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Your data matches 11 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)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St000418: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St000418: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 2
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 4
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 5
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 4
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 5
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 4
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 4
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 8
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 10
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 5
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 10
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 13
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 14
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 5
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 8
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> 10
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> 10
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 13
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 14
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> 8
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 10
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> 8
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 8
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 16
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> 20
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> 10
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 20
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 26
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 28
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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00116: Perfect matchings —Kasraoui-Zeng⟶ Perfect matchings
St001832: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [(1,2)]
=> [(1,2)]
=> 1
[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
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: St000421
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St000421: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St000421: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 7 = 8 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 9 = 10 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 9 = 10 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 12 = 13 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 13 = 14 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 7 = 8 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> 9 = 10 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> 9 = 10 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 12 = 13 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 13 = 14 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> 7 = 8 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 9 = 10 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> 7 = 8 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 7 = 8 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 15 = 16 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> 19 = 20 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> 9 = 10 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 19 = 20 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 25 = 26 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 27 = 28 - 1
Description
The number of Dyck paths that are weakly below a Dyck path, except for the path itself.
Matching statistic: St000883
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000883: Permutations ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 29%
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000883: Permutations ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 29%
Values
[1,0]
=> [(1,2)]
=> [2,1] => [1,2] => 1
[1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [1,2,3,4] => 1
[1,1,0,0]
=> [(1,4),(2,3)]
=> [3,4,2,1] => [1,3,2,4] => 2
[1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [1,2,3,4,5,6] => 1
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [1,2,3,5,4,6] => 2
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [1,3,2,4,5,6] => 2
[1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [1,3,2,5,4,6] => 4
[1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [1,4,3,6,2,5] => 5
[1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [1,2,3,4,5,6,7,8] => 1
[1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => [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)]
=> [2,1,5,6,4,3,8,7] => [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)]
=> [2,1,5,7,4,8,6,3] => [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)]
=> [2,1,6,7,8,5,4,3] => [1,2,3,6,5,8,4,7] => ? = 5
[1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [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)]
