Processing math: 100%

Your data matches 8 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001837
Mapping
St001837: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
 => 0
[(1,2),(3,4)]
 => 0
[(1,3),(2,4)]
 => 0
[(1,4),(2,3)]
 => 0
[(1,2),(3,4),(5,6)]
 => 0
[(1,3),(2,4),(5,6)]
 => 0
[(1,4),(2,3),(5,6)]
 => 0
[(1,5),(2,3),(4,6)]
 => 0
[(1,6),(2,3),(4,5)]
 => 0
[(1,6),(2,4),(3,5)]
 => 0
[(1,5),(2,4),(3,6)]
 => 0
[(1,4),(2,5),(3,6)]
 => 1
[(1,3),(2,5),(4,6)]
 => 0
[(1,2),(3,5),(4,6)]
 => 0
[(1,2),(3,6),(4,5)]
 => 0
[(1,3),(2,6),(4,5)]
 => 0
[(1,4),(2,6),(3,5)]
 => 1
[(1,5),(2,6),(3,4)]
 => 2
[(1,6),(2,5),(3,4)]
 => 0
[(1,2),(3,4),(5,6),(7,8)]
 => 0
[(1,3),(2,4),(5,6),(7,8)]
 => 0
[(1,4),(2,3),(5,6),(7,8)]
 => 0
[(1,5),(2,3),(4,6),(7,8)]
 => 0
[(1,6),(2,3),(4,5),(7,8)]
 => 0
[(1,7),(2,3),(4,5),(6,8)]
 => 0
[(1,8),(2,3),(4,5),(6,7)]
 => 0
[(1,8),(2,4),(3,5),(6,7)]
 => 0
[(1,7),(2,4),(3,5),(6,8)]
 => 0
[(1,6),(2,4),(3,5),(7,8)]
 => 0
[(1,5),(2,4),(3,6),(7,8)]
 => 0
[(1,4),(2,5),(3,6),(7,8)]
 => 1
[(1,3),(2,5),(4,6),(7,8)]
 => 0
[(1,2),(3,5),(4,6),(7,8)]
 => 0
[(1,2),(3,6),(4,5),(7,8)]
 => 0
[(1,3),(2,6),(4,5),(7,8)]
 => 0
[(1,4),(2,6),(3,5),(7,8)]
 => 1
[(1,5),(2,6),(3,4),(7,8)]
 => 2
[(1,6),(2,5),(3,4),(7,8)]
 => 0
[(1,7),(2,5),(3,4),(6,8)]
 => 0
[(1,8),(2,5),(3,4),(6,7)]
 => 0
[(1,8),(2,6),(3,4),(5,7)]
 => 0
[(1,7),(2,6),(3,4),(5,8)]
 => 0
[(1,6),(2,7),(3,4),(5,8)]
 => 3
[(1,5),(2,7),(3,4),(6,8)]
 => 2
[(1,4),(2,7),(3,5),(6,8)]
 => 1
[(1,3),(2,7),(4,5),(6,8)]
 => 0
[(1,2),(3,7),(4,5),(6,8)]
 => 0
[(1,2),(3,8),(4,5),(6,7)]
 => 0
[(1,3),(2,8),(4,5),(6,7)]
 => 0
[(1,4),(2,8),(3,5),(6,7)]
 => 1
Description
The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching.
Matching statistic: St000220
Mapping
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000220: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 19%
Values
[(1,2)]
 => [2,1] => [1,2] => [2,1] => 0
[(1,2),(3,4)]
 => [2,1,4,3] => [1,2,3,4] => [4,3,2,1] => 0
[(1,3),(2,4)]
 => [3,4,1,2] => [1,3,2,4] => [4,2,3,1] => 0
[(1,4),(2,3)]
 => [4,3,2,1] => [1,4,2,3] => [3,2,4,1] => 0
[(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 0
[(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => [1,3,2,4,5,6] => [6,5,4,2,3,1] => 0
[(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => [1,4,2,3,5,6] => [6,5,3,2,4,1] => 0
[(1,5),(2,3),(4,6)]
 => [5,3,2,6,1,4] => [1,5,2,3,4,6] => [6,4,3,2,5,1] => 0
[(1,6),(2,3),(4,5)]
 => [6,3,2,5,4,1] => [1,6,2,3,4,5] => [5,4,3,2,6,1] => 0
[(1,6),(2,4),(3,5)]
 => [6,4,5,2,3,1] => [1,6,2,4,3,5] => [5,3,4,2,6,1] => 0
[(1,5),(2,4),(3,6)]
 => [5,4,6,2,1,3] => [1,5,2,4,3,6] => [6,3,4,2,5,1] => 0
[(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => [1,4,2,5,3,6] => [6,3,5,2,4,1] => 1
[(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => [1,3,2,5,4,6] => [6,4,5,2,3,1] => 0
[(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => [1,2,3,5,4,6] => [6,4,5,3,2,1] => 0
[(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => [1,2,3,6,4,5] => [5,4,6,3,2,1] => 0
[(1,3),(2,6),(4,5)]
 => [3,6,1,5,4,2] => [1,3,2,6,4,5] => [5,4,6,2,3,1] => 0
[(1,4),(2,6),(3,5)]
 => [4,6,5,1,3,2] => [1,4,2,6,3,5] => [5,3,6,2,4,1] => 1
[(1,5),(2,6),(3,4)]
 => [5,6,4,3,1,2] => [1,5,2,6,3,4] => [4,3,6,2,5,1] => 2
[(1,6),(2,5),(3,4)]
 => [6,5,4,3,2,1] => [1,6,2,5,3,4] => [4,3,5,2,6,1] => 0
[(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => [1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1] => 0
[(1,3),(2,4),(5,6),(7,8)]
 => [3,4,1,2,6,5,8,7] => [1,3,2,4,5,6,7,8] => [8,7,6,5,4,2,3,1] => ? = 0
[(1,4),(2,3),(5,6),(7,8)]
 => [4,3,2,1,6,5,8,7] => [1,4,2,3,5,6,7,8] => [8,7,6,5,3,2,4,1] => ? = 0
[(1,5),(2,3),(4,6),(7,8)]
 => [5,3,2,6,1,4,8,7] => [1,5,2,3,4,6,7,8] => [8,7,6,4,3,2,5,1] => ? = 0
[(1,6),(2,3),(4,5),(7,8)]
 => [6,3,2,5,4,1,8,7] => [1,6,2,3,4,5,7,8] => [8,7,5,4,3,2,6,1] => ? = 0
[(1,7),(2,3),(4,5),(6,8)]
 => [7,3,2,5,4,8,1,6] => [1,7,2,3,4,5,6,8] => [8,6,5,4,3,2,7,1] => ? = 0
[(1,8),(2,3),(4,5),(6,7)]
 => [8,3,2,5,4,7,6,1] => [1,8,2,3,4,5,6,7] => [7,6,5,4,3,2,8,1] => 0
[(1,8),(2,4),(3,5),(6,7)]
 => [8,4,5,2,3,7,6,1] => [1,8,2,4,3,5,6,7] => [7,6,5,3,4,2,8,1] => ? = 0
[(1,7),(2,4),(3,5),(6,8)]
 => [7,4,5,2,3,8,1,6] => [1,7,2,4,3,5,6,8] => [8,6,5,3,4,2,7,1] => ? = 0
[(1,6),(2,4),(3,5),(7,8)]
 => [6,4,5,2,3,1,8,7] => [1,6,2,4,3,5,7,8] => [8,7,5,3,4,2,6,1] => ? = 0
[(1,5),(2,4),(3,6),(7,8)]
 => [5,4,6,2,1,3,8,7] => [1,5,2,4,3,6,7,8] => [8,7,6,3,4,2,5,1] => ? = 0
[(1,4),(2,5),(3,6),(7,8)]
 => [4,5,6,1,2,3,8,7] => [1,4,2,5,3,6,7,8] => [8,7,6,3,5,2,4,1] => 1
[(1,3),(2,5),(4,6),(7,8)]
 => [3,5,1,6,2,4,8,7] => [1,3,2,5,4,6,7,8] => [8,7,6,4,5,2,3,1] => 0
[(1,2),(3,5),(4,6),(7,8)]
 => [2,1,5,6,3,4,8,7] => [1,2,3,5,4,6,7,8] => [8,7,6,4,5,3,2,1] => ? = 0
[(1,2),(3,6),(4,5),(7,8)]
 => [2,1,6,5,4,3,8,7] => [1,2,3,6,4,5,7,8] => [8,7,5,4,6,3,2,1] => ? = 0
[(1,3),(2,6),(4,5),(7,8)]
 => [3,6,1,5,4,2,8,7] => [1,3,2,6,4,5,7,8] => [8,7,5,4,6,2,3,1] => ? = 0
[(1,4),(2,6),(3,5),(7,8)]
 => [4,6,5,1,3,2,8,7] => [1,4,2,6,3,5,7,8] => [8,7,5,3,6,2,4,1] => ? = 1
[(1,5),(2,6),(3,4),(7,8)]
 => [5,6,4,3,1,2,8,7] => [1,5,2,6,3,4,7,8] => [8,7,4,3,6,2,5,1] => ? = 2
[(1,6),(2,5),(3,4),(7,8)]
 => [6,5,4,3,2,1,8,7] => [1,6,2,5,3,4,7,8] => [8,7,4,3,5,2,6,1] => ? = 0
[(1,7),(2,5),(3,4),(6,8)]
 => [7,5,4,3,2,8,1,6] => [1,7,2,5,3,4,6,8] => [8,6,4,3,5,2,7,1] => ? = 0
[(1,8),(2,5),(3,4),(6,7)]
 => [8,5,4,3,2,7,6,1] => [1,8,2,5,3,4,6,7] => [7,6,4,3,5,2,8,1] => ? = 0
[(1,8),(2,6),(3,4),(5,7)]
 => [8,6,4,3,7,2,5,1] => [1,8,2,6,3,4,5,7] => [7,5,4,3,6,2,8,1] => ? = 0
[(1,7),(2,6),(3,4),(5,8)]
 => [7,6,4,3,8,2,1,5] => [1,7,2,6,3,4,5,8] => [8,5,4,3,6,2,7,1] => ? = 0
[(1,6),(2,7),(3,4),(5,8)]
 => [6,7,4,3,8,1,2,5] => [1,6,2,7,3,4,5,8] => [8,5,4,3,7,2,6,1] => ? = 3
[(1,5),(2,7),(3,4),(6,8)]
 => [5,7,4,3,1,8,2,6] => [1,5,2,7,3,4,6,8] => [8,6,4,3,7,2,5,1] => 2
[(1,4),(2,7),(3,5),(6,8)]
 => [4,7,5,1,3,8,2,6] => [1,4,2,7,3,5,6,8] => [8,6,5,3,7,2,4,1] => ? = 1
[(1,3),(2,7),(4,5),(6,8)]
 => [3,7,1,5,4,8,2,6] => [1,3,2,7,4,5,6,8] => [8,6,5,4,7,2,3,1] => ? = 0
[(1,2),(3,7),(4,5),(6,8)]
 => [2,1,7,5,4,8,3,6] => [1,2,3,7,4,5,6,8] => [8,6,5,4,7,3,2,1] => ? = 0
[(1,2),(3,8),(4,5),(6,7)]
 => [2,1,8,5,4,7,6,3] => [1,2,3,8,4,5,6,7] => [7,6,5,4,8,3,2,1] => 0
[(1,3),(2,8),(4,5),(6,7)]
 => [3,8,1,5,4,7,6,2] => [1,3,2,8,4,5,6,7] => [7,6,5,4,8,2,3,1] => ? = 0
[(1,4),(2,8),(3,5),(6,7)]
 => [4,8,5,1,3,7,6,2] => [1,4,2,8,3,5,6,7] => [7,6,5,3,8,2,4,1] => ? = 1
[(1,5),(2,8),(3,4),(6,7)]
 => [5,8,4,3,1,7,6,2] => [1,5,2,8,3,4,6,7] => [7,6,4,3,8,2,5,1] => ? = 2
[(1,6),(2,8),(3,4),(5,7)]
 => [6,8,4,3,7,1,5,2] => [1,6,2,8,3,4,5,7] => [7,5,4,3,8,2,6,1] => ? = 3
[(1,7),(2,8),(3,4),(5,6)]
 => [7,8,4,3,6,5,1,2] => [1,7,2,8,3,4,5,6] => [6,5,4,3,8,2,7,1] => ? = 4
[(1,8),(2,7),(3,4),(5,6)]
 => [8,7,4,3,6,5,2,1] => [1,8,2,7,3,4,5,6] => [6,5,4,3,7,2,8,1] => 0
