Your data matches 5 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000661
Mapping
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000661: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 0
[1,0,1,1,0,0]
 => [1,1]
 => [1,1,0,0]
 => [1,1,0,0]
 => 0
[1,1,0,0,1,0]
 => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 0
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 0
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 0
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 0
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 0
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,1,0,0]
 => [1,1,0,0]
 => 0
[1,1,1,0,0,0,1,0]
 => [3]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 0
[1,1,1,0,0,1,0,0]
 => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 0
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 1
[1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 0
Description
The number of rises of length 3 of a Dyck path.
Matching statistic: St000782
Mapping
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Values
[1,0,1,0,1,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => ? = 0 + 1
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => ? = 0 + 1
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 0 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => ? = 0 + 1
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => ? = 1 + 1
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => ? = 1 + 1
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => ? = 0 + 1
[1,1,1,0,0,1,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9)]
 => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,10)]
 => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,10),(6,7),(8,9)]
 => ? = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8)]
 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,10)]
 => ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [(1,2),(3,6),(4,5),(7,10),(8,9)]
 => ? = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,10)]
 => ? = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,10),(4,5),(6,7),(8,9)]
 => ? = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [(1,2),(3,10),(4,5),(6,9),(7,8)]
 => ? = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [(1,2),(3,8),(4,7),(5,6),(9,10)]
 => ? = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9)]
 => ? = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,10),(4,9),(5,6),(7,8)]
 => ? = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => ? = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10)]
 => ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9)]
 => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10)]
 => ? = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [(1,4),(2,3),(5,10),(6,7),(8,9)]
 => ? = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8)]
 => ? = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,10)]
 => ? = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [(1,6),(2,3),(4,5),(7,10),(8,9)]
 => ? = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [(1,8),(2,3),(4,5),(6,7),(9,10)]
 => ? = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => ? = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => ? = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [(1,8),(2,3),(4,7),(5,6),(9,10)]
 => ? = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => ? = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => ? = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,10)]
 => ? = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [(1,6),(2,5),(3,4),(7,10),(8,9)]
 => ? = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [(1,8),(2,5),(3,4),(6,7),(9,10)]
 => ? = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => ? = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => ? = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [(1,8),(2,7),(3,4),(5,6),(9,10)]
 => ? = 0 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10)]
 => ? = 0 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => ? = 0 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
 => ? = 1 + 1
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1
[1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1
[1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[1,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
Description
The indicator function of whether a given perfect matching is an L & P matching. An L&P matching is built inductively as follows: starting with either a single edge, or a hairpin $([1,3],[2,4])$, insert a noncrossing matching or inflate an edge by a ladder, that is, a number of nested edges. The number of L&P matchings is (see [thm. 1, 2]) $$\frac{1}{2} \cdot 4^{n} + \frac{1}{n + 1}{2 \, n \choose n} - {2 \, n + 1 \choose n} + {2 \, n - 1 \choose n - 1}$$
Matching statistic: St001722
Mapping
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Values
[1,0,1,0,1,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[1,1,0,0,1,0]
 => [2]
 => [1,1,0,0,1,0]
 => 110010 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => ? = 1 + 1
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 0 + 1
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => ? = 0 + 1
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 0 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => ? = 0 + 1
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => ? = 1 + 1
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => ? = 1 + 1
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 0 + 1
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => ? = 0 + 1
[1,1,1,0,0,1,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => 110010 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => ? = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => ? = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => ? = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => ? = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => ? = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => ? = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => ? = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => ? = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => ? = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => ? = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => ? = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => ? = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1101010010 => ? = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => ? = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => ? = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => ? = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => ? = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => ? = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => ? = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => ? = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? = 0 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 0 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => ? = 0 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => ? = 0 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => 110010 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 101010101100 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 101010110100 => ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 101010111000 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 101011001100 => ? = 1 + 1
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => 110010 => 1 = 0 + 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => 110010 => 1 = 0 + 1
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => 110010 => 1 = 0 + 1
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => 110010 => 1 = 0 + 1
[1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => 110010 => 1 = 0 + 1
[1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => 110010 => 1 = 0 + 1
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[1,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
Description
The number of minimal chains with small intervals between a binary word and the top element. A valley in a binary word is a subsequence $01$, or a trailing $0$. A peak is a subsequence $10$ or a trailing $1$. Let $P$ be the lattice on binary words of length $n$, where the covering elements of a word are obtained by replacing a valley with a peak. An interval $[w_1, w_2]$ in $P$ is small if $w_2$ is obtained from $w_1$ by replacing some valleys with peaks. This statistic counts the number of chains $w = w_1 < \dots < w_d = 1\dots 1$ to the top element of minimal length. For example, there are two such chains for the word $0110$: $$ 0110 < 1011 < 1101 < 1110 < 1111 $$ and $$ 0110 < 1010 < 1101 < 1110 < 1111. $$
Matching statistic: St001583
Mapping
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Values
[1,0,1,0,1,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[1,0,1,1,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 0 + 3
[1,1,0,0,1,0]
 => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 0 + 3