=> [3,4,2,1,7,8,6,5] => [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)]
=> [3,5,2,6,4,1,8,7] => [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)]
=> [3,5,2,7,4,8,6,1] => [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)]
=> [3,6,2,7,8,5,4,1] => [1,3,2,6,5,8,4,7] => ? = 10
[1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [1,4,3,6,2,5,7,8] => ? = 5
[1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [1,4,3,7,6,8,2,5] => ? = 10
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [1,4,3,7,2,6,5,8] => ? = 13
[1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => [1,5,4,8,2,6,3,7] => ? = 14
[1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => [1,2,3,4,5,6,7,8,9,10] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,9,10,8,7] => [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)]
=> [2,1,4,3,7,8,6,5,10,9] => [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)]
=> [2,1,4,3,7,9,6,10,8,5] => [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)]
=> [2,1,4,3,8,9,10,7,6,5] => [1,2,3,4,5,8,7,10,6,9] => ? = 5
[1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [2,1,5,6,4,3,8,7,10,9] => [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)]
=> [2,1,5,6,4,3,9,10,8,7] => [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)]
=> [2,1,5,7,4,8,6,3,10,9] => [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)]
=> [2,1,5,7,4,9,6,10,8,3] => [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)]
=> [2,1,5,8,4,9,10,7,6,3] => [1,2,3,5,4,8,7,10,6,9] => ? = 10
[1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,6,7,8,5,4,3,10,9] => [1,2,3,6,5,8,4,7,9,10] => ? = 5
[1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,6,7,9,5,4,10,8,3] => [1,2,3,6,5,9,8,10,4,7] => ? = 10
[1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,6,8,9,5,10,7,4,3] => [1,2,3,6,5,9,4,8,7,10] => ? = 13
[1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,7,8,9,10,6,5,4,3] => [1,2,3,7,6,10,4,8,5,9] => ? = 14
[1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [3,4,2,1,6,5,8,7,10,9] => [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)]
=> [3,4,2,1,6,5,9,10,8,7] => [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)]
=> [3,4,2,1,7,8,6,5,10,9] => [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)]
=> [3,4,2,1,7,9,6,10,8,5] => [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)]
=> [3,4,2,1,8,9,10,7,6,5] => [1,3,2,4,5,8,7,10,6,9] => ? = 10
[1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [3,5,2,6,4,1,8,7,10,9] => [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)]
=> [3,5,2,6,4,1,9,10,8,7] => [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)]
=> [3,5,2,7,4,8,6,1,10,9] => [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)]
=> [3,5,2,7,4,9,6,10,8,1] => [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)]
=> [3,5,2,8,4,9,10,7,6,1] => [1,3,2,5,4,8,7,10,6,9] => ? = 20
[1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [3,6,2,7,8,5,4,1,10,9] => [1,3,2,6,5,8,4,7,9,10] => ? = 10
[1,1,0,1,1,0,0,1,0,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9)]
=> [3,6,2,7,9,5,4,10,8,1] => [1,3,2,6,5,9,8,10,4,7] => ? = 20
[1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [3,6,2,8,9,5,10,7,4,1] => [1,3,2,6,5,9,4,8,7,10] => ? = 26
[1,1,0,1,1,1,0,0,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7)]
=> [3,7,2,8,9,10,6,5,4,1] => [1,3,2,7,6,10,4,8,5,9] => ? = 28
[1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [4,5,6,3,2,1,8,7,10,9] => [1,4,3,6,2,5,7,8,9,10] => ? = 5
[1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [4,5,6,3,2,1,9,10,8,7] => [1,4,3,6,2,5,7,9,8,10] => ? = 10
[1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [4,5,7,3,2,8,6,1,10,9] => [1,4,3,7,6,8,2,5,9,10] => ? = 10
[1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [4,5,7,3,2,9,6,10,8,1] => [1,4,3,7,6,9,8,10,2,5] => ? = 20
[1,1,1,0,0,1,1,0,0,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8)]
=> [4,5,8,3,2,9,10,7,6,1] => [1,4,3,8,7,10,2,5,6,9] => ? = 25
[1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [4,6,7,3,8,5,2,1,10,9] => [1,4,3,7,2,6,5,8,9,10] => ? = 13
[1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [4,6,7,3,9,5,2,10,8,1] => [1,4,3,7,2,6,5,9,8,10] => ? = 26
[1,1,1,0,1,0,1,0,0,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [4,6,8,3,9,5,10,7,2,1] => [1,4,3,8,7,10,2,6,5,9] => ? = 34
[1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> [4,7,8,3,9,10,6,5,2,1] => [1,4,3,8,5,9,2,7,6,10] => ? = 37
[1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [5,6,7,8,4,3,2,1,10,9] => [1,5,4,8,2,6,3,7,9,10] => ? = 14
[1,1,1,1,0,0,0,1,0,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9)]
=> [5,6,7,9,4,3,2,10,8,1] => [1,5,4,9,8,10,2,6,3,7] => ? = 28
[1,1,1,1,0,0,1,0,0,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8)]
=> [5,6,8,9,4,3,10,7,2,1] => [1,5,4,9,2,6,3,8,7,10] => ? = 37
[1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> [5,7,8,9,4,10,6,3,2,1] => [1,5,4,9,2,7,6,10,3,8] => ? = 41
[1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [6,7,8,9,10,5,4,3,2,1] => [1,6,5,10,2,7,4,9,3,8] => ? = 42
Description
The number of longest increasing subsequences of a permutation.
