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Your data matches 6 different statistics following compositions of up to 3 maps.
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Matching statistic: St000421
(load all 13 compositions to match this statistic)
(load all 13 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
[1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> 3
[1,1,1,0,0,0]
=> 4
[1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> 4
[1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,0]
=> 7
[1,1,0,1,1,0,0,0]
=> 9
[1,1,1,0,0,0,1,0]
=> 4
[1,1,1,0,0,1,0,0]
=> 9
[1,1,1,0,1,0,0,0]
=> 12
[1,1,1,1,0,0,0,0]
=> 13
[1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> 7
[1,0,1,1,0,1,1,0,0,0]
=> 9
[1,0,1,1,1,0,0,0,1,0]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> 9
[1,0,1,1,1,0,1,0,0,0]
=> 12
[1,0,1,1,1,1,0,0,0,0]
=> 13
[1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> 7
[1,1,0,0,1,1,1,0,0,0]
=> 9
[1,1,0,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> 7
[1,1,0,1,0,1,0,0,1,0]
=> 7
[1,1,0,1,0,1,0,1,0,0]
=> 15
[1,1,0,1,0,1,1,0,0,0]
=> 19
[1,1,0,1,1,0,0,0,1,0]
=> 9
[1,1,0,1,1,0,0,1,0,0]
=> 19
[1,1,0,1,1,0,1,0,0,0]
=> 25
[1,1,0,1,1,1,0,0,0,0]
=> 27
[1,1,1,0,0,0,1,0,1,0]
=> 4
Description
The number of Dyck paths that are weakly below a Dyck path, except for the path itself.
Matching statistic: St000418
(load all 10 compositions to match this statistic)
(load all 10 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 = 0 + 1
[1,1,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> 4 = 3 + 1
[1,1,1,0,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0]
=> 8 = 7 + 1
[1,1,0,1,1,0,0,0]
=> 10 = 9 + 1
[1,1,1,0,0,0,1,0]
=> 5 = 4 + 1
[1,1,1,0,0,1,0,0]
=> 10 = 9 + 1
[1,1,1,0,1,0,0,0]
=> 13 = 12 + 1
[1,1,1,1,0,0,0,0]
=> 14 = 13 + 1
[1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
=> 8 = 7 + 1
[1,0,1,1,0,1,1,0,0,0]
=> 10 = 9 + 1
[1,0,1,1,1,0,0,0,1,0]
=> 5 = 4 + 1
[1,0,1,1,1,0,0,1,0,0]
=> 10 = 9 + 1
[1,0,1,1,1,0,1,0,0,0]
=> 13 = 12 + 1
[1,0,1,1,1,1,0,0,0,0]
=> 14 = 13 + 1
[1,1,0,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
=> 8 = 7 + 1
[1,1,0,0,1,1,1,0,0,0]
=> 10 = 9 + 1
[1,1,0,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
=> 8 = 7 + 1
[1,1,0,1,0,1,0,0,1,0]
=> 8 = 7 + 1
[1,1,0,1,0,1,0,1,0,0]
=> 16 = 15 + 1
[1,1,0,1,0,1,1,0,0,0]
=> 20 = 19 + 1
[1,1,0,1,1,0,0,0,1,0]
=> 10 = 9 + 1
[1,1,0,1,1,0,0,1,0,0]
=> 20 = 19 + 1
[1,1,0,1,1,0,1,0,0,0]
=> 26 = 25 + 1
[1,1,0,1,1,1,0,0,0,0]
=> 28 = 27 + 1
[1,1,1,0,0,0,1,0,1,0]
=> 5 = 4 + 1
Description
The number of Dyck paths that are weakly below a Dyck path.