[(1,8),(2,7),(3,5),(4,6)]
 => [8,7,5,6,3,4,2,1] => [1,8,2,7,3,5,4,6] => [6,4,5,3,7,2,8,1] => ? = 0
[(1,7),(2,8),(3,5),(4,6)]
 => [7,8,5,6,3,4,1,2] => [1,7,2,8,3,5,4,6] => [6,4,5,3,8,2,7,1] => ? = 4
[(1,6),(2,8),(3,5),(4,7)]
 => [6,8,5,7,3,1,4,2] => [1,6,2,8,3,5,4,7] => [7,4,5,3,8,2,6,1] => ? = 3
[(1,5),(2,8),(3,6),(4,7)]
 => [5,8,6,7,1,3,4,2] => [1,5,2,8,3,6,4,7] => [7,4,6,3,8,2,5,1] => ? = 3
[(1,4),(2,8),(3,6),(5,7)]
 => [4,8,6,1,7,3,5,2] => [1,4,2,8,3,6,5,7] => [7,5,6,3,8,2,4,1] => ? = 1
[(1,3),(2,8),(4,6),(5,7)]
 => [3,8,1,6,7,4,5,2] => [1,3,2,8,4,6,5,7] => [7,5,6,4,8,2,3,1] => ? = 0
[(1,2),(3,8),(4,6),(5,7)]
 => [2,1,8,6,7,4,5,3] => [1,2,3,8,4,6,5,7] => [7,5,6,4,8,3,2,1] => ? = 0
[(1,2),(3,7),(4,6),(5,8)]
 => [2,1,7,6,8,4,3,5] => [1,2,3,7,4,6,5,8] => [8,5,6,4,7,3,2,1] => ? = 0
[(1,3),(2,7),(4,6),(5,8)]
 => [3,7,1,6,8,4,2,5] => [1,3,2,7,4,6,5,8] => [8,5,6,4,7,2,3,1] => ? = 0
[(1,4),(2,7),(3,6),(5,8)]
 => [4,7,6,1,8,3,2,5] => [1,4,2,7,3,6,5,8] => [8,5,6,3,7,2,4,1] => ? = 1
[(1,5),(2,7),(3,6),(4,8)]
 => [5,7,6,8,1,3,2,4] => [1,5,2,7,3,6,4,8] => [8,4,6,3,7,2,5,1] => ? = 3
[(1,6),(2,7),(3,5),(4,8)]
 => [6,7,5,8,3,1,2,4] => [1,6,2,7,3,5,4,8] => [8,4,5,3,7,2,6,1] => ? = 3
[(1,7),(2,6),(3,5),(4,8)]
 => [7,6,5,8,3,2,1,4] => [1,7,2,6,3,5,4,8] => [8,4,5,3,6,2,7,1] => ? = 0
[(1,8),(2,6),(3,5),(4,7)]
 => [8,6,5,7,3,2,4,1] => [1,8,2,6,3,5,4,7] => [7,4,5,3,6,2,8,1] => ? = 0
[(1,8),(2,5),(3,6),(4,7)]
 => [8,5,6,7,2,3,4,1] => [1,8,2,5,3,6,4,7] => [7,4,6,3,5,2,8,1] => ? = 1
[(1,7),(2,5),(3,6),(4,8)]
 => [7,5,6,8,2,3,1,4] => [1,7,2,5,3,6,4,8] => [8,4,6,3,5,2,7,1] => ? = 1
[(1,6),(2,5),(3,7),(4,8)]
 => [6,5,7,8,2,1,3,4] => [1,6,2,5,3,7,4,8] => [8,4,7,3,5,2,6,1] => ? = 2
[(1,5),(2,6),(3,7),(4,8)]
 => [5,6,7,8,1,2,3,4] => [1,5,2,6,3,7,4,8] => [8,4,7,3,6,2,5,1] => ? = 4
[(1,4),(2,6),(3,7),(5,8)]
 => [4,6,7,1,8,2,3,5] => [1,4,2,6,3,7,5,8] => [8,5,7,3,6,2,4,1] => ? = 2
[(1,3),(2,6),(4,7),(5,8)]
 => [3,6,1,7,8,2,4,5] => [1,3,2,6,4,7,5,8] => [8,5,7,4,6,2,3,1] => ? = 1
[(1,2),(3,6),(4,7),(5,8)]
 => [2,1,6,7,8,3,4,5] => [1,2,3,6,4,7,5,8] => [8,5,7,4,6,3,2,1] => ? = 1
[(1,2),(3,5),(4,7),(6,8)]
 => [2,1,5,7,3,8,4,6] => [1,2,3,5,4,7,6,8] => [8,6,7,4,5,3,2,1] => 0
[(1,3),(2,5),(4,7),(6,8)]
 => [3,5,1,7,2,8,4,6] => [1,3,2,5,4,7,6,8] => [8,6,7,4,5,2,3,1] => ? = 0
[(1,6),(2,3),(4,7),(5,8)]
 => [6,3,2,7,8,1,4,5] => [1,6,2,3,4,7,5,8] => [8,5,7,4,3,2,6,1] => 1
[(1,3),(2,4),(5,7),(6,8)]
 => [3,4,1,2,7,8,5,6] => [1,3,2,4,5,7,6,8] => [8,6,7,5,4,2,3,1] => 0
[(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,8,7,6,5] => [1,2,3,4,5,8,6,7] => [7,6,8,5,4,3,2,1] => 0
[(1,8),(2,3),(4,7),(5,6)]
 => [8,3,2,7,6,5,4,1] => [1,8,2,3,4,7,5,6] => [6,5,7,4,3,2,8,1] => 0
[(1,2),(3,8),(4,7),(5,6)]
 => [2,1,8,7,6,5,4,3] => [1,2,3,8,4,7,5,6] => [6,5,7,4,8,3,2,1] => 0
[(1,8),(2,7),(3,6),(4,5)]
 => [8,7,6,5,4,3,2,1] => [1,8,2,7,3,6,4,5] => [5,4,6,3,7,2,8,1] => 0
[(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => [1,2,3,4,5,6,7,8,9,10] => [10,9,8,7,6,5,4,3,2,1] => 0
Description
The number of occurrences of the pattern 132 in a permutation.
Matching statistic: St000233
Mapping
Mp00092: Perfect matchings —to set partition⟶ Set partitions
Mp00216: Set partitions —inverse Wachs-White⟶ Set partitions
Mp00218: Set partitions —inverse Wachs-White-rho⟶ Set partitions
St000233: Set partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 19%
Values
[(1,2)]
 => {{1,2}}
 => {{1,2}}
 => {{1,2}}
 => 0
[(1,2),(3,4)]
 => {{1,2},{3,4}}
 => {{1,2},{3,4}}
 => {{1,2},{3,4}}
 => 0
[(1,3),(2,4)]
 => {{1,3},{2,4}}
 => {{1,2,4},{3}}
 => {{1,2,4},{3}}
 => 0
[(1,4),(2,3)]
 => {{1,4},{2,3}}
 => {{1,4},{2,3}}
 => {{1,3},{2,4}}
 => 0
[(1,2),(3,4),(5,6)]
 => {{1,2},{3,4},{5,6}}
 => {{1,2},{3,4},{5,6}}
 => {{1,2},{3,4},{5,6}}
 => 0
[(1,3),(2,4),(5,6)]
 => {{1,3},{2,4},{5,6}}
 => {{1,2},{3,4,6},{5}}
 => {{1,2},{3,4,6},{5}}
 => 0
[(1,4),(2,3),(5,6)]
 => {{1,4},{2,3},{5,6}}
 => {{1,2},{3,6},{4,5}}
 => {{1,2},{3,5},{4,6}}
 => 0
[(1,5),(2,3),(4,6)]
 => {{1,5},{2,3},{4,6}}
 => {{1,2,6},{3},{4,5}}
 => {{1,2,5},{3},{4,6}}
 => 0
[(1,6),(2,3),(4,5)]
 => {{1,6},{2,3},{4,5}}
 => {{1,6},{2,3},{4,5}}
 => {{1,3},{2,5},{4,6}}
 => 0
[(1,6),(2,4),(3,5)]
 => {{1,6},{2,4},{3,5}}
 => {{1,6},{2,3,5},{4}}
 => {{1,3,6},{2,5},{4}}
 => 0
[(1,5),(2,4),(3,6)]
 => {{1,5},{2,4},{3,6}}
 => {{1,2,6},{3,5},{4}}
 => {{1,2,5},{3,6},{4}}
 => 0
[(1,4),(2,5),(3,6)]
 => {{1,4},{2,5},{3,6}}
 => {{1,2,4},{3,6},{5}}
 => {{1,2,6},{3,4},{5}}
 => 1
[(1,3),(2,5),(4,6)]
 => {{1,3},{2,5},{4,6}}
 => {{1,2,4,6},{3},{5}}
 => {{1,2,4,6},{3},{5}}
 => 0
[(1,2),(3,5),(4,6)]
 => {{1,2},{3,5},{4,6}}
 => {{1,2,4},{3},{5,6}}
 => {{1,2,4},{3},{5,6}}
 => 0
[(1,2),(3,6),(4,5)]
 => {{1,2},{3,6},{4,5}}
 => {{1,4},{2,3},{5,6}}
 => {{1,3},{2,4},{5,6}}
 => 0
[(1,3),(2,6),(4,5)]
 => {{1,3},{2,6},{4,5}}
 => {{1,4,6},{2,3},{5}}
 => {{1,3},{2,4,6},{5}}
 => 0
[(1,4),(2,6),(3,5)]
 => {{1,4},{2,6},{3,5}}
 => {{1,4},{2,3,6},{5}}
 => {{1,3,4},{2,6},{5}}
 => 1
[(1,5),(2,6),(3,4)]
 => {{1,5},{2,6},{3,4}}
 => {{1,3,4},{2,6},{5}}
 => {{1,6},{2,3,4},{5}}
 => 2
[(1,6),(2,5),(3,4)]
 => {{1,6},{2,5},{3,4}}
 => {{1,6},{2,5},{3,4}}
 => {{1,4},{2,5},{3,6}}
 => 0
[(1,2),(3,4),(5,6),(7,8)]
 => {{1,2},{3,4},{5,6},{7,8}}
 => {{1,2},{3,4},{5,6},{7,8}}
 => {{1,2},{3,4},{5,6},{7,8}}
 => 0
[(1,3),(2,4),(5,6),(7,8)]
 => {{1,3},{2,4},{5,6},{7,8}}
 => {{1,2},{3,4},{5,6,8},{7}}
 => {{1,2},{3,4},{5,6,8},{7}}
 => ? = 0
[(1,4),(2,3),(5,6),(7,8)]
 => {{1,4},{2,3},{5,6},{7,8}}
 => {{1,2},{3,4},{5,8},{6,7}}
 => {{1,2},{3,4},{5,7},{6,8}}
 => 0
[(1,5),(2,3),(4,6),(7,8)]
 => {{1,5},{2,3},{4,6},{7,8}}
 => {{1,2},{3,4,8},{5},{6,7}}
 => {{1,2},{3,4,7},{5},{6,8}}
 => ? = 0
[(1,6),(2,3),(4,5),(7,8)]
 => {{1,6},{2,3},{4,5},{7,8}}
 => {{1,2},{3,8},{4,5},{6,7}}
 => {{1,2},{3,5},{4,7},{6,8}}
 => 0
[(1,7),(2,3),(4,5),(6,8)]
 => {{1,7},{2,3},{4,5},{6,8}}
 => {{1,2,8},{3},{4,5},{6,7}}
 => {{1,2,5},{3},{4,7},{6,8}}
 => ? = 0
[(1,8),(2,3),(4,5),(6,7)]
 => {{1,8},{2,3},{4,5},{6,7}}
 => {{1,8},{2,3},{4,5},{6,7}}
 => {{1,3},{2,5},{4,7},{6,8}}
 => 0
[(1,8),(2,4),(3,5),(6,7)]
 => {{1,8},{2,4},{3,5},{6,7}}
 => {{1,8},{2,3},{4,5,7},{6}}
 => {{1,3},{2,5,8},{4,7},{6}}
 => ? = 0
[(1,7),(2,4),(3,5),(6,8)]
 => {{1,7},{2,4},{3,5},{6,8}}
 => {{1,2,8},{3},{4,5,7},{6}}
 => {{1,2,5,8},{3},{4,7},{6}}
 => ? = 0
[(1,6),(2,4),(3,5),(7,8)]
 => {{1,6},{2,4},{3,5},{7,8}}
 => {{1,2},{3,8},{4,5,7},{6}}
 => {{1,2},{3,5,8},{4,7},{6}}
 => ? = 0
[(1,5),(2,4),(3,6),(7,8)]
 => {{1,5},{2,4},{3,6},{7,8}}
 => {{1,2},{3,4,8},{5,7},{6}}
 => {{1,2},{3,4,7},{5,8},{6}}
 => ? = 0
[(1,4),(2,5),(3,6),(7,8)]
 => {{1,4},{2,5},{3,6},{7,8}}
 => {{1,2},{3,4,6},{5,8},{7}}
 => {{1,2},{3,4,8},{5,6},{7}}
 => ? = 1
[(1,3),(2,5),(4,6),(7,8)]
 => {{1,3},{2,5},{4,6},{7,8}}
 => {{1,2},{3,4,6,8},{5},{7}}
 => {{1,2},{3,4,6,8},{5},{7}}
