,2,2,2,2],[1]]
 => ? = 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [[4,4,4],[3,1]]
 => [[3,3,3,3],[2,1,1]]
 => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [[4,3,3],[2,1]]
 => [[3,3,3,1],[2,1]]
 => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [[4,4,3],[2,1]]
 => [[3,3,3,2],[2,1]]
 => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [[4,4,4],[2,1]]
 => [[3,3,3,3],[2,1]]
 => ? = 0
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [[3,3,3,3],[2,2,1]]
 => [[4,4,4],[3,2]]
 => ? = 3
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [[4,4,3],[1,1]]
 => [[3,3,3,2],[2]]
 => ? = 0
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [[4,4,4],[1,1]]
 => [[3,3,3,3],[2]]
 => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [[5,1],[]]
 => [[2,1,1,1,1],[]]
 => ? = 0
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [[4,4,1],[3]]
 => [[3,2,2,2],[1,1,1]]
 => ? = 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [[4,3,1],[2]]
 => [[3,2,2,1],[1,1]]
 => ? = 1
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => [[4,4,1],[2]]
 => [[3,2,2,2],[1,1]]
 => ? = 1
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3,1],[2,2]]
 => [[4,3,3],[2,2]]
 => ? = 2
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [[4,2,1],[1]]
 => [[3,2,1,1],[1]]
 => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2,1],[2,1]]
 => [[4,3,2],[2,1]]
 => ? = 2
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [[4,3,1],[1]]
 => [[3,2,2,1],[1]]
 => ? = 1
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [[4,4,1],[1]]
 => [[3,2,2,2],[1]]
 => ? = 0
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3,1],[2,1]]
 => [[4,3,3],[2,1]]
 => ? = 2
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2,1],[1,1]]
 => [[4,3,1],[2]]
 => ? = 2
Description
The number of missing boxes in the first row.
Matching statistic: St001862
(load all 2 compositions to match this statistic)
Mapping
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001862: Signed permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 75%
Values
[1,0]
 => [.,.]
 => [1] => [1] => 0
[1,0,1,0]
 => [.,[.,.]]
 => [2,1] => [2,1] => 0
[1,1,0,0]
 => [[.,.],.]
 => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => [3,2,1] => [3,2,1] => 0
[1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => [2,3,1] => [2,3,1] => 1
[1,1,0,0,1,0]
 => [[.,.],[.,.]]
 => [1,3,2] => [1,3,2] => 0
[1,1,0,1,0,0]
 => [[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => 0
[1,1,1,0,0,0]
 => [[[.,.],.],.]
 => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [4,3,2,1] => 0
[1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => [3,4,2,1] => 1
[1,0,1,1,0,0,1,0]
 => [.,[[.,.],[.,.]]]
 => [2,4,3,1] => [2,4,3,1] => 1
[1,0,1,1,0,1,0,0]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => [3,2,4,1] => 1
[1,0,1,1,1,0,0,0]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => [2,3,4,1] => 2
[1,1,0,0,1,0,1,0]
 => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => [1,4,3,2] => 0
[1,1,0,0,1,1,0,0]
 => [[.,.],[[.,.],.]]
 => [1,3,4,2] => [1,3,4,2] => 1
[1,1,0,1,0,0,1,0]
 => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => [2,1,4,3] => 0
[1,1,0,1,0,1,0,0]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => [3,2,1,4] => 0
[1,1,0,1,1,0,0,0]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => [2,3,1,4] => 1
[1,1,1,0,0,0,1,0]
 => [[[.,.],.],[.,.]]
 => [1,2,4,3] => [1,2,4,3] => 0
[1,1,1,0,0,1,0,0]
 => [[[.,.],[.,.]],.]
 => [1,3,2,4] => [1,3,2,4] => 0
[1,1,1,0,1,0,0,0]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3,4] => 0
[1,1,1,1,0,0,0,0]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
[1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [4,5,3,2,1] => ? = 1
[1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [3,5,4,2,1] => ? = 1
[1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [4,3,5,2,1] => ? = 1
[1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [3,4,5,2,1] => ? = 2
[1,0,1,1,0,0,1,0,1,0]
 => [.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [2,5,4,3,1] => ? = 1
[1,0,1,1,0,0,1,1,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [2,4,5,3,1] => ? = 2
[1,0,1,1,0,1,0,0,1,0]
 => [.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [3,2,5,4,1] => ? = 1
[1,0,1,1,0,1,0,1,0,0]
 => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [4,3,2,5,1] => ? = 0
[1,0,1,1,0,1,1,0,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [3,4,2,5,1] => ? = 2
[1,0,1,1,1,0,0,0,1,0]
 => [.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [2,3,5,4,1] => ? = 2
[1,0,1,1,1,0,0,1,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [2,4,3,5,1] => ? = 2
[1,0,1,1,1,0,1,0,0,0]
 => [.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [3,2,4,5,1] => ? = 2
[1,0,1,1,1,1,0,0,0,0]
 => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [2,3,4,5,1] => ? = 3
[1,1,0,0,1,0,1,0,1,0]
 => [[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [1,5,4,3,2] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [1,4,5,3,2] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [1,3,5,4,2] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [1,4,3,5,2] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [1,3,4,5,2] => 2
[1,1,0,1,0,0,1,0,1,0]
 => [[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
[1,1,0,1,0,0,1,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [2,1,4,5,3] => ? = 1
[1,1,0,1,0,1,0,0,1,0]
 => [[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [3,2,1,5,4] => ? = 0
[1,1,0,1,0,1,0,1,0,0]
 => [[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[1,1,0,1,0,1,1,0,0,0]
 => [[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => [3,4,2,1,5] => ? = 1
[1,1,0,1,1,0,0,0,1,0]
 => [[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [2,3,1,5,4] => ? = 1
[1,1,0,1,1,0,0,1,0,0]
 => [[.,[[.,.],[.,.]]],.]