Matching statistic: St001464
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001464: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 24%
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001464: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 24%
Values
[1,0]
=> [(1,2)]
=> [2,1] => [1,2] => 1
[1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [1,2,3,4] => 1
[1,1,0,0]
=> [(1,4),(2,3)]
=> [3,4,2,1] => [1,3,2,4] => 2
[1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [1,2,3,4,5,6] => 1
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [1,2,3,5,4,6] => 2
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [1,3,2,4,5,6] => 2
[1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [1,3,2,5,4,6] => 4
[1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [1,4,3,6,2,5] => 5
[1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [1,2,3,4,5,6,7,8] => ? = 1
[1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => [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)]
=> [2,1,5,6,4,3,8,7] => [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)]
=> [2,1,5,7,4,8,6,3] => [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)]
=> [2,1,6,7,8,5,4,3] => [1,2,3,6,5,8,4,7] => ? = 5
[1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [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)]
=> [3,4,2,1,7,8,6,5] => [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)]
=> [3,5,2,6,4,1,8,7] => [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)]
=> [3,5,2,7,4,8,6,1] => [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)]
=> [3,6,2,7,8,5,4,1] => [1,3,2,6,5,8,4,7] => ? = 10
[1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [1,4,3,6,2,5,7,8] => ? = 5
[1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [1,4,3,7,6,8,2,5] => ? = 10
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [1,4,3,7,2,6,5,8] => ? = 13
[1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => [1,5,4,8,2,6,3,7] => ? = 14
[1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => [1,2,3,4,5,6,7,8,9,10] => ? = 1
[1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,9,10,8,7] => [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)]
=> [2,1,4,3,7,8,6,5,10,9] => [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)]
=> [2,1,4,3,7,9,6,10,8,5] => [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)]
=> [2,1,4,3,8,9,10,7,6,5] => [1,2,3,4,5,8,7,10,6,9] => ? = 5
[1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [2,1,5,6,4,3,8,7,10,9] => [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)]
=> [2,1,5,6,4,3,9,10,8,7] => [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)]
=> [2,1,5,7,4,8,6,3,10,9] => [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)]
=> [2,1,5,7,4,9,6,10,8,3] => [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)]
=> [2,1,5,8,4,9,10,7,6,3] => [1,2,3,5,4,8,7,10,6,9] => ? = 10
[1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,6,7,8,5,4,3,10,9] => [1,2,3,6,5,8,4,7,9,10] => ? = 5
[1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,6,7,9,5,4,10,8,3] => [1,2,3,6,5,9,8,10,4,7] => ? = 10
[1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,6,8,9,5,10,7,4,3] => [1,2,3,6,5,9,4,8,7,10] => ? = 13
[1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,7,8,9,10,6,5,4,3] => [1,2,3,7,6,10,4,8,5,9] => ? = 14
[1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [3,4,2,1,6,5,8,7,10,9] => [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)]
=> [3,4,2,1,6,5,9,10,8,7] => [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)]
=> [3,4,2,1,7,8,6,5,10,9] => [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)]
=> [3,4,2,1,7,9,6,10,8,5] => [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)]
=> [3,4,2,1,8,9,10,7,6,5] => [1,3,2,4,5,8,7,10,6,9] => ? = 10
[1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [3,5,2,6,4,1,8,7,10,9] => [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)]
=> [3,5,2,6,4,1,9,10,8,7] => [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)]
=> [3,5,2,7,4,8,6,1,10,9] => [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)]
=> [3,5,2,7,4,9,6,10,8,1] => [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)]
=> [3,5,2,8,4,9,10,7,6,1] => [1,3,2,5,4,8,7,10,6,9] => ? = 20
[1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [3,6,2,7,8,5,4,1,10,9] => [1,3,2,6,5,8,4,7,9,10] => ? = 10
[1,1,0,1,1,0,0,1,0,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9)]
=> [3,6,2,7,9,5,4,10,8,1] => [1,3,2,6,5,9,8,10,4,7] => ? = 20
[1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [3,6,2,8,9,5,10,7,4,1] => [1,3,2,6,5,9,4,8,7,10] => ? = 26
[1,1,0,1,1,1,0,0,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7)]
=> [3,7,2,8,9,10,6,5,4,1] => [1,3,2,7,6,10,4,8,5,9] => ? = 28
[1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [4,5,6,3,2,1,8,7,10,9] => [1,4,3,6,2,5,7,8,9,10] => ? = 5
[1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [4,5,6,3,2,1,9,10,8,7] => [1,4,3,6,2,5,7,9,8,10] => ? = 10
[1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [4,5,7,3,2,8,6,1,10,9] => [1,4,3,7,6,8,2,5,9,10] => ? = 10
[1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [4,5,7,3,2,9,6,10,8,1] => [1,4,3,7,6,9,8,10,2,5] => ? = 20
[1,1,1,0,0,1,1,0,0,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8)]
=> [4,5,8,3,2,9,10,7,6,1] => [1,4,3,8,7,10,2,5,6,9] => ? = 25
[1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [4,6,7,3,8,5,2,1,10,9] => [1,4,3,7,2,6,5,8,9,10] => ? = 13
[1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [4,6,7,3,9,5,2,10,8,1] => [1,4,3,7,2,6,5,9,8,10] => ? = 26
[1,1,1,0,1,0,1,0,0,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [4,6,8,3,9,5,10,7,2,1] => [1,4,3,8,7,10,2,6,5,9] => ? = 34
Description
The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise.