 => ? = 0
[(1,2),(3,5),(4,6),(7,8)]
 => {{1,2},{3,5},{4,6},{7,8}}
 => {{1,2},{3,4,6},{5},{7,8}}
 => {{1,2},{3,4,6},{5},{7,8}}
 => ? = 0
[(1,2),(3,6),(4,5),(7,8)]
 => {{1,2},{3,6},{4,5},{7,8}}
 => {{1,2},{3,6},{4,5},{7,8}}
 => {{1,2},{3,5},{4,6},{7,8}}
 => 0
[(1,3),(2,6),(4,5),(7,8)]
 => {{1,3},{2,6},{4,5},{7,8}}
 => {{1,2},{3,6,8},{4,5},{7}}
 => {{1,2},{3,5},{4,6,8},{7}}
 => ? = 0
[(1,4),(2,6),(3,5),(7,8)]
 => {{1,4},{2,6},{3,5},{7,8}}
 => {{1,2},{3,6},{4,5,8},{7}}
 => {{1,2},{3,5,6},{4,8},{7}}
 => ? = 1
[(1,5),(2,6),(3,4),(7,8)]
 => {{1,5},{2,6},{3,4},{7,8}}
 => {{1,2},{3,5,6},{4,8},{7}}
 => {{1,2},{3,8},{4,5,6},{7}}
 => ? = 2
[(1,6),(2,5),(3,4),(7,8)]
 => {{1,6},{2,5},{3,4},{7,8}}
 => {{1,2},{3,8},{4,7},{5,6}}
 => {{1,2},{3,6},{4,7},{5,8}}
 => 0
[(1,7),(2,5),(3,4),(6,8)]
 => {{1,7},{2,5},{3,4},{6,8}}
 => {{1,2,8},{3},{4,7},{5,6}}
 => {{1,2,6},{3},{4,7},{5,8}}
 => ? = 0
[(1,8),(2,5),(3,4),(6,7)]
 => {{1,8},{2,5},{3,4},{6,7}}
 => {{1,8},{2,3},{4,7},{5,6}}
 => {{1,3},{2,6},{4,7},{5,8}}
 => 0
[(1,8),(2,6),(3,4),(5,7)]
 => {{1,8},{2,6},{3,4},{5,7}}
 => {{1,8},{2,3,7},{4},{5,6}}
 => {{1,3,7},{2,6},{4},{5,8}}
 => ? = 0
[(1,7),(2,6),(3,4),(5,8)]
 => {{1,7},{2,6},{3,4},{5,8}}
 => {{1,2,8},{3,7},{4},{5,6}}
 => {{1,2,6},{3,7},{4},{5,8}}
 => ? = 0
[(1,6),(2,7),(3,4),(5,8)]
 => {{1,6},{2,7},{3,4},{5,8}}
 => {{1,2,5,6},{3,8},{4},{7}}
 => {{1,2,8},{3,5,6},{4},{7}}
 => ? = 3
[(1,5),(2,7),(3,4),(6,8)]
 => {{1,5},{2,7},{3,4},{6,8}}
 => {{1,2,5,6},{3},{4,8},{7}}
 => {{1,2,8},{3},{4,5,6},{7}}
 => ? = 2
[(1,4),(2,7),(3,5),(6,8)]
 => {{1,4},{2,7},{3,5},{6,8}}
 => {{1,2,6},{3},{4,5,8},{7}}
 => {{1,2,5,6},{3},{4,8},{7}}
 => ? = 1
[(1,3),(2,7),(4,5),(6,8)]
 => {{1,3},{2,7},{4,5},{6,8}}
 => {{1,2,6,8},{3},{4,5},{7}}
 => {{1,2,5},{3},{4,6,8},{7}}
 => ? = 0
[(1,2),(3,7),(4,5),(6,8)]
 => {{1,2},{3,7},{4,5},{6,8}}
 => {{1,2,6},{3},{4,5},{7,8}}
 => {{1,2,5},{3},{4,6},{7,8}}
 => ? = 0
[(1,2),(3,8),(4,5),(6,7)]
 => {{1,2},{3,8},{4,5},{6,7}}
 => {{1,6},{2,3},{4,5},{7,8}}
 => {{1,3},{2,5},{4,6},{7,8}}
 => 0
[(1,3),(2,8),(4,5),(6,7)]
 => {{1,3},{2,8},{4,5},{6,7}}
 => {{1,6,8},{2,3},{4,5},{7}}
 => {{1,3},{2,5},{4,6,8},{7}}
 => ? = 0
[(1,4),(2,8),(3,5),(6,7)]
 => {{1,4},{2,8},{3,5},{6,7}}
 => {{1,6},{2,3},{4,5,8},{7}}
 => {{1,3},{2,5,6},{4,8},{7}}
 => ? = 1
[(1,5),(2,8),(3,4),(6,7)]
 => {{1,5},{2,8},{3,4},{6,7}}
 => {{1,5,6},{2,3},{4,8},{7}}
 => {{1,3},{2,8},{4,5,6},{7}}
 => ? = 2
[(1,6),(2,8),(3,4),(5,7)]
 => {{1,6},{2,8},{3,4},{5,7}}
 => {{1,5,6},{2,3,8},{4},{7}}
 => {{1,3,5,6},{2,8},{4},{7}}
 => ? = 3
[(1,7),(2,8),(3,4),(5,6)]
 => {{1,7},{2,8},{3,4},{5,6}}
 => {{1,5,6},{2,8},{3,4},{7}}
 => {{1,4},{2,8},{3,5,6},{7}}
 => ? = 4
[(1,8),(2,7),(3,4),(5,6)]
 => {{1,8},{2,7},{3,4},{5,6}}
 => {{1,8},{2,7},{3,4},{5,6}}
 => {{1,4},{2,6},{3,7},{5,8}}
 => 0
[(1,8),(2,7),(3,5),(4,6)]
 => {{1,8},{2,7},{3,5},{4,6}}
 => {{1,8},{2,7},{3,4,6},{5}}
 => {{1,4,8},{2,6},{3,7},{5}}
 => ? = 0
[(1,7),(2,8),(3,5),(4,6)]
 => {{1,7},{2,8},{3,5},{4,6}}
 => {{1,5},{2,8},{3,4,6},{7}}
 => {{1,4,5},{2,6},{3,8},{7}}
 => ? = 4
[(1,6),(2,8),(3,5),(4,7)]
 => {{1,6},{2,8},{3,5},{4,7}}
 => {{1,5},{2,3,8},{4,6},{7}}
 => {{1,3,8},{2,6},{4,5},{7}}
 => ? = 3
[(1,5),(2,8),(3,6),(4,7)]
 => {{1,5},{2,8},{3,6},{4,7}}
 => {{1,6},{2,3,5},{4,8},{7}}
 => {{1,3,5},{2,8},{4,6},{7}}
 => ? = 3
[(1,4),(2,8),(3,6),(5,7)]
 => {{1,4},{2,8},{3,6},{5,7}}
 => {{1,6},{2,3,5,8},{4},{7}}
 => {{1,3,8},{2,5,6},{4},{7}}
 => ? = 1
[(1,3),(2,8),(4,6),(5,7)]
 => {{1,3},{2,8},{4,6},{5,7}}
 => {{1,6,8},{2,3,5},{4},{7}}
 => {{1,3,6,8},{2,5},{4},{7}}
 => ? = 0
[(1,2),(3,8),(4,6),(5,7)]
 => {{1,2},{3,8},{4,6},{5,7}}
 => {{1,6},{2,3,5},{4},{7,8}}
 => {{1,3,6},{2,5},{4},{7,8}}
 => ? = 0
[(1,2),(3,7),(4,6),(5,8)]
 => {{1,2},{3,7},{4,6},{5,8}}
 => {{1,2,6},{3,5},{4},{7,8}}
 => {{1,2,5},{3,6},{4},{7,8}}
 => ? = 0
[(1,3),(2,7),(4,6),(5,8)]
 => {{1,3},{2,7},{4,6},{5,8}}
 => {{1,2,6,8},{3,5},{4},{7}}
 => {{1,2,5},{3,6,8},{4},{7}}
 => ? = 0
[(1,4),(2,7),(3,6),(5,8)]
 => {{1,4},{2,7},{3,6},{5,8}}
 => {{1,2,6},{3,5,8},{4},{7}}
 => {{1,2,5,6},{3,8},{4},{7}}
 => ? = 1
[(1,5),(2,7),(3,6),(4,8)]
 => {{1,5},{2,7},{3,6},{4,8}}
 => {{1,2,6},{3,5},{4,8},{7}}
 => {{1,2,8},{3,5},{4,6},{7}}
 => ? = 3
[(1,6),(2,7),(3,5),(4,8)]
 => {{1,6},{2,7},{3,5},{4,8}}
 => {{1,2,5},{3,8},{4,6},{7}}
 => {{1,2,6},{3,8},{4,5},{7}}
 => ? = 3
[(1,7),(2,6),(3,5),(4,8)]
 => {{1,7},{2,6},{3,5},{4,8}}
 => {{1,2,8},{3,7},{4,6},{5}}
 => {{1,2,6},{3,7},{4,8},{5}}
 => ? = 0
[(1,8),(2,6),(3,5),(4,7)]
 => {{1,8},{2,6},{3,5},{4,7}}
 => {{1,8},{2,3,7},{4,6},{5}}
 => {{1,3,7},{2,6},{4,8},{5}}
 => ? = 0
[(1,8),(2,5),(3,6),(4,7)]
 => {{1,8},{2,5},{3,6},{4,7}}
 => {{1,8},{2,3,5},{4,7},{6}}
 => {{1,3,5},{2,7},{4,8},{6}}
 => ? = 1
[(1,7),(2,5),(3,6),(4,8)]
 => {{1,7},{2,5},{3,6},{4,8}}
 => {{1,2,8},{3,5},{4,7},{6}}
 => {{1,2,7},{3,5},{4,8},{6}}
 => ? = 1
[(1,6),(2,5),(3,7),(4,8)]
 => {{1,6},{2,5},{3,7},{4,8}}
 => {{1,2,4,7},{3,8},{5},{6}}
 => {{1,2,8},{3,4,7},{5},{6}}
 => ? = 2
[(1,5),(2,6),(3,7),(4,8)]
 => {{1,5},{2,6},{3,7},{4,8}}
 => {{1,2,4,8},{3,6},{5},{7}}
 => {{1,2,6},{3,4,8},{5},{7}}
 => ? = 4
[(1,4),(2,6),(3,7),(5,8)]
 => {{1,4},{2,6},{3,7},{5,8}}
 => {{1,2,4},{3,6},{5,8},{7}}
 => {{1,2,8},{3,4},{5,6},{7}}
 => ? = 2
[(1,3),(2,6),(4,7),(5,8)]
 => {{1,3},{2,6},{4,7},{5,8}}
 => {{1,2,4},{3,6,8},{5},{7}}
 => {{1,2,6,8},{3,4},{5},{7}}
 => ? = 1
[(1,2),(3,6),(4,7),(5,8)]
 => {{1,2},{3,6},{4,7},{5,8}}
 => {{1,2,4},{3,6},{5},{7,8}}
 => {{1,2,6},{3,4},{5},{7,8}}
 => ? = 1
[(1,2),(3,5),(4,7),(6,8)]
 => {{1,2},{3,5},{4,7},{6,8}}
 => {{1,2,4,6},{3},{5},{7,8}}
 => {{1,2,4,6},{3},{5},{7,8}}
 => ? = 0
[(1,3),(2,5),(4,7),(6,8)]
 => {{1,3},{2,5},{4,7},{6,8}}
 => {{1,2,4,6,8},{3},{5},{7}}
 => {{1,2,4,6,8},{3},{5},{7}}
 => ? = 0
[(1,4),(2,5),(3,7),(6,8)]
 => {{1,4},{2,5},{3,7},{6,8}}
 => {{1,2,4,6},{3},{5,8},{7}}
 => {{1,2,4,8},{3},{5,6},{7}}
 => ? = 1
[(1,2),(3,4),(5,8),(6,7)]
 => {{1,2},{3,4},{5,8},{6,7}}
 => {{1,4},{2,3},{5,6},{7,8}}
 => {{1,3},{2,4},{5,6},{7,8}}
 => 0
[(1,4),(2,3),(5,8),(6,7)]
 => {{1,4},{2,3},{5,8},{6,7}}
 => {{1,4},{2,3},{5,8},{6,7}}
 => {{1,3},{2,4},{5,7},{6,8}}
 => 0
[(1,8),(2,3),(4,7),(5,6)]
 => {{1,8},{2,3},{4,7},{5,6}}
 => {{1,8},{2,5},{3,4},{6,7}}
 => {{1,4},{2,5},{3,7},{6,8}}
 => 0
[(1,2),(3,8),(4,7),(5,6)]
 => {{1,2},{3,8},{4,7},{5,6}}
 => {{1,6},{2,5},{3,4},{7,8}}
 => {{1,4},{2,5},{3,6},{7,8}}
 => 0
[(1,8),(2,7),(3,6),(4,5)]
 => {{1,8},{2,7},{3,6},{4,5}}
 => {{1,8},{2,7},{3,6},{4,5}}
 => {{1,5},{2,6},{3,7},{4,8}}
 => 0
[(1,2),(3,4),(5,6),(7,8),(9,10)]
 => {{1,2},{3,4},{5,6},{7,8},{9,10}}
 => {{1,2},{3,4},{5,6},{7,8},{9,10}}
 => {{1,2},{3,4},{5,6},{7,8},{9,10}}
 => 0
Description
The number of nestings of a set partition. This is given by the number of i<i′<j′<j such that i,j are two consecutive entries on one block, and i′,j′ are consecutive entries in another block.