 => [2,4,3,1,5] => [2,4,3,1,5] => ? = 1
[1,1,0,1,1,0,1,0,0,0]
 => [[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => [3,2,4,1,5] => ? = 0
[1,1,0,1,1,1,0,0,0,0]
 => [[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => [2,3,4,1,5] => ? = 2
[1,1,1,0,0,0,1,0,1,0]
 => [[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [1,2,5,4,3] => 0
[1,1,1,0,0,0,1,1,0,0]
 => [[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [1,2,4,5,3] => 1
[1,1,1,0,0,1,0,0,1,0]
 => [[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,1,1,0,0,1,0,1,0,0]
 => [[[.,.],[.,[.,.]]],.]
 => [1,4,3,2,5] => [1,4,3,2,5] => 0
[1,1,1,0,0,1,1,0,0,0]
 => [[[.,.],[[.,.],.]],.]
 => [1,3,4,2,5] => [1,3,4,2,5] => 1
[1,1,1,0,1,0,0,0,1,0]
 => [[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[1,1,1,0,1,0,0,1,0,0]
 => [[[.,[.,.]],[.,.]],.]
 => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[1,1,1,0,1,0,1,0,0,0]
 => [[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => [3,2,1,4,5] => ? = 1
[1,1,1,0,1,1,0,0,0,0]
 => [[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => [2,3,1,4,5] => ? = 1
[1,1,1,1,0,0,0,0,1,0]
 => [[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,1,1,1,0,0,0,1,0,0]
 => [[[[.,.],.],[.,.]],.]
 => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,1,1,1,0,0,1,0,0,0]
 => [[[[.,.],[.,.]],.],.]
 => [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,1,1,1,0,1,0,0,0,0]
 => [[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => [2,1,3,4,5] => ? = 0
[1,1,1,1,1,0,0,0,0,0]
 => [[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [.,[[.,[.,.]],[.,[.,.]]]]
 => [3,2,6,5,4,1] => [3,2,6,5,4,1] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [.,[[.,[.,.]],[[.,.],.]]]
 => [3,2,5,6,4,1] => [3,2,5,6,4,1] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [.,[[.,[.,[.,.]]],[.,.]]]
 => [4,3,2,6,5,1] => [4,3,2,6,5,1] => ? = 0
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [.,[[.,[.,[.,[.,.]]]],.]]
 => [5,4,3,2,6,1] => [5,4,3,2,6,1] => ? = 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [.,[[.,[.,[[.,.],.]]],.]]
 => [4,5,3,2,6,1] => [4,5,3,2,6,1] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [.,[[.,[[.,.],.]],[.,.]]]
 => [3,4,2,6,5,1] => [3,4,2,6,5,1] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [.,[[.,[[.,.],[.,.]]],.]]
 => [3,5,4,2,6,1] => [3,5,4,2,6,1] => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [.,[[.,[[.,[.,.]],.]],.]]
 => [4,3,5,2,6,1] => [4,3,5,2,6,1] => ? = 0
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [.,[[.,[[[.,.],.],.]],.]]
 => [3,4,5,2,6,1] => [3,4,5,2,6,1] => ? = 3
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [.,[[[.,[.,.]],[.,.]],.]]
 => [3,2,5,4,6,1] => [3,2,5,4,6,1] => ? = 0
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [.,[[[.,[.,[.,.]]],.],.]]
 => [4,3,2,5,6,1] => [4,3,2,5,6,1] => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [[.,.],[.,[.,[.,[.,.]]]]]
 => [1,6,5,4,3,2] => [1,6,5,4,3,2] => ? = 0
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [[.,.],[.,[.,[[.,.],.]]]]
 => [1,5,6,4,3,2] => [1,5,6,4,3,2] => ? = 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [[.,.],[.,[[.,.],[.,.]]]]
 => [1,4,6,5,3,2] => [1,4,6,5,3,2] => ? = 1
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [[.,.],[.,[[.,[.,.]],.]]]