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: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 19%
Mp00116: Perfect matchings —Kasraoui-Zeng⟶ Perfect matchings
St001832: Perfect matchings ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 19%
Values
[1,0,1,0]
=> [(1,2),(3,4)]
=> [(1,2),(3,4)]
=> 1 = 0 + 1
[1,1,0,0]
=> [(1,4),(2,3)]
=> [(1,3),(2,4)]
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [(1,2),(3,5),(4,6)]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [(1,3),(2,4),(5,6)]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [(1,3),(2,5),(4,6)]
=> 4 = 3 + 1
[1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [(1,4),(2,5),(3,6)]
=> 5 = 4 + 1
[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 = 0 + 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 + 1
[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 + 1
[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 = 3 + 1
[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 = 4 + 1
[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
[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 = 3 + 1
[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 = 3 + 1
[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 = 7 + 1
[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 = 9 + 1
[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 = 4 + 1
[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 = 9 + 1
[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 = 12 + 1
[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 = 13 + 1
[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 = 0 + 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 + 1
[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 + 1
[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 = 3 + 1
[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 = 4 + 1
[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 + 1
[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 = 3 + 1
[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 = 3 + 1
[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 = 7 + 1
[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 = 9 + 1
[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 = 4 + 1
[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 = 9 + 1
[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 = 12 + 1
[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 = 13 + 1
[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
[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 = 3 + 1
[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 = 3 + 1
[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 = 7 + 1
[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 = 9 + 1
[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 = 3 + 1
[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 = 7 + 1
[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 = 7 + 1
[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 = 15 + 1
[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 = 19 + 1
[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 = 9 + 1
[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 = 19 + 1
[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 = 25 + 1
[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 = 27 + 1
[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 = 4 + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 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)]
=> [(1,2),(3,4),(5,6),(7,9),(8,11),(10,12)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 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)]
=> [(1,2),(3,4),(5,6),(7,10),(8,11),(9,12)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 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)]
=> [(1,2),(3,4),(5,7),(6,8),(9,11),(10,12)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[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)]
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
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.
Matching statistic: St001583
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 5%●distinct values known / distinct values provided: 3%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 5%●distinct values known / distinct values provided: 3%
Values
[1,0,1,0]
=> [1]
=> [1,0]
=> [2,1] => 0
[1,1,0,0]
=> []
=> []
=> [1] => ? = 1
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3
[1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[1,1,0,1,0,0]
=> [1]
=> [1,0]
=> [2,1] => 0
[1,1,1,0,0,0]
=> []
=> []
=> [1] => ? = 4
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? ∊ {4,4,7,9,9,12,13}
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => ? ∊ {4,4,7,9,9,12,13}
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? ∊ {4,4,7,9,9,12,13}
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? ∊ {4,4,7,9,9,12,13}
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? ∊ {4,4,7,9,9,12,13}
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? ∊ {4,4,7,9,9,12,13}
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> [2,1] => 0
[1,1,1,1,0,0,0,0]
=> []
=> []
=> [1] => ? ∊ {4,4,7,9,9,12,13}
[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]
=> [8,1,4,5,2,7,3,6] => ? ∊ {1,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[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]
=> [7,3,4,1,6,2,5] => ? ∊ {1,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[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]
=> [4,1,2,5,8,7,3,6] => ? ∊ {1,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[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]
=> [3,1,4,7,6,2,5] => ? ∊ {1,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => ? ∊ {1,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[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]
=> [7,1,4,8,2,3,5,6] => ? ∊ {1,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[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]
=> [6,3,7,1,2,4,5] => ? ∊ {1,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[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]
=> [4,1,2,8,7,3,5,6] => ? ∊ {1,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[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]
=> [3,1,7,6,2,4,5] => ? ∊ {1,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => ? ∊ {1,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[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]
=> [8,1,2,3,7,4,5,6] => ? ∊ {1,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[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]
=> [7,1,2,6,3,4,5] => ? ∊ {1,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? ∊ {1,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ? ∊ {1,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[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]
=> [6,1,4,5,2,7,3] => ? ∊ {1,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => ? ∊ {1,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[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]
=> [4,1,2,5,6,7,3] => ? ∊ {1,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? ∊ {1,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? ∊ {1,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[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]
=> [7,1,4,6,2,3,5] => ? ∊ {1,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => ? ∊ {1,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[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]
=> [4,1,2,7,6,3,5] => ? ∊ {1,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? ∊ {1,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => ? ∊ {1,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[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]
=> [7,1,2,3,6,4,5] => ? ∊ {1,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? ∊ {1,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[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] => ? ∊ {1,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? ∊ {1,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => ? ∊ {1,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ? ∊ {1,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? ∊ {1,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? ∊ {1,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? ∊ {1,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? ∊ {1,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0]