Matching statistic: St001083
Mapping
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001083: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 19%
Values
[(1,2)]
 => [2,1] => [1,2] => [2,1] => 0
[(1,2),(3,4)]
 => [2,1,4,3] => [1,2,3,4] => [4,3,2,1] => 0
[(1,3),(2,4)]
 => [3,4,1,2] => [1,3,2,4] => [4,2,3,1] => 0
[(1,4),(2,3)]
 => [4,3,2,1] => [1,4,2,3] => [3,2,4,1] => 0
[(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 0
[(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => [1,3,2,4,5,6] => [6,5,4,2,3,1] => 0
[(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => [1,4,2,3,5,6] => [6,5,3,2,4,1] => 0
[(1,5),(2,3),(4,6)]
 => [5,3,2,6,1,4] => [1,5,2,3,4,6] => [6,4,3,2,5,1] => 0
[(1,6),(2,3),(4,5)]
 => [6,3,2,5,4,1] => [1,6,2,3,4,5] => [5,4,3,2,6,1] => 0
[(1,6),(2,4),(3,5)]
 => [6,4,5,2,3,1] => [1,6,2,4,3,5] => [5,3,4,2,6,1] => 0
[(1,5),(2,4),(3,6)]
 => [5,4,6,2,1,3] => [1,5,2,4,3,6] => [6,3,4,2,5,1] => 0
[(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => [1,4,2,5,3,6] => [6,3,5,2,4,1] => 1
[(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => [1,3,2,5,4,6] => [6,4,5,2,3,1] => 0
[(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => [1,2,3,5,4,6] => [6,4,5,3,2,1] => 0
[(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => [1,2,3,6,4,5] => [5,4,6,3,2,1] => 0
[(1,3),(2,6),(4,5)]
 => [3,6,1,5,4,2] => [1,3,2,6,4,5] => [5,4,6,2,3,1] => 0
[(1,4),(2,6),(3,5)]
 => [4,6,5,1,3,2] => [1,4,2,6,3,5] => [5,3,6,2,4,1] => 1
[(1,5),(2,6),(3,4)]
 => [5,6,4,3,1,2] => [1,5,2,6,3,4] => [4,3,6,2,5,1] => 2
[(1,6),(2,5),(3,4)]
 => [6,5,4,3,2,1] => [1,6,2,5,3,4] => [4,3,5,2,6,1] => 0
[(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => [1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1] => 0
[(1,3),(2,4),(5,6),(7,8)]
 => [3,4,1,2,6,5,8,7] => [1,3,2,4,5,6,7,8] => [8,7,6,5,4,2,3,1] => ? = 0
[(1,4),(2,3),(5,6),(7,8)]
 => [4,3,2,1,6,5,8,7] => [1,4,2,3,5,6,7,8] => [8,7,6,5,3,2,4,1] => ? = 0
[(1,5),(2,3),(4,6),(7,8)]
 => [5,3,2,6,1,4,8,7] => [1,5,2,3,4,6,7,8] => [8,7,6,4,3,2,5,1] => ? = 0
[(1,6),(2,3),(4,5),(7,8)]
 => [6,3,2,5,4,1,8,7] => [1,6,2,3,4,5,7,8] => [8,7,5,4,3,2,6,1] => ? = 0
[(1,7),(2,3),(4,5),(6,8)]
 => [7,3,2,5,4,8,1,6] => [1,7,2,3,4,5,6,8] => [8,6,5,4,3,2,7,1] => ? = 0
[(1,8),(2,3),(4,5),(6,7)]
 => [8,3,2,5,4,7,6,1] => [1,8,2,3,4,5,6,7] => [7,6,5,4,3,2,8,1] => 0
[(1,8),(2,4),(3,5),(6,7)]
 => [8,4,5,2,3,7,6,1] => [1,8,2,4,3,5,6,7] => [7,6,5,3,4,2,8,1] => ? = 0
[(1,7),(2,4),(3,5),(6,8)]
 => [7,4,5,2,3,8,1,6] => [1,7,2,4,3,5,6,8] => [8,6,5,3,4,2,7,1] => ? = 0
[(1,6),(2,4),(3,5),(7,8)]
 => [6,4,5,2,3,1,8,7] => [1,6,2,4,3,5,7,8] => [8,7,5,3,4,2,6,1] => ? = 0
[(1,5),(2,4),(3,6),(7,8)]
 => [5,4,6,2,1,3,8,7] => [1,5,2,4,3,6,7,8] => [8,7,6,3,4,2,5,1] => ? = 0
[(1,4),(2,5),(3,6),(7,8)]
 => [4,5,6,1,2,3,8,7] => [1,4,2,5,3,6,7,8] => [8,7,6,3,5,2,4,1] => 1
[(1,3),(2,5),(4,6),(7,8)]
 => [3,5,1,6,2,4,8,7] => [1,3,2,5,4,6,7,8] => [8,7,6,4,5,2,3,1] => 0
[(1,2),(3,5),(4,6),(7,8)]
 => [2,1,5,6,3,4,8,7] => [1,2,3,5,4,6,7,8] => [8,7,6,4,5,3,2,1] => ? = 0
[(1,2),(3,6),(4,5),(7,8)]
 => [2,1,6,5,4,3,8,7] => [1,2,3,6,4,5,7,8] => [8,7,5,4,6,3,2,1] => ? = 0
[(1,3),(2,6),(4,5),(7,8)]
 => [3,6,1,5,4,2,8,7] => [1,3,2,6,4,5,7,8] => [8,7,5,4,6,2,3,1] => ? = 0
[(1,4),(2,6),(3,5),(7,8)]
 => [4,6,5,1,3,2,8,7] => [1,4,2,6,3,5,7,8] => [8,7,5,3,6,2,4,1] => ? = 1
[(1,5),(2,6),(3,4),(7,8)]
 => [5,6,4,3,1,2,8,7] => [1,5,2,6,3,4,7,8] => [8,7,4,3,6,2,5,1] => ? = 2
[(1,6),(2,5),(3,4),(7,8)]
 => [6,5,4,3,2,1,8,7] => [1,6,2,5,3,4,7,8] => [8,7,4,3,5,2,6,1] => ? = 0
[(1,7),(2,5),(3,4),(6,8)]
 => [7,5,4,3,2,8,1,6] => [1,7,2,5,3,4,6,8] => [8,6,4,3,5,2,7,1] => ? = 0
[(1,8),(2,5),(3,4),(6,7)]
 => [8,5,4,3,2,7,6,1] => [1,8,2,5,3,4,6,7] => [7,6,4,3,5,2,8,1] => ? = 0
[(1,8),(2,6),(3,4),(5,7)]
 => [8,6,4,3,7,2,5,1] => [1,8,2,6,3,4,5,7] => [7,5,4,3,6,2,8,1] => ? = 0
[(1,7),(2,6),(3,4),(5,8)]
 => [7,6,4,3,8,2,1,5] => [1,7,2,6,3,4,5,8] => [8,5,4,3,6,2,7,1] => ? = 0
[(1,6),(2,7),(3,4),(5,8)]
 => [6,7,4,3,8,1,2,5] => [1,6,2,7,3,4,5,8] => [8,5,4,3,7,2,6,1] => ? = 3
[(1,5),(2,7),(3,4),(6,8)]
 => [5,7,4,3,1,8,2,6] => [1,5,2,7,3,4,6,8] => [8,6,4,3,7,2,5,1] => 2
[(1,4),(2,7),(3,5),(6,8)]
 => [4,7,5,1,3,8,2,6] => [1,4,2,7,3,5,6,8] => [8,6,5,3,7,2,4,1] => ? = 1
[(1,3),(2,7),(4,5),(6,8)]
 => [3,7,1,5,4,8,2,6] => [1,3,2,7,4,5,6,8] => [8,6,5,4,7,2,3,1] => ? = 0
[(1,2),(3,7),(4,5),(6,8)]
 => [2,1,7,5,4,8,3,6] => [1,2,3,7,4,5,6,8] => [8,6,5,4,7,3,2,1] => ? = 0
[(1,2),(3,8),(4,5),(6,7)]
 => [2,1,8,5,4,7,6,3] => [1,2,3,8,4,5,6,7] => [7,6,5,4,8,3,2,1] => 0
[(1,3),(2,8),(4,5),(6,7)]
 => [3,8,1,5,4,7,6,2] => [1,3,2,8,4,5,6,7] => [7,6,5,4,8,2,3,1] => ? = 0
[(1,4),(2,8),(3,5),(6,7)]
 => [4,8,5,1,3,7,6,2] => [1,4,2,8,3,5,6,7] => [7,6,5,3,8,2,4,1] => ? = 1
[(1,5),(2,8),(3,4),(6,7)]
 => [5,8,4,3,1,7,6,2] => [1,5,2,8,3,4,6,7] => [7,6,4,3,8,2,5,1] => ? = 2
[(1,6),(2,8),(3,4),(5,7)]
 => [6,8,4,3,7,1,5,2] => [1,6,2,8,3,4,5,7] => [7,5,4,3,8,2,6,1] => ? = 3
[(1,7),(2,8),(3,4),(5,6)]
 => [7,8,4,3,6,5,1,2] => [1,7,2,8,3,4,5,6] => [6,5,4,3,8,2,7,1] => ? = 4
[(1,8),(2,7),(3,4),(5,6)]
 => [8,7,4,3,6,5,2,1] => [1,8,2,7,3,4,5,6] => [6,5,4,3,7,2,8,1] => 0
[(1,8),(2,7),(3,5),(4,6)]
 => [8,7,5,6,3,4,2,1] => [1,8,2,7,3,5,4,6] => [6,4,5,3,7,2,8,1] => ? = 0
[(1,7),(2,8),(3,5),(4,6)]
 => [7,8,5,6,3,4,1,2] => [1,7,2,8,3,5,4,6] => [6,4,5,3,8,2,7,1] => ? = 4
[(1,6),(2,8),(3,5),(4,7)]
 => [6,8,5,7,3,1,4,2] => [1,6,2,8,3,5,4,7] => [7,4,5,3,8,2,6,1] => ? = 3
[(1,5),(2,8),(3,6),(4,7)]
 => [5,8,6,7,1,3,4,2] => [1,5,2,8,3,6,4,7] => [7,4,6,3,8,2,5,1] => ? = 3
[(1,4),(2,8),(3,6),(5,7)]
 => [4,8,6,1,7,3,5,2] => [1,4,2,8,3,6,5,7] => [7,5,6,3,8,2,4,1] => ? = 1
[(1,3),(2,8),(4,6),(5,7)]
 => [3,8,1,6,7,4,5,2] => [1,3,2,8,4,6,5,7] => [7,5,6,4,8,2,3,1] => ? = 0
[(1,2),(3,8),(4,6),(5,7)]
 => [2,1,8,6,7,4,5,3] => [1,2,3,8,4,6,5,7] => [7,5,6,4,8,3,2,1] => ? = 0
[(1,2),(3,7),(4,6),(5,8)]
 => [2,1,7,6,8,4,3,5] => [1,2,3,7,4,6,5,8] => [8,5,6,4,7,3,2,1] => ? = 0
[(1,3),(2,7),(4,6),(5,8)]
 => [3,7,1,6,8,4,2,5] => [1,3,2,7,4,6,5,8] => [8,5,6,4,7,2,3,1] => ? = 0
[(1,4),(2,7),(3,6),(5,8)]
 => [4,7,6,1,8,3,2,5] => [1,4,2,7,3,6,5,8] => [8,5,6,3,7,2,4,1] => ? = 1
[(1,5),(2,7),(3,6),(4,8)]
 => [5,7,6,8,1,3,2,4] => [1,5,2,7,3,6,4,8] => [8,4,6,3,7,2,5,1] => ? = 3
[(1,6),(2,7),(3,5),(4,8)]
 => [6,7,5,8,3,1,2,4] => [1,6,2,7,3,5,4,8] => [8,4,5,3,7,2,6,1] => ? = 3
[(1,7),(2,6),(3,5),(4,8)]
 => [7,6,5,8,3,2,1,4] => [1,7,2,6,3,5,4,8] => [8,4,5,3,6,2,7,1] => ? = 0
[(1,8),(2,6),(3,5),(4,7)]
 => [8,6,5,7,3,2,4,1] => [1,8,2,6,3,5,4,7] => [7,4,5,3,6,2,8,1] => ? = 0
[(1,8),(2,5),(3,6),(4,7)]
 => [8,5,6,7,2,3,4,1] => [1,8,2,5,3,6,4,7] => [7,4,6,3,5,2,8,1] => ? = 1
[(1,7),(2,5),(3,6),(4,8)]
 => [7,5,6,8,2,3,1,4] => [1,7,2,5,3,6,4,8] => [8,4,6,3,5,2,7,1] => ? = 1
[(1,6),(2,5),(3,7),(4,8)]
 => [6,5,7,8,2,1,3,4] => [1,6,2,5,3,7,4,8] => [8,4,7,3,5,2,6,1] => ? = 2
[(1,5),(2,6),(3,7),(4,8)]
 => [5,6,7,8,1,2,3,4] => [1,5,2,6,3,7,4,8] => [8,4,7,3,6,2,5,1] => ? = 4
[(1,4),(2,6),(3,7),(5,8)]
 => [4,6,7,1,8,2,3,5] => [1,4,2,6,3,7,5,8] => [8,5,7,3,6,2,4,1] => ? = 2
[(1,3),(2,6),(4,7),(5,8)]
 => [3,6,1,7,8,2,4,5] => [1,3,2,6,4,7,5,8] => [8,5,7,4,6,2,3,1] => ? = 1
[(1,2),(3,6),(4,7),(5,8)]
 => [2,1,6,7,8,3,4,5] => [1,2,3,6,4,7,5,8] => [8,5,7,4,6,3,2,1] => ? = 1
[(1,2),(3,5),(4,7),(6,8)]
 => [2,1,5,7,3,8,4,6] => [1,2,3,5,4,7,6,8] => [8,6,7,4,5,3,2,1] => 0
[(1,3),(2,5),(4,7),(6,8)]
 => [3,5,1,7,2,8,4,6] => [1,3,2,5,4,7,6,8] => [8,6,7,4,5,2,3,1] => ? = 0
[(1,6),(2,3),(4,7),(5,8)]
 => [6,3,2,7,8,1,4,5] => [1,6,2,3,4,7,5,8] => [8,5,7,4,3,2,6,1] => 1
[(1,3),(2,4),(5,7),(6,8)]
 => [3,4,1,2,7,8,5,6] => [1,3,2,4,5,7,6,8] => [8,6,7,5,4,2,3,1] => 0
[(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,8,7,6,5] => [1,2,3,4,5,8,6,7] => [7,6,8,5,4,3,2,1] => 0
[(1,8),(2,3),(4,7),(5,6)]
 => [8,3,2,7,6,5,4,1] => [1,8,2,3,4,7,5,6] => [6,5,7,4,3,2,8,1] => 0
[(1,2),(3,8),(4,7),(5,6)]
 => [2,1,8,7,6,5,4,3] => [1,2,3,8,4,7,5,6] => [6,5,7,4,8,3,2,1] => 0
[(1,8),(2,7),(3,6),(4,5)]
 => [8,7,6,5,4,3,2,1] => [1,8,2,7,3,6,4,5] => [5,4,6,3,7,2,8,1] => 0
[(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => [1,2,3,4,5,6,7,8,9,10] => [10,9,8,7,6,5,4,3,2,1] => 0
Description
The number of boxed occurrences of 132 in a permutation. This is the number of occurrences of the pattern 132 such that any entry between the three matched entries is either larger than the largest matched entry or smaller than the smallest matched entry.