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ? = 1 + 3
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => ? = 0 + 3
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => ? = 0 + 3
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 0 + 3
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 0 + 3
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => ? = 1 + 3
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 1 + 3
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 0 + 3
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 0 + 3
[1,1,1,0,0,0,1,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 0 + 3
[1,1,1,0,0,1,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 0 + 3
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => ? = 1 + 3
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => ? = 0 + 3
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => ? = 0 + 3
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => ? = 0 + 3
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => ? = 1 + 3
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => ? = 0 + 3
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => ? = 0 + 3
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => ? = 1 + 3
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => ? = 0 + 3
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ? = 0 + 3
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => ? = 0 + 3
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => ? = 0 + 3
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ? = 0 + 3
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ? = 0 + 3
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => ? = 1 + 3
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => ? = 1 + 3
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => ? = 0 + 3
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => ? = 0 + 3
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 0 + 3
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => ? = 1 + 3
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => ? = 0 + 3
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => ? = 1 + 3
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ? = 1 + 3
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => ? = 0 + 3
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0 + 3
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => ? = 0 + 3
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 0 + 3
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 0 + 3
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => ? = 1 + 3
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1 + 3
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 1 + 3
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => ? = 1 + 3
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 1 + 3
[1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => ? = 0 + 3
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 0 + 3
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 0 + 3
[1,1,1,1,0,0,0,0,1,0]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => ? = 0 + 3
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 0 + 3
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 0 + 3
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,1,2,3,4,7,5] => ? = 0 + 3
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [7,1,2,3,6,4,5] => ? = 0 + 3
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => ? = 1 + 3
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,1,2,6,3,7,5] => ? = 1 + 3
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 0 + 3
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 0 + 3
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 0 + 3
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 0 + 3
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 0 + 3
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 0 + 3
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 0 + 3
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 0 + 3
[1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 0 + 3
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 0 + 3
[1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 0 + 3
[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 0 + 3
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[1,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 0 + 3
[1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3 = 0 + 3
Description
The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
Matching statistic: St001896
Mapping
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001896: Signed permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 67%
Values
[1,0,1,0,1,0]
 => [2,1,3] => [2,1,3] => [2,1,3] => 1 = 0 + 1
[1,0,1,1,0,0]
 => [2,3,1] => [3,1,2] => [3,1,2] => 1 = 0 + 1
[1,1,0,0,1,0]
 => [3,1,2] => [2,3,1] => [2,3,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [3,4,1,2] => [3,4,1,2] => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [4,2,3,1] => [4,2,3,1] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,4,1,3] => [2,4,1,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [2,3,1,4] => [2,3,1,4] => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,3,4,1] => [2,3,4,1] => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [1,3,4,2] => [1,3,4,2] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => [5,3,4,1,2] => [5,3,4,1,2] => ? = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => [4,2,3,1,5] => [4,2,3,1,5] => ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => [5,2,3,1,4] => [5,2,3,1,4] => ? = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => [5,2,4,1,3] => [5,2,4,1,3] => ? = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => [4,5,2,3,1] => [4,5,2,3,1] => ? = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [1,4,5,2,3] => [1,4,5,2,3] => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => [3,5,2,4,1] => [3,5,2,4,1] => ? = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,5,3] => [2,5,3,4,1] => [2,5,3,4,1] => ? = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,4,2,5,3] => [1,5,3,4,2] => [1,5,3,4,2] => 2 = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,4,5,2,3] => [1,3,5,2,4] => [1,3,5,2,4] => 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,4,2,3,5] => [1,3,4,2,5] => [1,3,4,2,5] => 1 = 0 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1 = 0 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,5,2,3,4] => [1,3,4,5,2] => [1,3,4,5,2] => 1 = 0 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,2,5,3,4] => [1,2,4,5,3] => [1,2,4,5,3] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,6,5] => [3,4,1,2,6,5] => [3,4,1,2,6,5] => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,6,3,5] => [5,6,3,4,1,2] => [5,6,3,4,1,2] => ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [2,4,6,1,3,5] => [3,5,6,1,2,4] => [3,5,6,1,2,4] => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [2,4,1,3,5,6] => [3,4,1,2,5,6] => [3,4,1,2,5,6] => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [2,4,1,5,3,6] => [5,3,4,1,2,6] => [5,3,4,1,2,6] => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [2,4,5,1,3,6] => [3,5,1,2,4,6] => [3,5,1,2,4,6] => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [2,4,1,5,6,3] => [6,3,4,1,2,5] => [6,3,4,1,2,5] => ? = 0 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [2,4,5,1,6,3] => [6,3,5,1,2,4] => [6,3,5,1,2,4] => ? = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,4,5,6,1,3] => [3,6,1,2,4,5] => [3,6,1,2,4,5] => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [2,5,1,6,3,4] => [4,6,3,5,1,2] => [4,6,3,5,1,2] => ? = 0 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [2,5,6,1,3,4] => [3,4,6,1,2,5] => [3,4,6,1,2,5] => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [2,3,1,5,4,6] => [3,1,2,5,4,6] => [3,1,2,5,4,6] => ? = 0 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4,6] => [4,5,1,2,3,6] => [4,5,1,2,3,6] => ? = 0 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [2,3,1,5,6,4] => [3,1,2,6,4,5] => [3,1,2,6,4,5] => ? = 0 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [2,3,5,1,6,4] => [6,4,5,1,2,3] => [6,4,5,1,2,3] => ? = 1 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,3,5,6,1,4] => [4,6,1,2,3,5] => [4,6,1,2,3,5] => ? = 1 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [2,6,1,3,4,5] => [3,4,5,6,1,2] => [3,4,5,6,1,2] => ? = 0 + 1
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [2,3,1,6,4,5] => [3,1,2,5,6,4] => [3,1,2,5,6,4] => ? = 0 + 1
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [2,3,6,1,4,5] => [4,5,6,1,2,3] => [4,5,6,1,2,3] => ? = 0 + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [2,3,1,4,6,5] => [3,1,2,4,6,5] => [3,1,2,4,6,5] => ? = 0 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [2,3,4,1,6,5] => [4,1,2,3,6,5] => [4,1,2,3,6,5] => ? = 0 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,3,4,6,1,5] => [5,6,1,2,3,4] => [5,6,1,2,3,4] => ? = 0 + 1
Description
The number of right descents of a signed permutations. An index is a right descent if it is a left descent of the inverse signed permutation.