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Your data matches 3 different statistics following compositions of up to 3 maps.
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Matching statistic: St000418
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
St000418: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> 1
[1,1,0,0]
=> 2
[1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> 4
[1,1,1,0,0,0]
=> 5
[1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> 4
[1,0,1,1,1,0,0,0]
=> 5
[1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> 4
[1,1,0,1,0,0,1,0]
=> 4
[1,1,0,1,0,1,0,0]
=> 8
[1,1,0,1,1,0,0,0]
=> 10
[1,1,1,0,0,0,1,0]
=> 5
[1,1,1,0,0,1,0,0]
=> 10
[1,1,1,0,1,0,0,0]
=> 13
[1,1,1,1,0,0,0,0]
=> 14
[1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> 5
[1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> 8
[1,0,1,1,0,1,1,0,0,0]
=> 10
[1,0,1,1,1,0,0,0,1,0]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> 10
[1,0,1,1,1,0,1,0,0,0]
=> 13
[1,0,1,1,1,1,0,0,0,0]
=> 14
[1,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> 8
[1,1,0,0,1,1,1,0,0,0]
=> 10
[1,1,0,1,0,0,1,0,1,0]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> 8
[1,1,0,1,0,1,0,0,1,0]
=> 8
[1,1,0,1,0,1,0,1,0,0]
=> 16
[1,1,0,1,0,1,1,0,0,0]
=> 20
[1,1,0,1,1,0,0,0,1,0]
=> 10
[1,1,0,1,1,0,0,1,0,0]
=> 20
[1,1,0,1,1,0,1,0,0,0]
=> 26
[1,1,0,1,1,1,0,0,0,0]
=> 28
[1,1,1,0,0,0,1,0,1,0]
=> 5
Description
The number of Dyck paths that are weakly below a Dyck path.
Matching statistic: St000421
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
St000421: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,1,0,0,0]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> 7 = 8 - 1
[1,1,0,1,1,0,0,0]
=> 9 = 10 - 1
[1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[1,1,1,0,0,1,0,0]
=> 9 = 10 - 1
[1,1,1,0,1,0,0,0]
=> 12 = 13 - 1
[1,1,1,1,0,0,0,0]
=> 13 = 14 - 1
[1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> 7 = 8 - 1
[1,0,1,1,0,1,1,0,0,0]
=> 9 = 10 - 1
[1,0,1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
=> 9 = 10 - 1
[1,0,1,1,1,0,1,0,0,0]
=> 12 = 13 - 1
[1,0,1,1,1,1,0,0,0,0]
=> 13 = 14 - 1
[1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> 7 = 8 - 1
[1,1,0,0,1,1,1,0,0,0]
=> 9 = 10 - 1
[1,1,0,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> 7 = 8 - 1
[1,1,0,1,0,1,0,0,1,0]
=> 7 = 8 - 1
[1,1,0,1,0,1,0,1,0,0]
=> 15 = 16 - 1
[1,1,0,1,0,1,1,0,0,0]
=> 19 = 20 - 1
[1,1,0,1,1,0,0,0,1,0]
=> 9 = 10 - 1
[1,1,0,1,1,0,0,1,0,0]
=> 19 = 20 - 1
[1,1,0,1,1,0,1,0,0,0]
=> 25 = 26 - 1
[1,1,0,1,1,1,0,0,0,0]
=> 27 = 28 - 1
[1,1,1,0,0,0,1,0,1,0]
=> 4 = 5 - 1
Description
The number of Dyck paths that are weakly below a Dyck path, except for the path itself.