Matching statistic: St000218
Mapping
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000218: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 19%
Values
[(1,2)]
 => [2,1] => [1,2] => [2,1] => 0
[(1,2),(3,4)]
 => [2,1,4,3] => [1,2,3,4] => [4,3,2,1] => 0
[(1,3),(2,4)]
 => [3,4,1,2] => [1,3,2,4] => [4,2,3,1] => 0
[(1,4),(2,3)]
 => [4,3,2,1] => [1,4,2,3] => [4,1,3,2] => 0
[(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 0
[(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => [1,3,2,4,5,6] => [6,4,5,3,2,1] => 0
[(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => [1,4,2,3,5,6] => [6,3,5,4,2,1] => 0
[(1,5),(2,3),(4,6)]
 => [5,3,2,6,1,4] => [1,5,2,3,4,6] => [6,2,5,4,3,1] => 0
[(1,6),(2,3),(4,5)]
 => [6,3,2,5,4,1] => [1,6,2,3,4,5] => [6,1,5,4,3,2] => 0
[(1,6),(2,4),(3,5)]
 => [6,4,5,2,3,1] => [1,6,2,4,3,5] => [6,1,5,3,4,2] => 0
[(1,5),(2,4),(3,6)]
 => [5,4,6,2,1,3] => [1,5,2,4,3,6] => [6,2,5,3,4,1] => 0
[(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => [1,4,2,5,3,6] => [6,3,5,2,4,1] => 1
[(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => [1,3,2,5,4,6] => [6,4,5,2,3,1] => 0
[(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => [1,2,3,5,4,6] => [6,5,4,2,3,1] => 0
[(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => [1,2,3,6,4,5] => [6,5,4,1,3,2] => 0
[(1,3),(2,6),(4,5)]
 => [3,6,1,5,4,2] => [1,3,2,6,4,5] => [6,4,5,1,3,2] => 0
[(1,4),(2,6),(3,5)]
 => [4,6,5,1,3,2] => [1,4,2,6,3,5] => [6,3,5,1,4,2] => 1
[(1,5),(2,6),(3,4)]
 => [5,6,4,3,1,2] => [1,5,2,6,3,4] => [6,2,5,1,4,3] => 2
[(1,6),(2,5),(3,4)]
 => [6,5,4,3,2,1] => [1,6,2,5,3,4] => [6,1,5,2,4,3] => 0
[(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => [1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1] => 0
[(1,3),(2,4),(5,6),(7,8)]
 => [3,4,1,2,6,5,8,7] => [1,3,2,4,5,6,7,8] => [8,6,7,5,4,3,2,1] => ? = 0
[(1,4),(2,3),(5,6),(7,8)]
 => [4,3,2,1,6,5,8,7] => [1,4,2,3,5,6,7,8] => [8,5,7,6,4,3,2,1] => ? = 0
[(1,5),(2,3),(4,6),(7,8)]
 => [5,3,2,6,1,4,8,7] => [1,5,2,3,4,6,7,8] => [8,4,7,6,5,3,2,1] => ? = 0
[(1,6),(2,3),(4,5),(7,8)]
 => [6,3,2,5,4,1,8,7] => [1,6,2,3,4,5,7,8] => [8,3,7,6,5,4,2,1] => ? = 0
[(1,7),(2,3),(4,5),(6,8)]
 => [7,3,2,5,4,8,1,6] => [1,7,2,3,4,5,6,8] => [8,2,7,6,5,4,3,1] => ? = 0
[(1,8),(2,3),(4,5),(6,7)]
 => [8,3,2,5,4,7,6,1] => [1,8,2,3,4,5,6,7] => [8,1,7,6,5,4,3,2] => ? = 0
[(1,8),(2,4),(3,5),(6,7)]
 => [8,4,5,2,3,7,6,1] => [1,8,2,4,3,5,6,7] => [8,1,7,5,6,4,3,2] => 0
[(1,7),(2,4),(3,5),(6,8)]
 => [7,4,5,2,3,8,1,6] => [1,7,2,4,3,5,6,8] => [8,2,7,5,6,4,3,1] => ? = 0
[(1,6),(2,4),(3,5),(7,8)]
 => [6,4,5,2,3,1,8,7] => [1,6,2,4,3,5,7,8] => [8,3,7,5,6,4,2,1] => ? = 0
[(1,5),(2,4),(3,6),(7,8)]
 => [5,4,6,2,1,3,8,7] => [1,5,2,4,3,6,7,8] => [8,4,7,5,6,3,2,1] => ? = 0
[(1,4),(2,5),(3,6),(7,8)]
 => [4,5,6,1,2,3,8,7] => [1,4,2,5,3,6,7,8] => [8,5,7,4,6,3,2,1] => ? = 1
[(1,3),(2,5),(4,6),(7,8)]
 => [3,5,1,6,2,4,8,7] => [1,3,2,5,4,6,7,8] => [8,6,7,4,5,3,2,1] => 0
[(1,2),(3,5),(4,6),(7,8)]
 => [2,1,5,6,3,4,8,7] => [1,2,3,5,4,6,7,8] => [8,7,6,4,5,3,2,1] => ? = 0
[(1,2),(3,6),(4,5),(7,8)]
 => [2,1,6,5,4,3,8,7] => [1,2,3,6,4,5,7,8] => [8,7,6,3,5,4,2,1] => ? = 0
[(1,3),(2,6),(4,5),(7,8)]
 => [3,6,1,5,4,2,8,7] => [1,3,2,6,4,5,7,8] => [8,6,7,3,5,4,2,1] => ? = 0
[(1,4),(2,6),(3,5),(7,8)]
 => [4,6,5,1,3,2,8,7] => [1,4,2,6,3,5,7,8] => [8,5,7,3,6,4,2,1] => ? = 1
[(1,5),(2,6),(3,4),(7,8)]
 => [5,6,4,3,1,2,8,7] => [1,5,2,6,3,4,7,8] => [8,4,7,3,6,5,2,1] => ? = 2
[(1,6),(2,5),(3,4),(7,8)]
 => [6,5,4,3,2,1,8,7] => [1,6,2,5,3,4,7,8] => [8,3,7,4,6,5,2,1] => 0
[(1,7),(2,5),(3,4),(6,8)]
 => [7,5,4,3,2,8,1,6] => [1,7,2,5,3,4,6,8] => [8,2,7,4,6,5,3,1] => 0
[(1,8),(2,5),(3,4),(6,7)]
 => [8,5,4,3,2,7,6,1] => [1,8,2,5,3,4,6,7] => [8,1,7,4,6,5,3,2] => ? = 0
[(1,8),(2,6),(3,4),(5,7)]
 => [8,6,4,3,7,2,5,1] => [1,8,2,6,3,4,5,7] => [8,1,7,3,6,5,4,2] => ? = 0
[(1,7),(2,6),(3,4),(5,8)]
 => [7,6,4,3,8,2,1,5] => [1,7,2,6,3,4,5,8] => [8,2,7,3,6,5,4,1] => ? = 0
[(1,6),(2,7),(3,4),(5,8)]
 => [6,7,4,3,8,1,2,5] => [1,6,2,7,3,4,5,8] => [8,3,7,2,6,5,4,1] => ? = 3
[(1,5),(2,7),(3,4),(6,8)]
 => [5,7,4,3,1,8,2,6] => [1,5,2,7,3,4,6,8] => [8,4,7,2,6,5,3,1] => 2
[(1,4),(2,7),(3,5),(6,8)]
 => [4,7,5,1,3,8,2,6] => [1,4,2,7,3,5,6,8] => [8,5,7,2,6,4,3,1] => ? = 1
[(1,3),(2,7),(4,5),(6,8)]
 => [3,7,1,5,4,8,2,6] => [1,3,2,7,4,5,6,8] => [8,6,7,2,5,4,3,1] => ? = 0
[(1,2),(3,7),(4,5),(6,8)]
 => [2,1,7,5,4,8,3,6] => [1,2,3,7,4,5,6,8] => [8,7,6,2,5,4,3,1] => ? = 0
[(1,2),(3,8),(4,5),(6,7)]
 => [2,1,8,5,4,7,6,3] => [1,2,3,8,4,5,6,7] => [8,7,6,1,5,4,3,2] => ? = 0
[(1,3),(2,8),(4,5),(6,7)]
 => [3,8,1,5,4,7,6,2] => [1,3,2,8,4,5,6,7] => [8,6,7,1,5,4,3,2] => ? = 0
[(1,4),(2,8),(3,5),(6,7)]
 => [4,8,5,1,3,7,6,2] => [1,4,2,8,3,5,6,7] => [8,5,7,1,6,4,3,2] => ? = 1
[(1,5),(2,8),(3,4),(6,7)]
 => [5,8,4,3,1,7,6,2] => [1,5,2,8,3,4,6,7] => [8,4,7,1,6,5,3,2] => ? = 2
[(1,6),(2,8),(3,4),(5,7)]
 => [6,8,4,3,7,1,5,2] => [1,6,2,8,3,4,5,7] => [8,3,7,1,6,5,4,2] => ? = 3
[(1,7),(2,8),(3,4),(5,6)]
 => [7,8,4,3,6,5,1,2] => [1,7,2,8,3,4,5,6] => [8,2,7,1,6,5,4,3] => ? = 4
[(1,8),(2,7),(3,4),(5,6)]
 => [8,7,4,3,6,5,2,1] => [1,8,2,7,3,4,5,6] => [8,1,7,2,6,5,4,3] => ? = 0
[(1,8),(2,7),(3,5),(4,6)]
 => [8,7,5,6,3,4,2,1] => [1,8,2,7,3,5,4,6] => [8,1,7,2,6,4,5,3] => ? = 0
[(1,7),(2,8),(3,5),(4,6)]
 => [7,8,5,6,3,4,1,2] => [1,7,2,8,3,5,4,6] => [8,2,7,1,6,4,5,3] => ? = 4
[(1,6),(2,8),(3,5),(4,7)]
 => [6,8,5,7,3,1,4,2] => [1,6,2,8,3,5,4,7] => [8,3,7,1,6,4,5,2] => ? = 3
[(1,5),(2,8),(3,6),(4,7)]
 => [5,8,6,7,1,3,4,2] => [1,5,2,8,3,6,4,7] => [8,4,7,1,6,3,5,2] => ? = 3
[(1,4),(2,8),(3,6),(5,7)]
 => [4,8,6,1,7,3,5,2] => [1,4,2,8,3,6,5,7] => [8,5,7,1,6,3,4,2] => ? = 1
[(1,3),(2,8),(4,6),(5,7)]
 => [3,8,1,6,7,4,5,2] => [1,3,2,8,4,6,5,7] => [8,6,7,1,5,3,4,2] => ? = 0
[(1,2),(3,8),(4,6),(5,7)]
 => [2,1,8,6,7,4,5,3] => [1,2,3,8,4,6,5,7] => [8,7,6,1,5,3,4,2] => 0
[(1,2),(3,7),(4,6),(5,8)]
 => [2,1,7,6,8,4,3,5] => [1,2,3,7,4,6,5,8] => [8,7,6,2,5,3,4,1] => ? = 0
[(1,3),(2,7),(4,6),(5,8)]
 => [3,7,1,6,8,4,2,5] => [1,3,2,7,4,6,5,8] => [8,6,7,2,5,3,4,1] => ? = 0
[(1,4),(2,7),(3,6),(5,8)]
 => [4,7,6,1,8,3,2,5] => [1,4,2,7,3,6,5,8] => [8,5,7,2,6,3,4,1] => ? = 1
[(1,5),(2,7),(3,6),(4,8)]
 => [5,7,6,8,1,3,2,4] => [1,5,2,7,3,6,4,8] => [8,4,7,2,6,3,5,1] => ? = 3
[(1,6),(2,7),(3,5),(4,8)]
 => [6,7,5,8,3,1,2,4] => [1,6,2,7,3,5,4,8] => [8,3,7,2,6,4,5,1] => ? = 3
[(1,7),(2,6),(3,5),(4,8)]
 => [7,6,5,8,3,2,1,4] => [1,7,2,6,3,5,4,8] => [8,2,7,3,6,4,5,1] => ? = 0
[(1,8),(2,6),(3,5),(4,7)]
 => [8,6,5,7,3,2,4,1] => [1,8,2,6,3,5,4,7] => [8,1,7,3,6,4,5,2] => ? = 0
[(1,8),(2,5),(3,6),(4,7)]
 => [8,5,6,7,2,3,4,1] => [1,8,2,5,3,6,4,7] => [8,1,7,4,6,3,5,2] => ? = 1
[(1,7),(2,5),(3,6),(4,8)]
 => [7,5,6,8,2,3,1,4] => [1,7,2,5,3,6,4,8] => [8,2,7,4,6,3,5,1] => ? = 1
[(1,6),(2,5),(3,7),(4,8)]
 => [6,5,7,8,2,1,3,4] => [1,6,2,5,3,7,4,8] => [8,3,7,4,6,2,5,1] => ? = 2
[(1,5),(2,6),(3,7),(4,8)]
 => [5,6,7,8,1,2,3,4] => [1,5,2,6,3,7,4,8] => [8,4,7,3,6,2,5,1] => ? = 4
[(1,4),(2,6),(3,7),(5,8)]
 => [4,6,7,1,8,2,3,5] => [1,4,2,6,3,7,5,8] => [8,5,7,3,6,2,4,1] => ? = 2
[(1,3),(2,6),(4,7),(5,8)]
 => [3,6,1,7,8,2,4,5] => [1,3,2,6,4,7,5,8] => [8,6,7,3,5,2,4,1] => ? = 1
[(1,2),(3,6),(4,7),(5,8)]
 => [2,1,6,7,8,3,4,5] => [1,2,3,6,4,7,5,8] => [8,7,6,3,5,2,4,1] => 1
[(1,2),(3,5),(4,7),(6,8)]
 => [2,1,5,7,3,8,4,6] => [1,2,3,5,4,7,6,8] => [8,7,6,4,5,2,3,1] => 0
[(1,3),(2,5),(4,7),(6,8)]
 => [3,5,1,7,2,8,4,6] => [1,3,2,5,4,7,6,8] => [8,6,7,4,5,2,3,1] => ? = 0
[(1,4),(2,5),(3,7),(6,8)]
 => [4,5,7,1,2,8,3,6] => [1,4,2,5,3,7,6,8] => [8,5,7,4,6,2,3,1] => ? = 1
[(1,3),(2,4),(5,7),(6,8)]
 => [3,4,1,2,7,8,5,6] => [1,3,2,4,5,7,6,8] => [8,6,7,5,4,2,3,1] => 0
[(1,5),(2,3),(4,8),(6,7)]
 => [5,3,2,8,1,7,6,4] => [1,5,2,3,4,8,6,7] => [8,4,7,6,5,1,3,2] => 0
[(1,8),(2,3),(4,7),(5,6)]
 => [8,3,2,7,6,5,4,1] => [1,8,2,3,4,7,5,6] => [8,1,7,6,5,2,4,3] => 0
[(1,2),(3,6),(4,8),(5,7)]
 => [2,1,6,8,7,3,5,4] => [1,2,3,6,4,8,5,7] => [8,7,6,3,5,1,4,2] => 1
[(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => [1,2,3,4,5,6,7,8,9,10] => [10,9,8,7,6,5,4,3,2,1] => 0
Description
The number of occurrences of the pattern 213 in a permutation.