Matching statistic: St000358
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 24%
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 24%
Values
[1,0]
=> [(1,2)]
=> [2,1] => [1,2] => 0 = 1 - 1
[1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [1,2,3,4] => 0 = 1 - 1
[1,1,0,0]
=> [(1,4),(2,3)]
=> [4,3,2,1] => [1,4,2,3] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [1,2,3,4,5,6] => 0 = 1 - 1
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [1,2,3,6,4,5] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [1,4,2,3,5,6] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [1,6,2,3,4,5] => 3 = 4 - 1
[1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [1,6,2,5,3,4] => 4 = 5 - 1
[1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [1,2,3,4,5,6,7,8] => ? = 1 - 1
[1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,8,7,6,5] => [1,2,3,4,5,8,6,7] => ? = 2 - 1
[1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => [1,2,3,6,4,5,7,8] => ? = 2 - 1
[1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [1,2,3,8,4,5,6,7] => ? = 4 - 1
[1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,8,7,6,5,4,3] => [1,2,3,8,4,7,5,6] => ? = 5 - 1
[1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => [1,4,2,3,5,6,7,8] => ? = 2 - 1
[1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [4,3,2,1,8,7,6,5] => [1,4,2,3,5,8,6,7] => ? = 4 - 1
[1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [1,6,2,3,4,5,7,8] => ? = 4 - 1
[1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => [1,8,2,3,4,5,6,7] => ? = 8 - 1
[1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [8,3,2,7,6,5,4,1] => [1,8,2,3,4,7,5,6] => ? = 10 - 1
[1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [1,6,2,5,3,4,7,8] => ? = 5 - 1
[1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => [1,8,2,5,3,4,6,7] => ? = 10 - 1
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [1,8,2,7,3,4,5,6] => ? = 13 - 1
[1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => [1,8,2,7,3,6,4,5] => ? = 14 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => [1,2,3,4,5,6,7,8,9,10] => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,10,9,8,7] => [1,2,3,4,5,6,7,10,8,9] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [2,1,4,3,8,7,6,5,10,9] => [1,2,3,4,5,8,6,7,9,10] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,10,7,6,9,8,5] => [1,2,3,4,5,10,6,7,8,9] => ? = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,10,9,8,7,6,5] => [1,2,3,4,5,10,6,9,7,8] => ? = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [2,1,6,5,4,3,8,7,10,9] => [1,2,3,6,4,5,7,8,9,10] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> [2,1,6,5,4,3,10,9,8,7] => [1,2,3,6,4,5,7,10,8,9] => ? = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [2,1,8,5,4,7,6,3,10,9] => [1,2,3,8,4,5,6,7,9,10] => ? = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,10,5,4,7,6,9,8,3] => [1,2,3,10,4,5,6,7,8,9] => ? = 8 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,10,5,4,9,8,7,6,3] => [1,2,3,10,4,5,6,9,7,8] => ? = 10 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,8,7,6,5,4,3,10,9] => [1,2,3,8,4,7,5,6,9,10] => ? = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,10,7,6,5,4,9,8,3] => [1,2,3,10,4,7,5,6,8,9] => ? = 10 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,10,9,6,5,8,7,4,3] => [1,2,3,10,4,9,5,6,7,8] => ? = 13 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,10,9,8,7,6,5,4,3] => [1,2,3,10,4,9,5,8,6,7] => ? = 14 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [4,3,2,1,6,5,8,7,10,9] => [1,4,2,3,5,6,7,8,9,10] => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> [4,3,2,1,6,5,10,9,8,7] => [1,4,2,3,5,6,7,10,8,9] => ? = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [4,3,2,1,8,7,6,5,10,9] => [1,4,2,3,5,8,6,7,9,10] => ? = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [4,3,2,1,10,7,6,9,8,5] => [1,4,2,3,5,10,6,7,8,9] => ? = 8 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [4,3,2,1,10,9,8,7,6,5] => [1,4,2,3,5,10,6,9,7,8] => ? = 10 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [6,3,2,5,4,1,8,7,10,9] => [1,6,2,3,4,5,7,8,9,10] => ? = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [6,3,2,5,4,1,10,9,8,7] => [1,6,2,3,4,5,7,10,8,9] => ? = 8 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [8,3,2,5,4,7,6,1,10,9] => [1,8,2,3,4,5,6,7,9,10] => ? = 8 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [10,3,2,5,4,7,6,9,8,1] => [1,10,2,3,4,5,6,7,8,9] => ? = 16 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> [10,3,2,5,4,9,8,7,6,1] => [1,10,2,3,4,5,6,9,7,8] => ? = 20 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [8,3,2,7,6,5,4,1,10,9] => [1,8,2,3,4,7,5,6,9,10] => ? = 10 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9)]