Matching statistic: St001832
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00116: Perfect matchings —Kasraoui-Zeng⟶ Perfect matchings
St001832: Perfect matchings ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 19%
Mp00116: Perfect matchings —Kasraoui-Zeng⟶ Perfect matchings
St001832: Perfect matchings ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 19%
Values
[1,0,1,0]
=> [(1,2),(3,4)]
=> [(1,2),(3,4)]
=> 1
[1,1,0,0]
=> [(1,4),(2,3)]
=> [(1,3),(2,4)]
=> 2
[1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [(1,2),(3,4),(5,6)]
=> 1
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [(1,2),(3,5),(4,6)]
=> 2
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [(1,3),(2,4),(5,6)]
=> 2
[1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [(1,3),(2,5),(4,6)]
=> 4
[1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [(1,4),(2,5),(3,6)]
=> 5
[1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 1
[1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [(1,2),(3,4),(5,7),(6,8)]
=> 2
[1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [(1,2),(3,5),(4,6),(7,8)]
=> 2
[1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [(1,2),(3,5),(4,7),(6,8)]
=> 4
[1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [(1,2),(3,6),(4,7),(5,8)]
=> 5
[1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [(1,3),(2,4),(5,6),(7,8)]
=> 2
[1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [(1,3),(2,4),(5,7),(6,8)]
=> 4
[1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [(1,3),(2,5),(4,6),(7,8)]
=> 4
[1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [(1,3),(2,5),(4,7),(6,8)]
=> 8
[1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [(1,3),(2,6),(4,7),(5,8)]
=> 10
[1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [(1,4),(2,5),(3,6),(7,8)]
=> 5
[1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [(1,4),(2,5),(3,7),(6,8)]
=> 10
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [(1,4),(2,6),(3,7),(5,8)]
=> 13
[1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [(1,5),(2,6),(3,7),(4,8)]
=> 14
[1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [(1,2),(3,4),(5,6),(7,9),(8,10)]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [(1,2),(3,4),(5,7),(6,8),(9,10)]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [(1,2),(3,4),(5,7),(6,9),(8,10)]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [(1,2),(3,4),(5,8),(6,9),(7,10)]
=> 5
[1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [(1,2),(3,5),(4,6),(7,8),(9,10)]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> [(1,2),(3,5),(4,6),(7,9),(8,10)]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [(1,2),(3,5),(4,7),(6,8),(9,10)]
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [(1,2),(3,5),(4,7),(6,9),(8,10)]
=> 8
[1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [(1,2),(3,5),(4,8),(6,9),(7,10)]
=> 10
[1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [(1,2),(3,6),(4,7),(5,8),(9,10)]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [(1,2),(3,6),(4,7),(5,9),(8,10)]
=> 10
[1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [(1,2),(3,6),(4,8),(5,9),(7,10)]
=> 13
[1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [(1,2),(3,7),(4,8),(5,9),(6,10)]
=> 14
[1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [(1,3),(2,4),(5,6),(7,8),(9,10)]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> [(1,3),(2,4),(5,6),(7,9),(8,10)]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [(1,3),(2,4),(5,7),(6,8),(9,10)]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [(1,3),(2,4),(5,7),(6,9),(8,10)]
=> 8
[1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [(1,3),(2,4),(5,8),(6,9),(7,10)]
=> 10
[1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [(1,3),(2,5),(4,6),(7,8),(9,10)]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [(1,3),(2,5),(4,6),(7,9),(8,10)]
=> 8
[1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [(1,3),(2,5),(4,7),(6,8),(9,10)]
=> 8
[1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [(1,3),(2,5),(4,7),(6,9),(8,10)]
=> 16
[1,1,0,1,0,1,1,0,0,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> [(1,3),(2,5),(4,8),(6,9),(7,10)]
=> 20
[1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [(1,3),(2,6),(4,7),(5,8),(9,10)]
=> 10
[1,1,0,1,1,0,0,1,0,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9)]
=> [(1,3),(2,6),(4,7),(5,9),(8,10)]
=> 20
[1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [(1,3),(2,6),(4,8),(5,9),(7,10)]
=> 26
[1,1,0,1,1,1,0,0,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7)]
=> [(1,3),(2,7),(4,8),(5,9),(6,10)]
=> 28
[1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [(1,4),(2,5),(3,6),(7,8),(9,10)]
=> 5
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 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)]
=> [(1,2),(3,4),(5,6),(7,8),(9,11),(10,12)]
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]
=> [(1,2),(3,4),(5,6),(7,9),(8,10),(11,12)]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> [(1,2),(3,4),(5,6),(7,9),(8,11),(10,12)]
=> ? = 4
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
=> [(1,2),(3,4),(5,6),(7,10),(8,11),(9,12)]
=> ? = 5
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> [(1,2),(3,4),(5,7),(6,8),(9,10),(11,12)]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> [(1,2),(3,4),(5,7),(6,8),(9,11),(10,12)]
=> ? = 4
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9),(11,12)]