Matching statistic: St000408
Mapping
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000408: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 19%
Values
[(1,2)]
 => [2,1] => [1,2] => [2,1] => 0
[(1,2),(3,4)]
 => [2,1,4,3] => [1,2,3,4] => [2,3,4,1] => 0
[(1,3),(2,4)]
 => [3,4,1,2] => [1,3,2,4] => [3,2,4,1] => 0
[(1,4),(2,3)]
 => [4,3,2,1] => [1,4,2,3] => [3,4,2,1] => 0
[(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => 0
[(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => [1,3,2,4,5,6] => [3,2,4,5,6,1] => 0
[(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => [1,4,2,3,5,6] => [3,4,2,5,6,1] => 0
[(1,5),(2,3),(4,6)]
 => [5,3,2,6,1,4] => [1,5,2,3,4,6] => [3,4,5,2,6,1] => 0
[(1,6),(2,3),(4,5)]
 => [6,3,2,5,4,1] => [1,6,2,3,4,5] => [3,4,5,6,2,1] => 0
[(1,6),(2,4),(3,5)]
 => [6,4,5,2,3,1] => [1,6,2,4,3,5] => [3,5,4,6,2,1] => 0
[(1,5),(2,4),(3,6)]
 => [5,4,6,2,1,3] => [1,5,2,4,3,6] => [3,5,4,2,6,1] => 0
[(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => [1,4,2,5,3,6] => [3,5,2,4,6,1] => 1
[(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => [1,3,2,5,4,6] => [3,2,5,4,6,1] => 0
[(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => [1,2,3,5,4,6] => [2,3,5,4,6,1] => 0
[(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => [1,2,3,6,4,5] => [2,3,5,6,4,1] => 0
[(1,3),(2,6),(4,5)]
 => [3,6,1,5,4,2] => [1,3,2,6,4,5] => [3,2,5,6,4,1] => 0
[(1,4),(2,6),(3,5)]
 => [4,6,5,1,3,2] => [1,4,2,6,3,5] => [3,5,2,6,4,1] => 1
[(1,5),(2,6),(3,4)]
 => [5,6,4,3,1,2] => [1,5,2,6,3,4] => [3,5,6,2,4,1] => 2
[(1,6),(2,5),(3,4)]
 => [6,5,4,3,2,1] => [1,6,2,5,3,4] => [3,5,6,4,2,1] => 0
[(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => [1,2,3,4,5,6,7,8] => [2,3,4,5,6,7,8,1] => 0
[(1,3),(2,4),(5,6),(7,8)]
 => [3,4,1,2,6,5,8,7] => [1,3,2,4,5,6,7,8] => [3,2,4,5,6,7,8,1] => 0
[(1,4),(2,3),(5,6),(7,8)]
 => [4,3,2,1,6,5,8,7] => [1,4,2,3,5,6,7,8] => [3,4,2,5,6,7,8,1] => 0
[(1,5),(2,3),(4,6),(7,8)]
 => [5,3,2,6,1,4,8,7] => [1,5,2,3,4,6,7,8] => [3,4,5,2,6,7,8,1] => 0
[(1,6),(2,3),(4,5),(7,8)]
 => [6,3,2,5,4,1,8,7] => [1,6,2,3,4,5,7,8] => [3,4,5,6,2,7,8,1] => ? = 0
[(1,7),(2,3),(4,5),(6,8)]
 => [7,3,2,5,4,8,1,6] => [1,7,2,3,4,5,6,8] => [3,4,5,6,7,2,8,1] => ? = 0
[(1,8),(2,3),(4,5),(6,7)]
 => [8,3,2,5,4,7,6,1] => [1,8,2,3,4,5,6,7] => [3,4,5,6,7,8,2,1] => ? = 0
[(1,8),(2,4),(3,5),(6,7)]
 => [8,4,5,2,3,7,6,1] => [1,8,2,4,3,5,6,7] => [3,5,4,6,7,8,2,1] => ? = 0
[(1,7),(2,4),(3,5),(6,8)]
 => [7,4,5,2,3,8,1,6] => [1,7,2,4,3,5,6,8] => [3,5,4,6,7,2,8,1] => ? = 0
[(1,6),(2,4),(3,5),(7,8)]
 => [6,4,5,2,3,1,8,7] => [1,6,2,4,3,5,7,8] => [3,5,4,6,2,7,8,1] => ? = 0
[(1,5),(2,4),(3,6),(7,8)]
 => [5,4,6,2,1,3,8,7] => [1,5,2,4,3,6,7,8] => [3,5,4,2,6,7,8,1] => ? = 0
[(1,4),(2,5),(3,6),(7,8)]
 => [4,5,6,1,2,3,8,7] => [1,4,2,5,3,6,7,8] => [3,5,2,4,6,7,8,1] => ? = 1
[(1,3),(2,5),(4,6),(7,8)]
 => [3,5,1,6,2,4,8,7] => [1,3,2,5,4,6,7,8] => [3,2,5,4,6,7,8,1] => ? = 0
[(1,2),(3,5),(4,6),(7,8)]
 => [2,1,5,6,3,4,8,7] => [1,2,3,5,4,6,7,8] => [2,3,5,4,6,7,8,1] => ? = 0
[(1,2),(3,6),(4,5),(7,8)]
 => [2,1,6,5,4,3,8,7] => [1,2,3,6,4,5,7,8] => [2,3,5,6,4,7,8,1] => ? = 0
[(1,3),(2,6),(4,5),(7,8)]
 => [3,6,1,5,4,2,8,7] => [1,3,2,6,4,5,7,8] => [3,2,5,6,4,7,8,1] => ? = 0
[(1,4),(2,6),(3,5),(7,8)]
 => [4,6,5,1,3,2,8,7] => [1,4,2,6,3,5,7,8] => [3,5,2,6,4,7,8,1] => ? = 1
[(1,5),(2,6),(3,4),(7,8)]
 => [5,6,4,3,1,2,8,7] => [1,5,2,6,3,4,7,8] => [3,5,6,2,4,7,8,1] => ? = 2
[(1,6),(2,5),(3,4),(7,8)]
 => [6,5,4,3,2,1,8,7] => [1,6,2,5,3,4,7,8] => [3,5,6,4,2,7,8,1] => ? = 0
[(1,7),(2,5),(3,4),(6,8)]
 => [7,5,4,3,2,8,1,6] => [1,7,2,5,3,4,6,8] => [3,5,6,4,7,2,8,1] => ? = 0
[(1,8),(2,5),(3,4),(6,7)]
 => [8,5,4,3,2,7,6,1] => [1,8,2,5,3,4,6,7] => [3,5,6,4,7,8,2,1] => ? = 0
[(1,8),(2,6),(3,4),(5,7)]
 => [8,6,4,3,7,2,5,1] => [1,8,2,6,3,4,5,7] => [3,5,6,7,4,8,2,1] => ? = 0
[(1,7),(2,6),(3,4),(5,8)]
 => [7,6,4,3,8,2,1,5] => [1,7,2,6,3,4,5,8] => [3,5,6,7,4,2,8,1] => ? = 0
[(1,6),(2,7),(3,4),(5,8)]
 => [6,7,4,3,8,1,2,5] => [1,6,2,7,3,4,5,8] => [3,5,6,7,2,4,8,1] => ? = 3
[(1,5),(2,7),(3,4),(6,8)]
 => [5,7,4,3,1,8,2,6] => [1,5,2,7,3,4,6,8] => [3,5,6,2,7,4,8,1] => ? = 2
[(1,4),(2,7),(3,5),(6,8)]
 => [4,7,5,1,3,8,2,6] => [1,4,2,7,3,5,6,8] => [3,5,2,6,7,4,8,1] => ? = 1
[(1,3),(2,7),(4,5),(6,8)]
 => [3,7,1,5,4,8,2,6] => [1,3,2,7,4,5,6,8] => [3,2,5,6,7,4,8,1] => ? = 0
[(1,2),(3,7),(4,5),(6,8)]
 => [2,1,7,5,4,8,3,6] => [1,2,3,7,4,5,6,8] => [2,3,5,6,7,4,8,1] => ? = 0
[(1,2),(3,8),(4,5),(6,7)]
 => [2,1,8,5,4,7,6,3] => [1,2,3,8,4,5,6,7] => [2,3,5,6,7,8,4,1] => ? = 0
[(1,3),(2,8),(4,5),(6,7)]
 => [3,8,1,5,4,7,6,2] => [1,3,2,8,4,5,6,7] => [3,2,5,6,7,8,4,1] => ? = 0
[(1,4),(2,8),(3,5),(6,7)]
 => [4,8,5,1,3,7,6,2] => [1,4,2,8,3,5,6,7] => [3,5,2,6,7,8,4,1] => ? = 1
[(1,5),(2,8),(3,4),(6,7)]
 => [5,8,4,3,1,7,6,2] => [1,5,2,8,3,4,6,7] => [3,5,6,2,7,8,4,1] => ? = 2
[(1,6),(2,8),(3,4),(5,7)]
 => [6,8,4,3,7,1,5,2] => [1,6,2,8,3,4,5,7] => [3,5,6,7,2,8,4,1] => ? = 3
[(1,7),(2,8),(3,4),(5,6)]
 => [7,8,4,3,6,5,1,2] => [1,7,2,8,3,4,5,6] => [3,5,6,7,8,2,4,1] => ? = 4
[(1,8),(2,7),(3,4),(5,6)]
 => [8,7,4,3,6,5,2,1] => [1,8,2,7,3,4,5,6] => [3,5,6,7,8,4,2,1] => ? = 0
[(1,8),(2,7),(3,5),(4,6)]
 => [8,7,5,6,3,4,2,1] => [1,8,2,7,3,5,4,6] => [3,5,7,6,8,4,2,1] => ? = 0
[(1,7),(2,8),(3,5),(4,6)]
 => [7,8,5,6,3,4,1,2] => [1,7,2,8,3,5,4,6] => [3,5,7,6,8,2,4,1] => ? = 4
[(1,6),(2,8),(3,5),(4,7)]
 => [6,8,5,7,3,1,4,2] => [1,6,2,8,3,5,4,7] => [3,5,7,6,2,8,4,1] => ? = 3
[(1,5),(2,8),(3,6),(4,7)]
 => [5,8,6,7,1,3,4,2] => [1,5,2,8,3,6,4,7] => [3,5,7,2,6,8,4,1] => ? = 3
[(1,4),(2,8),(3,6),(5,7)]
 => [4,8,6,1,7,3,5,2] => [1,4,2,8,3,6,5,7] => [3,5,2,7,6,8,4,1] => ? = 1
[(1,3),(2,8),(4,6),(5,7)]
 => [3,8,1,6,7,4,5,2] => [1,3,2,8,4,6,5,7] => [3,2,5,7,6,8,4,1] => ? = 0
[(1,2),(3,8),(4,6),(5,7)]
 => [2,1,8,6,7,4,5,3] => [1,2,3,8,4,6,5,7] => [2,3,5,7,6,8,4,1] => ? = 0
[(1,2),(3,7),(4,6),(5,8)]
 => [2,1,7,6,8,4,3,5] => [1,2,3,7,4,6,5,8] => [2,3,5,7,6,4,8,1] => ? = 0
[(1,3),(2,7),(4,6),(5,8)]
 => [3,7,1,6,8,4,2,5] => [1,3,2,7,4,6,5,8] => [3,2,5,7,6,4,8,1] => ? = 0
[(1,4),(2,7),(3,6),(5,8)]
 => [4,7,6,1,8,3,2,5] => [1,4,2,7,3,6,5,8] => [3,5,2,7,6,4,8,1] => ? = 1
[(1,5),(2,7),(3,6),(4,8)]
 => [5,7,6,8,1,3,2,4] => [1,5,2,7,3,6,4,8] => [3,5,7,2,6,4,8,1] => ? = 3
[(1,6),(2,7),(3,5),(4,8)]
 => [6,7,5,8,3,1,2,4] => [1,6,2,7,3,5,4,8] => [3,5,7,6,2,4,8,1] => ? = 3
[(1,7),(2,6),(3,5),(4,8)]
 => [7,6,5,8,3,2,1,4] => [1,7,2,6,3,5,4,8] => [3,5,7,6,4,2,8,1] => ? = 0
[(1,8),(2,6),(3,5),(4,7)]
 => [8,6,5,7,3,2,4,1] => [1,8,2,6,3,5,4,7] => [3,5,7,6,4,8,2,1] => ? = 0
[(1,8),(2,5),(3,6),(4,7)]
 => [8,5,6,7,2,3,4,1] => [1,8,2,5,3,6,4,7] => [3,5,7,4,6,8,2,1] => ? = 1
[(1,7),(2,5),(3,6),(4,8)]
 => [7,5,6,8,2,3,1,4] => [1,7,2,5,3,6,4,8] => [3,5,7,4,6,2,8,1] => ? = 1
[(1,6),(2,5),(3,7),(4,8)]
 => [6,5,7,8,2,1,3,4] => [1,6,2,5,3,7,4,8] => [3,5,7,4,2,6,8,1] => ? = 2
[(1,5),(2,6),(3,7),(4,8)]
 => [5,6,7,8,1,2,3,4] => [1,5,2,6,3,7,4,8] => [3,5,7,2,4,6,8,1] => ? = 4
[(1,4),(2,6),(3,7),(5,8)]
 => [4,6,7,1,8,2,3,5] => [1,4,2,6,3,7,5,8] => [3,5,2,7,4,6,8,1] => ? = 2
[(1,3),(2,5),(4,7),(6,8)]
 => [3,5,1,7,2,8,4,6] => [1,3,2,5,4,7,6,8] => [3,2,5,4,7,6,8,1] => 0
[(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => [1,2,3,4,5,6,7,8,9,10] => [2,3,4,5,6,7,8,9,10,1] => 0
[(1,3),(2,4),(5,6),(7,8),(9,10)]
 => [3,4,1,2,6,5,8,7,10,9] => [1,3,2,4,5,6,7,8,9,10] => [3,2,4,5,6,7,8,9,10,1] => 0
Description
The number of occurrences of the pattern 4231 in a permutation. It is a necessary condition that a permutation π avoids this pattern for the Schubert variety associated to π to be smooth [2].