=> [2,1] => 0
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> [1] => ? ∊ {1,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [10,1,4,6,2,7,9,3,5,8] => ? ∊ {1,1,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [9,3,5,1,6,8,2,4,7] => ? ∊ {1,1,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [4,1,2,10,6,7,9,3,5,8] => ? ∊ {1,1,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [3,1,9,5,6,8,2,4,7] => ? ∊ {1,1,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [2,8,4,5,7,1,3,6] => ? ∊ {1,1,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [7,1,10,6,2,3,9,4,5,8] => ? ∊ {1,1,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131}
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> [1,0]
=> [2,1] => 0
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1]
=> [1,0]
=> [2,1] => 0
Description
The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
Matching statistic: St001934
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001934: Integer partitions ⟶ ℤResult quality: 2% ●values known / values provided: 4%●distinct values known / distinct values provided: 2%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001934: Integer partitions ⟶ ℤResult quality: 2% ●values known / values provided: 4%●distinct values known / distinct values provided: 2%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {0,1}
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {0,1}
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,1,3,4}
[1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,1,3,4}
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,1,3,4}
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,1,3,4}
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,1,3,4}
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,1,1,3,3,3,4,4,7,9,9,12,13}
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,1,1,3,3,3,4,4,7,9,9,12,13}
[1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,1,1,3,3,3,4,4,7,9,9,12,13}
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,1,1,3,3,3,4,4,7,9,9,12,13}
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,1,1,3,3,3,4,4,7,9,9,12,13}
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,1,1,3,3,3,4,4,7,9,9,12,13}
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,1,1,3,3,3,4,4,7,9,9,12,13}
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,1,1,3,3,3,4,4,7,9,9,12,13}
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,1,1,3,3,3,4,4,7,9,9,12,13}
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,1,1,3,3,3,4,4,7,9,9,12,13}
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,1,1,3,3,3,4,4,7,9,9,12,13}
[1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,1,1,3,3,3,4,4,7,9,9,12,13}
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,1,1,3,3,3,4,4,7,9,9,12,13}
[1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,3,3,3,3,3,3,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,3,3,3,3,3,3,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,3,3,3,3,3,3,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,3,3,3,3,3,3,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,3,3,3,3,3,3,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,3,3,3,3,3,3,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,3,3,3,3,3,3,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,3,3,3,3,3,3,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,3,3,3,3,3,3,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,3,3,3,3,3,3,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,3,3,3,3,3,3,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,3,3,3,3,3,3,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,3,3,3,3,3,3,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,3,3,3,3,3,3,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,3,3,3,3,3,3,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,3,3,3,3,3,3,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,3,3,3,3,3,3,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,3,3,3,3,3,3,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,3,3,3,3,3,3,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,3,3,3,3,3,3,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,3,3,3,3,3,3,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,3,3,3,3,3,3,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,3,3,3,3,3,3,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,3,3,3,3,3,3,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,3,3,3,3,3,3,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,3,3,3,3,3,3,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 4
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,3,3,3,3,3,3,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,3,3,3,3,3,3,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,3,3,3,3,3,3,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,3,3,3,3,3,3,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41}
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 4
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 4
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 4
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 4
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 1
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 1
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [3,3]
=> 4
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [3,3]
=> 4
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [3,3]
=> 4
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,2,2]
=> 1
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,2,2]
=> 1
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,2,2]
=> 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,2]
=> 1
Description
The number of monotone factorisations of genus zero of a permutation of given cycle type.
A monotone factorisation of genus zero of a permutation $\pi\in\mathfrak S_n$ with $\ell$ cycles, including fixed points, is a tuple of $r = n - \ell$ transpositions
$$
(a_1, b_1),\dots,(a_r, b_r)
$$
with $b_1 \leq \dots \leq b_r$ and $a_i < b_i$ for all $i$, whose product, in this order, is $\pi$.
For example, the cycle $(2,3,1)$ has the two factorizations $(2,3)(1,3)$ and $(1,2)(2,3)$.
Matching statistic: St001232
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 3%●distinct values known / distinct values provided: 2%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 3%●distinct values known / distinct values provided: 2%
Values
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {3,4} + 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {3,4} + 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {3,3,3,4,4,7,9,9,12,13} + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,3,4,4,7,9,9,12,13} + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {3,3,3,4,4,7,9,9,12,13} + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,3,4,4,7,9,9,12,13} + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {3,3,3,4,4,7,9,9,12,13} + 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,3,4,4,7,9,9,12,13} + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {3,3,3,4,4,7,9,9,12,13} + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,3,4,4,7,9,9,12,13} + 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {3,3,3,4,4,7,9,9,12,13} + 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,3,4,4,7,9,9,12,13} + 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4,7,7,7,7,9,9,9,9,9,9,12,12,13,13,15,19,19,19,24,25,25,27,27,33,36,36,40,41} + 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,12,12,12,13,13,13,15,15,15,15,15,19,19,19,19,19,19,19,19,19,19,19,19,24,24,24,25,25,25,25,25,25,27,27,27,27,27,27,31,33,33,36,36,36,36,39,39,39,39,40,40,41,41,49,49,49,51,51,51,55,55,55,64,64,67,67,69,69,73,73,73,73,81,81,83,83,88,96,96,97,105,108,108,111,111,121,125,125,130,131} + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1 = 0 + 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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