=> [(1,2),(3,4),(5,7),(6,9),(8,10),(11,12)]
=> ? = 4
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
=> [(1,2),(3,4),(5,7),(6,9),(8,11),(10,12)]
=> ? = 8
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,11),(9,10)]
=> [(1,2),(3,4),(5,7),(6,10),(8,11),(9,12)]
=> ? = 10
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)]
=> [(1,2),(3,4),(5,8),(6,9),(7,10),(11,12)]
=> ? = 5
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)]
=> [(1,2),(3,4),(5,8),(6,9),(7,11),(10,12)]
=> ? = 10
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10)]
=> [(1,2),(3,4),(5,8),(6,10),(7,11),(9,12)]
=> ? = 13
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> [(1,2),(3,4),(5,9),(6,10),(7,11),(8,12)]
=> ? = 14
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
=> [(1,2),(3,5),(4,6),(7,8),(9,10),(11,12)]
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> [(1,2),(3,5),(4,6),(7,8),(9,11),(10,12)]
=> ? = 4
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> [(1,2),(3,5),(4,6),(7,9),(8,10),(11,12)]
=> ? = 4
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,9),(10,11)]
=> [(1,2),(3,5),(4,6),(7,9),(8,11),(10,12)]
=> ? = 8
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
=> [(1,2),(3,5),(4,6),(7,10),(8,11),(9,12)]
=> ? = 10
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]
=> [(1,2),(3,5),(4,7),(6,8),(9,10),(11,12)]
=> ? = 4
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
=> [(1,2),(3,5),(4,7),(6,8),(9,11),(10,12)]
=> ? = 8
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)]
=> [(1,2),(3,5),(4,7),(6,9),(8,10),(11,12)]
=> ? = 8
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
=> [(1,2),(3,5),(4,7),(6,9),(8,11),(10,12)]
=> ? = 16
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,11),(9,10)]
=> [(1,2),(3,5),(4,7),(6,10),(8,11),(9,12)]
=> ? = 20
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)]
=> [(1,2),(3,5),(4,8),(6,9),(7,10),(11,12)]
=> ? = 10
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,9),(7,8),(10,11)]
=> [(1,2),(3,5),(4,8),(6,9),(7,11),(10,12)]
=> ? = 20
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [(1,2),(3,12),(4,5),(6,11),(7,8),(9,10)]
=> [(1,2),(3,5),(4,8),(6,10),(7,11),(9,12)]
=> ? = 26
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,5),(6,11),(7,10),(8,9)]
=> [(1,2),(3,5),(4,9),(6,10),(7,11),(8,12)]
=> ? = 28
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10),(11,12)]
=> [(1,2),(3,6),(4,7),(5,8),(9,10),(11,12)]
=> ? = 5
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> [(1,2),(3,6),(4,7),(5,8),(9,11),(10,12)]
=> ? = 10
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9),(11,12)]
=> [(1,2),(3,6),(4,7),(5,9),(8,10),(11,12)]
=> ? = 10
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
=> [(1,2),(3,6),(4,7),(5,9),(8,11),(10,12)]
=> ? = 20
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [(1,2),(3,12),(4,7),(5,6),(8,11),(9,10)]
=> [(1,2),(3,6),(4,7),(5,10),(8,11),(9,12)]
=> ? = 25
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8),(11,12)]
=> [(1,2),(3,6),(4,8),(5,9),(7,10),(11,12)]
=> ? = 13
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [(1,2),(3,12),(4,9),(5,6),(7,8),(10,11)]
=> [(1,2),(3,6),(4,8),(5,9),(7,11),(10,12)]
=> ? = 26
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,8),(9,10)]
=> [(1,2),(3,6),(4,8),(5,10),(7,11),(9,12)]
=> ? = 34
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> [(1,2),(3,6),(4,9),(5,10),(7,11),(8,12)]
=> ? = 37
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7),(11,12)]
=> [(1,2),(3,7),(4,8),(5,9),(6,10),(11,12)]
=> ? = 14
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [(1,2),(3,12),(4,9),(5,8),(6,7),(10,11)]
=> [(1,2),(3,7),(4,8),(5,9),(6,11),(10,12)]
=> ? = 28
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
=> [(1,2),(3,7),(4,8),(5,10),(6,11),(9,12)]
=> ? = 37
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> [(1,2),(3,7),(4,9),(5,10),(6,11),(8,12)]
=> ? = 41
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> [(1,2),(3,8),(4,9),(5,10),(6,11),(7,12)]
=> ? = 42
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)]
=> [(1,3),(2,4),(5,6),(7,8),(9,10),(11,12)]
=> ? = 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)]
=> [(1,3),(2,4),(5,6),(7,8),(9,11),(10,12)]
=> ? = 4
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> [(1,3),(2,4),(5,6),(7,9),(8,10),(11,12)]
=> ? = 4
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,9),(10,11)]
=> [(1,3),(2,4),(5,6),(7,9),(8,11),(10,12)]
=> ? = 8
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10)]
=> [(1,3),(2,4),(5,6),(7,10),(8,11),(9,12)]
=> ? = 10
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
=> [(1,3),(2,4),(5,7),(6,8),(9,10),(11,12)]
=> ? = 4
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12)]
=> ? = 8
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9),(11,12)]
=> [(1,3),(2,4),(5,7),(6,9),(8,10),(11,12)]
=> ? = 8
Description
The number of non-crossing perfect matchings in the chord expansion of a perfect matching.
Given a perfect matching, we obtain a formal sum of non-crossing perfect matchings by replacing recursively every matching $M$ that has a crossing $(a, c), (b, d)$ with $a < b < c < d$ with the sum of the two matchings $(M\setminus \{(a,c), (b,d)\})\cup \{(a,b), (c,d)\}$ and $(M\setminus \{(a,c), (b,d)\})\cup \{(a,d), (b,c)\}$.
This statistic is the number of distinct non-crossing perfect matchings in the formal sum.
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