Matching statistic: St000440
Mapping
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000440: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 19%
Values
[(1,2)]
 => [2,1] => [1,2] => [2,1] => 0
[(1,2),(3,4)]
 => [2,1,4,3] => [1,2,3,4] => [2,3,4,1] => 0
[(1,3),(2,4)]
 => [3,4,1,2] => [1,3,2,4] => [3,2,4,1] => 0
[(1,4),(2,3)]
 => [4,3,2,1] => [1,4,2,3] => [3,4,2,1] => 0
[(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => 0
[(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => [1,3,2,4,5,6] => [3,2,4,5,6,1] => 0
[(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => [1,4,2,3,5,6] => [3,4,2,5,6,1] => 0
[(1,5),(2,3),(4,6)]
 => [5,3,2,6,1,4] => [1,5,2,3,4,6] => [3,4,5,2,6,1] => 0
[(1,6),(2,3),(4,5)]
 => [6,3,2,5,4,1] => [1,6,2,3,4,5] => [3,4,5,6,2,1] => 0
[(1,6),(2,4),(3,5)]
 => [6,4,5,2,3,1] => [1,6,2,4,3,5] => [3,5,4,6,2,1] => 0
[(1,5),(2,4),(3,6)]
 => [5,4,6,2,1,3] => [1,5,2,4,3,6] => [3,5,4,2,6,1] => 0
[(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => [1,4,2,5,3,6] => [3,5,2,4,6,1] => 1
[(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => [1,3,2,5,4,6] => [3,2,5,4,6,1] => 0
[(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => [1,2,3,5,4,6] => [2,3,5,4,6,1] => 0
[(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => [1,2,3,6,4,5] => [2,3,5,6,4,1] => 0
[(1,3),(2,6),(4,5)]
 => [3,6,1,5,4,2] => [1,3,2,6,4,5] => [3,2,5,6,4,1] => 0
[(1,4),(2,6),(3,5)]
 => [4,6,5,1,3,2] => [1,4,2,6,3,5] => [3,5,2,6,4,1] => 1
[(1,5),(2,6),(3,4)]
 => [5,6,4,3,1,2] => [1,5,2,6,3,4] => [3,5,6,2,4,1] => 2
[(1,6),(2,5),(3,4)]
 => [6,5,4,3,2,1] => [1,6,2,5,3,4] => [3,5,6,4,2,1] => 0
[(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => [1,2,3,4,5,6,7,8] => [2,3,4,5,6,7,8,1] => 0
[(1,3),(2,4),(5,6),(7,8)]
 => [3,4,1,2,6,5,8,7] => [1,3,2,4,5,6,7,8] => [3,2,4,5,6,7,8,1] => 0
[(1,4),(2,3),(5,6),(7,8)]
 => [4,3,2,1,6,5,8,7] => [1,4,2,3,5,6,7,8] => [3,4,2,5,6,7,8,1] => 0
[(1,5),(2,3),(4,6),(7,8)]
 => [5,3,2,6,1,4,8,7] => [1,5,2,3,4,6,7,8] => [3,4,5,2,6,7,8,1] => 0
[(1,6),(2,3),(4,5),(7,8)]
 => [6,3,2,5,4,1,8,7] => [1,6,2,3,4,5,7,8] => [3,4,5,6,2,7,8,1] => ? = 0
[(1,7),(2,3),(4,5),(6,8)]
 => [7,3,2,5,4,8,1,6] => [1,7,2,3,4,5,6,8] => [3,4,5,6,7,2,8,1] => ? = 0
[(1,8),(2,3),(4,5),(6,7)]
 => [8,3,2,5,4,7,6,1] => [1,8,2,3,4,5,6,7] => [3,4,5,6,7,8,2,1] => ? = 0
[(1,8),(2,4),(3,5),(6,7)]
 => [8,4,5,2,3,7,6,1] => [1,8,2,4,3,5,6,7] => [3,5,4,6,7,8,2,1] => ? = 0
[(1,7),(2,4),(3,5),(6,8)]
 => [7,4,5,2,3,8,1,6] => [1,7,2,4,3,5,6,8] => [3,5,4,6,7,2,8,1] => ? = 0
[(1,6),(2,4),(3,5),(7,8)]
 => [6,4,5,2,3,1,8,7] => [1,6,2,4,3,5,7,8] => [3,5,4,6,2,7,8,1] => ? = 0
[(1,5),(2,4),(3,6),(7,8)]
 => [5,4,6,2,1,3,8,7] => [1,5,2,4,3,6,7,8] => [3,5,4,2,6,7,8,1] => ? = 0
[(1,4),(2,5),(3,6),(7,8)]
 => [4,5,6,1,2,3,8,7] => [1,4,2,5,3,6,7,8] => [3,5,2,4,6,7,8,1] => ? = 1
[(1,3),(2,5),(4,6),(7,8)]
 => [3,5,1,6,2,4,8,7] => [1,3,2,5,4,6,7,8] => [3,2,5,4,6,7,8,1] => ? = 0
[(1,2),(3,5),(4,6),(7,8)]
 => [2,1,5,6,3,4,8,7] => [1,2,3,5,4,6,7,8] => [2,3,5,4,6,7,8,1] => ? = 0
[(1,2),(3,6),(4,5),(7,8)]
 => [2,1,6,5,4,3,8,7] => [1,2,3,6,4,5,7,8] => [2,3,5,6,4,7,8,1] => ? = 0
[(1,3),(2,6),(4,5),(7,8)]
 => [3,6,1,5,4,2,8,7] => [1,3,2,6,4,5,7,8] => [3,2,5,6,4,7,8,1] => ? = 0
[(1,4),(2,6),(3,5),(7,8)]
 => [4,6,5,1,3,2,8,7] => [1,4,2,6,3,5,7,8] => [3,5,2,6,4,7,8,1] => ? = 1
[(1,5),(2,6),(3,4),(7,8)]
 => [5,6,4,3,1,2,8,7] => [1,5,2,6,3,4,7,8] => [3,5,6,2,4,7,8,1] => ? = 2
[(1,6),(2,5),(3,4),(7,8)]
 => [6,5,4,3,2,1,8,7] => [1,6,2,5,3,4,7,8] => [3,5,6,4,2,7,8,1] => ? = 0
[(1,7),(2,5),(3,4),(6,8)]
 => [7,5,4,3,2,8,1,6] => [1,7,2,5,3,4,6,8] => [3,5,6,4,7,2,8,1] => ? = 0
[(1,8),(2,5),(3,4),(6,7)]
 => [8,5,4,3,2,7,6,1] => [1,8,2,5,3,4,6,7] => [3,5,6,4,7,8,2,1] => ? = 0
[(1,8),(2,6),(3,4),(5,7)]
 => [8,6,4,3,7,2,5,1] => [1,8,2,6,3,4,5,7] => [3,5,6,7,4,8,2,1] => ? = 0
[(1,7),(2,6),(3,4),(5,8)]
 => [7,6,4,3,8,2,1,5] => [1,7,2,6,3,4,5,8] => [3,5,6,7,4,2,8,1] => ? = 0
[(1,6),(2,7),(3,4),(5,8)]
 => [6,7,4,3,8,1,2,5] => [1,6,2,7,3,4,5,8] => [3,5,6,7,2,4,8,1] => ? = 3
[(1,5),(2,7),(3,4),(6,8)]
 => [5,7,4,3,1,8,2,6] => [1,5,2,7,3,4,6,8] => [3,5,6,2,7,4,8,1] => ? = 2
[(1,4),(2,7),(3,5),(6,8)]
 => [4,7,5,1,3,8,2,6] => [1,4,2,7,3,5,6,8] => [3,5,2,6,7,4,8,1] => ? = 1
[(1,3),(2,7),(4,5),(6,8)]
 => [3,7,1,5,4,8,2,6] => [1,3,2,7,4,5,6,8] => [3,2,5,6,7,4,8,1] => ? = 0
[(1,2),(3,7),(4,5),(6,8)]
 => [2,1,7,5,4,8,3,6] => [1,2,3,7,4,5,6,8] => [2,3,5,6,7,4,8,1] => ? = 0
[(1,2),(3,8),(4,5),(6,7)]
 => [2,1,8,5,4,7,6,3] => [1,2,3,8,4,5,6,7] => [2,3,5,6,7,8,4,1] => ? = 0
[(1,3),(2,8),(4,5),(6,7)]
 => [3,8,1,5,4,7,6,2] => [1,3,2,8,4,5,6,7] => [3,2,5,6,7,8,4,1] => ? = 0
[(1,4),(2,8),(3,5),(6,7)]
 => [4,8,5,1,3,7,6,2] => [1,4,2,8,3,5,6,7] => [3,5,2,6,7,8,4,1] => ? = 1
[(1,5),(2,8),(3,4),(6,7)]
 => [5,8,4,3,1,7,6,2] => [1,5,2,8,3,4,6,7] => [3,5,6,2,7,8,4,1] => ? = 2
[(1,6),(2,8),(3,4),(5,7)]
 => [6,8,4,3,7,1,5,2] => [1,6,2,8,3,4,5,7] => [3,5,6,7,2,8,4,1] => ? = 3
[(1,7),(2,8),(3,4),(5,6)]
 => [7,8,4,3,6,5,1,2] => [1,7,2,8,3,4,5,6] => [3,5,6,7,8,2,4,1] => ? = 4
[(1,8),(2,7),(3,4),(5,6)]
 => [8,7,4,3,6,5,2,1] => [1,8,2,7,3,4,5,6] => [3,5,6,7,8,4,2,1] => ? = 0
[(1,8),(2,7),(3,5),(4,6)]
 => [8,7,5,6,3,4,2,1] => [1,8,2,7,3,5,4,6] => [3,5,7,6,8,4,2,1] => ? = 0
[(1,7),(2,8),(3,5),(4,6)]
 => [7,8,5,6,3,4,1,2] => [1,7,2,8,3,5,4,6] => [3,5,7,6,8,2,4,1] => ? = 4
[(1,6),(2,8),(3,5),(4,7)]
 => [6,8,5,7,3,1,4,2] => [1,6,2,8,3,5,4,7] => [3,5,7,6,2,8,4,1] => ? = 3
[(1,5),(2,8),(3,6),(4,7)]
 => [5,8,6,7,1,3,4,2] => [1,5,2,8,3,6,4,7] => [3,5,7,2,6,8,4,1] => ? = 3
[(1,4),(2,8),(3,6),(5,7)]
 => [4,8,6,1,7,3,5,2] => [1,4,2,8,3,6,5,7] => [3,5,2,7,6,8,4,1] => ? = 1
[(1,3),(2,8),(4,6),(5,7)]
 => [3,8,1,6,7,4,5,2] => [1,3,2,8,4,6,5,7] => [3,2,5,7,6,8,4,1] => ? = 0
[(1,2),(3,8),(4,6),(5,7)]
 => [2,1,8,6,7,4,5,3] => [1,2,3,8,4,6,5,7] => [2,3,5,7,6,8,4,1] => ? = 0
[(1,2),(3,7),(4,6),(5,8)]
 => [2,1,7,6,8,4,3,5] => [1,2,3,7,4,6,5,8] => [2,3,5,7,6,4,8,1] => ? = 0
[(1,3),(2,7),(4,6),(5,8)]
 => [3,7,1,6,8,4,2,5] => [1,3,2,7,4,6,5,8] => [3,2,5,7,6,4,8,1] => ? = 0
[(1,4),(2,7),(3,6),(5,8)]
 => [4,7,6,1,8,3,2,5] => [1,4,2,7,3,6,5,8] => [3,5,2,7,6,4,8,1] => ? = 1
[(1,5),(2,7),(3,6),(4,8)]
 => [5,7,6,8,1,3,2,4] => [1,5,2,7,3,6,4,8] => [3,5,7,2,6,4,8,1] => ? = 3
[(1,6),(2,7),(3,5),(4,8)]
 => [6,7,5,8,3,1,2,4] => [1,6,2,7,3,5,4,8] => [3,5,7,6,2,4,8,1] => ? = 3
[(1,7),(2,6),(3,5),(4,8)]
 => [7,6,5,8,3,2,1,4] => [1,7,2,6,3,5,4,8] => [3,5,7,6,4,2,8,1] => ? = 0
[(1,8),(2,6),(3,5),(4,7)]
 => [8,6,5,7,3,2,4,1] => [1,8,2,6,3,5,4,7] => [3,5,7,6,4,8,2,1] => ? = 0
[(1,8),(2,5),(3,6),(4,7)]
 => [8,5,6,7,2,3,4,1] => [1,8,2,5,3,6,4,7] => [3,5,7,4,6,8,2,1] => ? = 1
[(1,7),(2,5),(3,6),(4,8)]
 => [7,5,6,8,2,3,1,4] => [1,7,2,5,3,6,4,8] => [3,5,7,4,6,2,8,1] => ? = 1
[(1,6),(2,5),(3,7),(4,8)]
 => [6,5,7,8,2,1,3,4] => [1,6,2,5,3,7,4,8] => [3,5,7,4,2,6,8,1] => ? = 2
[(1,5),(2,6),(3,7),(4,8)]
 => [5,6,7,8,1,2,3,4] => [1,5,2,6,3,7,4,8] => [3,5,7,2,4,6,8,1] => ? = 4
[(1,4),(2,6),(3,7),(5,8)]
 => [4,6,7,1,8,2,3,5] => [1,4,2,6,3,7,5,8] => [3,5,2,7,4,6,8,1] => ? = 2
[(1,3),(2,5),(4,7),(6,8)]
 => [3,5,1,7,2,8,4,6] => [1,3,2,5,4,7,6,8] => [3,2,5,4,7,6,8,1] => 0
[(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => [1,2,3,4,5,6,7,8,9,10] => [2,3,4,5,6,7,8,9,10,1] => 0
[(1,3),(2,4),(5,6),(7,8),(9,10)]
 => [3,4,1,2,6,5,8,7,10,9] => [1,3,2,4,5,6,7,8,9,10] => [3,2,4,5,6,7,8,9,10,1] => 0
Description
The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. There is a bijection between permutations avoiding these two pattern and Schröder paths [1,2].