=> [10,3,2,7,6,5,4,9,8,1] => [1,10,2,3,4,7,5,6,8,9] => ? = 20 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [10,3,2,9,6,5,8,7,4,1] => [1,10,2,3,4,9,5,6,7,8] => ? = 26 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7)]
=> [10,3,2,9,8,7,6,5,4,1] => [1,10,2,3,4,9,5,8,6,7] => ? = 28 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [6,5,4,3,2,1,8,7,10,9] => [1,6,2,5,3,4,7,8,9,10] => ? = 5 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [6,5,4,3,2,1,10,9,8,7] => [1,6,2,5,3,4,7,10,8,9] => ? = 10 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [8,5,4,3,2,7,6,1,10,9] => [1,8,2,5,3,4,6,7,9,10] => ? = 10 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [10,5,4,3,2,7,6,9,8,1] => [1,10,2,5,3,4,6,7,8,9] => ? = 20 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8)]
=> [10,5,4,3,2,9,8,7,6,1] => [1,10,2,5,3,4,6,9,7,8] => ? = 25 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [8,7,4,3,6,5,2,1,10,9] => [1,8,2,7,3,4,5,6,9,10] => ? = 13 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [10,7,4,3,6,5,2,9,8,1] => [1,10,2,7,3,4,5,6,8,9] => ? = 26 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [10,9,4,3,6,5,8,7,2,1] => [1,10,2,9,3,4,5,6,7,8] => ? = 34 - 1
Description
The number of occurrences of the pattern 31-2.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $31\!\!-\!\!2$.
Matching statistic: St001171
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001171: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 24%
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001171: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 24%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 3 = 4 - 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 4 = 5 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ? = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ? = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ? = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ? = 4 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ? = 5 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ? = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ? = 4 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ? = 4 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ? = 8 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ? = 10 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => ? = 5 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ? = 10 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => ? = 13 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ? = 14 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => ? = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => ? = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => ? = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => ? = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => ? = 8 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,5,3] => ? = 10 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,4,3,6] => ? = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => ? = 10 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,5,6,4,3] => ? = 13 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => ? = 14 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => ? = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => ? = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,3,2,5,6,4] => ? = 8 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,5,4] => ? = 10 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,4,2,5,6] => ? = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,4,2,6,5] => ? = 8 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,2,6] => ? = 8 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => ? = 16 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,4,6,5,2] => ? = 20 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,5,4,2,6] => ? = 10 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,5,4,6,2] => ? = 20 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,3,5,6,4,2] => ? = 26 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,6,5,4,2] => ? = 28 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => ? = 5 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,3,2,6,5] => ? = 10 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,3,5,2,6] => ? = 10 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,3,5,6,2] => ? = 20 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,4,3,6,5,2] => ? = 25 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,5,3,2,6] => ? = 13 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,4,5,3,6,2] => ? = 26 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,4,5,6,3,2] => ? = 34 - 1
Description
The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$.