 => [1,5,4,6,3,2] => [1,5,4,6,3,2] => ? = 1
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [[.,.],[.,[[[.,.],.],.]]]
 => [1,4,5,6,3,2] => [1,4,5,6,3,2] => ? = 2
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [[.,.],[[.,.],[.,[.,.]]]]
 => [1,3,6,5,4,2] => [1,3,6,5,4,2] => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [[.,.],[[.,.],[[.,.],.]]]
 => [1,3,5,6,4,2] => [1,3,5,6,4,2] => ? = 2
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [[.,.],[[.,[.,.]],[.,.]]]
 => [1,4,3,6,5,2] => [1,4,3,6,5,2] => ? = 1
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [[.,.],[[.,[.,[.,.]]],.]]
 => [1,5,4,3,6,2] => [1,5,4,3,6,2] => ? = 0
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [[.,.],[[.,[[.,.],.]],.]]
 => [1,4,5,3,6,2] => [1,4,5,3,6,2] => ? = 2
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [[.,.],[[[.,.],.],[.,.]]]
 => [1,3,4,6,5,2] => [1,3,4,6,5,2] => ? = 2
Description
The number of crossings of a signed permutation. A crossing of a signed permutation $\pi$ is a pair $(i, j)$ of indices such that * $i < j \leq \pi(i) < \pi(j)$, or * $-i < j \leq -\pi(i) < \pi(j)$, or * $i > j > \pi(i) > \pi(j)$.
Matching statistic: St001882
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001882: Signed permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 75%
Values
[1,0]
 => [1] => [1] => 0
[1,0,1,0]
 => [2,1] => [2,1] => 0
[1,1,0,0]
 => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
 => [3,2,1] => [3,2,1] => 0
[1,0,1,1,0,0]
 => [2,3,1] => [2,3,1] => 1
[1,1,0,0,1,0]
 => [3,1,2] => [3,1,2] => 0
[1,1,0,1,0,0]
 => [2,1,3] => [2,1,3] => 0
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4,3,2,1] => 0
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,4,2,1] => 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [4,2,3,1] => 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [3,2,4,1] => 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,3,4,1] => 2
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [4,3,1,2] => 0
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [3,4,1,2] => 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [4,2,1,3] => 0
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,2,1,4] => 0
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [2,3,1,4] => 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [4,1,2,3] => 0
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [3,1,2,4] => 0
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => 0
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [4,5,3,2,1] => ? = 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [5,3,4,2,1] => ? = 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [4,3,5,2,1] => ? = 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [3,4,5,2,1] => ? = 2
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [4,5,2,3,1] => ? = 2
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [5,3,2,4,1] => ? = 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [4,3,2,5,1] => ? = 0
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,4,2,5,1] => ? = 2
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [5,2,3,4,1] => ? = 2
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [4,2,3,5,1] => ? = 2
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [3,2,4,5,1] => ? = 2
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,3,4,5,1] => ? = 3
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [5,4,3,1,2] => ? = 0
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [4,5,3,1,2] => ? = 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [5,3,4,1,2] => ? = 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [4,3,5,1,2] => ? = 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [3,4,5,1,2] => ? = 2
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [5,4,2,1,3] => ? = 0
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [4,5,2,1,3] => ? = 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [5,3,2,1,4] => ? = 0
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [3,4,2,1,5] => ? = 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [5,2,3,1,4] => ? = 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [4,2,3,1,5] => ? = 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [3,2,4,1,5] => ? = 0
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [2,3,4,1,5] => ? = 2
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [5,4,1,2,3] => ? = 0
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [4,5,1,2,3] => ? = 1
[1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [5,3,1,2,4] => ? = 0
[1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => [4,3,1,2,5] => ? = 0
[1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => [3,4,1,2,5] => ? = 1
[1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => [5,2,1,3,4] => ? = 0
[1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,3,5] => [4,2,1,3,5] => ? = 0
[1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => [3,2,1,4,5] => ? = 1
[1,1,1,0,1,1,0,0,0,0]
 => [2,3,1,4,5] => [2,3,1,4,5] => ? = 1
[1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [5,1,2,3,4] => ? = 0
[1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => [4,1,2,3,5] => ? = 0
[1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => [3,1,2,4,5] => ? = 0
[1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => ? = 0
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,4,1] => [6,5,3,2,4,1] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [5,6,3,2,4,1] => [5,6,3,2,4,1] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [6,4,3,2,5,1] => [6,4,3,2,5,1] => ? = 0
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2,6,1] => [5,4,3,2,6,1] => ? = 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [4,5,3,2,6,1] => [4,5,3,2,6,1] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [6,3,4,2,5,1] => [6,3,4,2,5,1] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [5,3,4,2,6,1] => [5,3,4,2,6,1] => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [4,3,5,2,6,1] => [4,3,5,2,6,1] => ? = 0
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [3,4,5,2,6,1] => [3,4,5,2,6,1] => ? = 3
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.
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St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module