Matching statistic: St000036
Mapping
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000036: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 19%
Values
[(1,2)]
 => [2,1] => [1,2] => [2,1] => 1 = 0 + 1
[(1,2),(3,4)]
 => [2,1,4,3] => [1,2,3,4] => [2,3,4,1] => 1 = 0 + 1
[(1,3),(2,4)]
 => [3,4,1,2] => [1,3,2,4] => [3,2,4,1] => 1 = 0 + 1
[(1,4),(2,3)]
 => [4,3,2,1] => [1,4,2,3] => [3,4,2,1] => 1 = 0 + 1
[(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => 1 = 0 + 1
[(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => [1,3,2,4,5,6] => [3,2,4,5,6,1] => 1 = 0 + 1
[(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => [1,4,2,3,5,6] => [3,4,2,5,6,1] => 1 = 0 + 1
[(1,5),(2,3),(4,6)]
 => [5,3,2,6,1,4] => [1,5,2,3,4,6] => [3,4,5,2,6,1] => 1 = 0 + 1
[(1,6),(2,3),(4,5)]
 => [6,3,2,5,4,1] => [1,6,2,3,4,5] => [3,4,5,6,2,1] => 1 = 0 + 1
[(1,6),(2,4),(3,5)]
 => [6,4,5,2,3,1] => [1,6,2,4,3,5] => [3,5,4,6,2,1] => 1 = 0 + 1
[(1,5),(2,4),(3,6)]
 => [5,4,6,2,1,3] => [1,5,2,4,3,6] => [3,5,4,2,6,1] => 1 = 0 + 1
[(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => [1,4,2,5,3,6] => [3,5,2,4,6,1] => 2 = 1 + 1
[(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => [1,3,2,5,4,6] => [3,2,5,4,6,1] => 1 = 0 + 1
[(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => [1,2,3,5,4,6] => [2,3,5,4,6,1] => 1 = 0 + 1
[(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => [1,2,3,6,4,5] => [2,3,5,6,4,1] => 1 = 0 + 1
[(1,3),(2,6),(4,5)]
 => [3,6,1,5,4,2] => [1,3,2,6,4,5] => [3,2,5,6,4,1] => 1 = 0 + 1
[(1,4),(2,6),(3,5)]
 => [4,6,5,1,3,2] => [1,4,2,6,3,5] => [3,5,2,6,4,1] => 2 = 1 + 1
[(1,5),(2,6),(3,4)]
 => [5,6,4,3,1,2] => [1,5,2,6,3,4] => [3,5,6,2,4,1] => 3 = 2 + 1
[(1,6),(2,5),(3,4)]
 => [6,5,4,3,2,1] => [1,6,2,5,3,4] => [3,5,6,4,2,1] => 1 = 0 + 1
[(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => [1,2,3,4,5,6,7,8] => [2,3,4,5,6,7,8,1] => 1 = 0 + 1
[(1,3),(2,4),(5,6),(7,8)]
 => [3,4,1,2,6,5,8,7] => [1,3,2,4,5,6,7,8] => [3,2,4,5,6,7,8,1] => 1 = 0 + 1
[(1,4),(2,3),(5,6),(7,8)]
 => [4,3,2,1,6,5,8,7] => [1,4,2,3,5,6,7,8] => [3,4,2,5,6,7,8,1] => 1 = 0 + 1
[(1,5),(2,3),(4,6),(7,8)]
 => [5,3,2,6,1,4,8,7] => [1,5,2,3,4,6,7,8] => [3,4,5,2,6,7,8,1] => 1 = 0 + 1
[(1,6),(2,3),(4,5),(7,8)]
 => [6,3,2,5,4,1,8,7] => [1,6,2,3,4,5,7,8] => [3,4,5,6,2,7,8,1] => ? = 0 + 1
[(1,7),(2,3),(4,5),(6,8)]
 => [7,3,2,5,4,8,1,6] => [1,7,2,3,4,5,6,8] => [3,4,5,6,7,2,8,1] => ? = 0 + 1
[(1,8),(2,3),(4,5),(6,7)]
 => [8,3,2,5,4,7,6,1] => [1,8,2,3,4,5,6,7] => [3,4,5,6,7,8,2,1] => ? = 0 + 1
[(1,8),(2,4),(3,5),(6,7)]
 => [8,4,5,2,3,7,6,1] => [1,8,2,4,3,5,6,7] => [3,5,4,6,7,8,2,1] => ? = 0 + 1
[(1,7),(2,4),(3,5),(6,8)]
 => [7,4,5,2,3,8,1,6] => [1,7,2,4,3,5,6,8] => [3,5,4,6,7,2,8,1] => ? = 0 + 1
[(1,6),(2,4),(3,5),(7,8)]
 => [6,4,5,2,3,1,8,7] => [1,6,2,4,3,5,7,8] => [3,5,4,6,2,7,8,1] => ? = 0 + 1
[(1,5),(2,4),(3,6),(7,8)]
 => [5,4,6,2,1,3,8,7] => [1,5,2,4,3,6,7,8] => [3,5,4,2,6,7,8,1] => ? = 0 + 1
[(1,4),(2,5),(3,6),(7,8)]
 => [4,5,6,1,2,3,8,7] => [1,4,2,5,3,6,7,8] => [3,5,2,4,6,7,8,1] => ? = 1 + 1
[(1,3),(2,5),(4,6),(7,8)]
 => [3,5,1,6,2,4,8,7] => [1,3,2,5,4,6,7,8] => [3,2,5,4,6,7,8,1] => ? = 0 + 1
[(1,2),(3,5),(4,6),(7,8)]
 => [2,1,5,6,3,4,8,7] => [1,2,3,5,4,6,7,8] => [2,3,5,4,6,7,8,1] => ? = 0 + 1
[(1,2),(3,6),(4,5),(7,8)]
 => [2,1,6,5,4,3,8,7] => [1,2,3,6,4,5,7,8] => [2,3,5,6,4,7,8,1] => ? = 0 + 1
[(1,3),(2,6),(4,5),(7,8)]
 => [3,6,1,5,4,2,8,7] => [1,3,2,6,4,5,7,8] => [3,2,5,6,4,7,8,1] => ? = 0 + 1
[(1,4),(2,6),(3,5),(7,8)]
 => [4,6,5,1,3,2,8,7] => [1,4,2,6,3,5,7,8] => [3,5,2,6,4,7,8,1] => ? = 1 + 1
[(1,5),(2,6),(3,4),(7,8)]
 => [5,6,4,3,1,2,8,7] => [1,5,2,6,3,4,7,8] => [3,5,6,2,4,7,8,1] => ? = 2 + 1
[(1,6),(2,5),(3,4),(7,8)]
 => [6,5,4,3,2,1,8,7] => [1,6,2,5,3,4,7,8] => [3,5,6,4,2,7,8,1] => ? = 0 + 1
[(1,7),(2,5),(3,4),(6,8)]
 => [7,5,4,3,2,8,1,6] => [1,7,2,5,3,4,6,8] => [3,5,6,4,7,2,8,1] => ? = 0 + 1
[(1,8),(2,5),(3,4),(6,7)]
 => [8,5,4,3,2,7,6,1] => [1,8,2,5,3,4,6,7] => [3,5,6,4,7,8,2,1] => ? = 0 + 1
[(1,8),(2,6),(3,4),(5,7)]
 => [8,6,4,3,7,2,5,1] => [1,8,2,6,3,4,5,7] => [3,5,6,7,4,8,2,1] => ? = 0 + 1
[(1,7),(2,6),(3,4),(5,8)]
 => [7,6,4,3,8,2,1,5] => [1,7,2,6,3,4,5,8] => [3,5,6,7,4,2,8,1] => ? = 0 + 1
[(1,6),(2,7),(3,4),(5,8)]
 => [6,7,4,3,8,1,2,5] => [1,6,2,7,3,4,5,8] => [3,5,6,7,2,4,8,1] => ? = 3 + 1
[(1,5),(2,7),(3,4),(6,8)]
 => [5,7,4,3,1,8,2,6] => [1,5,2,7,3,4,6,8] => [3,5,6,2,7,4,8,1] => ? = 2 + 1
[(1,4),(2,7),(3,5),(6,8)]
 => [4,7,5,1,3,8,2,6] => [1,4,2,7,3,5,6,8] => [3,5,2,6,7,4,8,1] => ? = 1 + 1
[(1,3),(2,7),(4,5),(6,8)]
 => [3,7,1,5,4,8,2,6] => [1,3,2,7,4,5,6,8] => [3,2,5,6,7,4,8,1] => ? = 0 + 1
[(1,2),(3,7),(4,5),(6,8)]
 => [2,1,7,5,4,8,3,6] => [1,2,3,7,4,5,6,8] => [2,3,5,6,7,4,8,1] => ? = 0 + 1
[(1,2),(3,8),(4,5),(6,7)]
 => [2,1,8,5,4,7,6,3] => [1,2,3,8,4,5,6,7] => [2,3,5,6,7,8,4,1] => ? = 0 + 1
[(1,3),(2,8),(4,5),(6,7)]
 => [3,8,1,5,4,7,6,2] => [1,3,2,8,4,5,6,7] => [3,2,5,6,7,8,4,1] => ? = 0 + 1
[(1,4),(2,8),(3,5),(6,7)]
 => [4,8,5,1,3,7,6,2] => [1,4,2,8,3,5,6,7] => [3,5,2,6,7,8,4,1] => ? = 1 + 1
[(1,5),(2,8),(3,4),(6,7)]
 => [5,8,4,3,1,7,6,2] => [1,5,2,8,3,4,6,7] => [3,5,6,2,7,8,4,1] => ? = 2 + 1
[(1,6),(2,8),(3,4),(5,7)]
 => [6,8,4,3,7,1,5,2] => [1,6,2,8,3,4,5,7] => [3,5,6,7,2,8,4,1] => ? = 3 + 1
[(1,7),(2,8),(3,4),(5,6)]
 => [7,8,4,3,6,5,1,2] => [1,7,2,8,3,4,5,6] => [3,5,6,7,8,2,4,1] => ? = 4 + 1
[(1,8),(2,7),(3,4),(5,6)]
 => [8,7,4,3,6,5,2,1] => [1,8,2,7,3,4,5,6] => [3,5,6,7,8,4,2,1] => ? = 0 + 1
[(1,8),(2,7),(3,5),(4,6)]
 => [8,7,5,6,3,4,2,1] => [1,8,2,7,3,5,4,6] => [3,5,7,6,8,4,2,1] => ? = 0 + 1
[(1,7),(2,8),(3,5),(4,6)]
 => [7,8,5,6,3,4,1,2] => [1,7,2,8,3,5,4,6] => [3,5,7,6,8,2,4,1] => ? = 4 + 1
[(1,6),(2,8),(3,5),(4,7)]
 => [6,8,5,7,3,1,4,2] => [1,6,2,8,3,5,4,7] => [3,5,7,6,2,8,4,1] => ? = 3 + 1
[(1,5),(2,8),(3,6),(4,7)]
 => [5,8,6,7,1,3,4,2] => [1,5,2,8,3,6,4,7] => [3,5,7,2,6,8,4,1] => ? = 3 + 1
[(1,4),(2,8),(3,6),(5,7)]
 => [4,8,6,1,7,3,5,2] => [1,4,2,8,3,6,5,7] => [3,5,2,7,6,8,4,1] => ? = 1 + 1
[(1,3),(2,8),(4,6),(5,7)]
 => [3,8,1,6,7,4,5,2] => [1,3,2,8,4,6,5,7] => [3,2,5,7,6,8,4,1] => ? = 0 + 1
[(1,2),(3,8),(4,6),(5,7)]
 => [2,1,8,6,7,4,5,3] => [1,2,3,8,4,6,5,7] => [2,3,5,7,6,8,4,1] => ? = 0 + 1
[(1,2),(3,7),(4,6),(5,8)]
 => [2,1,7,6,8,4,3,5] => [1,2,3,7,4,6,5,8] => [2,3,5,7,6,4,8,1] => ? = 0 + 1
[(1,3),(2,7),(4,6),(5,8)]
 => [3,7,1,6,8,4,2,5] => [1,3,2,7,4,6,5,8] => [3,2,5,7,6,4,8,1] => ? = 0 + 1
[(1,4),(2,7),(3,6),(5,8)]
 => [4,7,6,1,8,3,2,5] => [1,4,2,7,3,6,5,8] => [3,5,2,7,6,4,8,1] => ? = 1 + 1
[(1,5),(2,7),(3,6),(4,8)]
 => [5,7,6,8,1,3,2,4] => [1,5,2,7,3,6,4,8] => [3,5,7,2,6,4,8,1] => ? = 3 + 1
[(1,6),(2,7),(3,5),(4,8)]
 => [6,7,5,8,3,1,2,4] => [1,6,2,7,3,5,4,8] => [3,5,7,6,2,4,8,1] => ? = 3 + 1
[(1,7),(2,6),(3,5),(4,8)]
 => [7,6,5,8,3,2,1,4] => [1,7,2,6,3,5,4,8] => [3,5,7,6,4,2,8,1] => ? = 0 + 1
[(1,8),(2,6),(3,5),(4,7)]
 => [8,6,5,7,3,2,4,1] => [1,8,2,6,3,5,4,7] => [3,5,7,6,4,8,2,1] => ? = 0 + 1
[(1,8),(2,5),(3,6),(4,7)]
 => [8,5,6,7,2,3,4,1] => [1,8,2,5,3,6,4,7] => [3,5,7,4,6,8,2,1] => ? = 1 + 1
[(1,7),(2,5),(3,6),(4,8)]
 => [7,5,6,8,2,3,1,4] => [1,7,2,5,3,6,4,8] => [3,5,7,4,6,2,8,1] => ? = 1 + 1
[(1,6),(2,5),(3,7),(4,8)]
 => [6,5,7,8,2,1,3,4] => [1,6,2,5,3,7,4,8] => [3,5,7,4,2,6,8,1] => ? = 2 + 1
[(1,5),(2,6),(3,7),(4,8)]
 => [5,6,7,8,1,2,3,4] => [1,5,2,6,3,7,4,8] => [3,5,7,2,4,6,8,1] => ? = 4 + 1
[(1,4),(2,6),(3,7),(5,8)]
 => [4,6,7,1,8,2,3,5] => [1,4,2,6,3,7,5,8] => [3,5,2,7,4,6,8,1] => ? = 2 + 1
[(1,3),(2,5),(4,7),(6,8)]
 => [3,5,1,7,2,8,4,6] => [1,3,2,5,4,7,6,8] => [3,2,5,4,7,6,8,1] => 1 = 0 + 1
Description
The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. These are multiplicities of Verma modules.