Matching statistic: St001535
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001535: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 24%
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001535: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 24%
Values
[1,0]
=> [[1],[2]]
=> [[1,2]]
=> [1,2] => 0 = 1 - 1
[1,0,1,0]
=> [[1,3],[2,4]]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[1,1,0,0]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> [2,4,1,3] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 0 = 1 - 1
[1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> [[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => 3 = 4 - 1
[1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => 4 = 5 - 1
[1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 1 - 1
[1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> [[1,2],[3,4],[5,7],[6,8]]
=> [6,8,5,7,3,4,1,2] => ? = 2 - 1
[1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> [[1,2],[3,5],[4,6],[7,8]]
=> [7,8,4,6,3,5,1,2] => ? = 2 - 1
[1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [[1,2],[3,5],[4,7],[6,8]]
=> [6,8,4,7,3,5,1,2] => ? = 4 - 1
[1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> [[1,2],[3,6],[4,7],[5,8]]
=> [5,8,4,7,3,6,1,2] => ? = 5 - 1
[1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> [[1,3],[2,4],[5,6],[7,8]]
=> [7,8,5,6,2,4,1,3] => ? = 2 - 1
[1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> [[1,3],[2,4],[5,7],[6,8]]
=> [6,8,5,7,2,4,1,3] => ? = 4 - 1
[1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> [[1,3],[2,5],[4,6],[7,8]]
=> [7,8,4,6,2,5,1,3] => ? = 4 - 1
[1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [[1,3],[2,5],[4,7],[6,8]]
=> [6,8,4,7,2,5,1,3] => ? = 8 - 1
[1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> [[1,3],[2,6],[4,7],[5,8]]
=> [5,8,4,7,2,6,1,3] => ? = 10 - 1
[1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> [[1,4],[2,5],[3,6],[7,8]]
=> [7,8,3,6,2,5,1,4] => ? = 5 - 1
[1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> [[1,4],[2,5],[3,7],[6,8]]
=> [6,8,3,7,2,5,1,4] => ? = 10 - 1
[1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> [[1,4],[2,6],[3,7],[5,8]]
=> [5,8,3,7,2,6,1,4] => ? = 13 - 1
[1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8]]
=> [4,8,3,7,2,6,1,5] => ? = 14 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [9,10,7,8,5,6,3,4,1,2] => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> [[1,2],[3,4],[5,6],[7,9],[8,10]]
=> [8,10,7,9,5,6,3,4,1,2] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,6,9],[2,4,7,8,10]]
=> [[1,2],[3,4],[5,7],[6,8],[9,10]]
=> [9,10,6,8,5,7,3,4,1,2] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> [[1,2],[3,4],[5,7],[6,9],[8,10]]
=> [8,10,6,9,5,7,3,4,1,2] => ? = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> [[1,2],[3,4],[5,8],[6,9],[7,10]]
=> [7,10,6,9,5,8,3,4,1,2] => ? = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [[1,3,4,7,9],[2,5,6,8,10]]
=> [[1,2],[3,5],[4,6],[7,8],[9,10]]
=> [9,10,7,8,4,6,3,5,1,2] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> [[1,2],[3,5],[4,6],[7,9],[8,10]]
=> [8,10,7,9,4,6,3,5,1,2] => ? = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [[1,3,4,6,9],[2,5,7,8,10]]
=> [[1,2],[3,5],[4,7],[6,8],[9,10]]
=> [9,10,6,8,4,7,3,5,1,2] => ? = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> [[1,2],[3,5],[4,7],[6,9],[8,10]]
=> [8,10,6,9,4,7,3,5,1,2] => ? = 8 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> [[1,2],[3,5],[4,8],[6,9],[7,10]]
=> [7,10,6,9,4,8,3,5,1,2] => ? = 10 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [[1,3,4,5,9],[2,6,7,8,10]]
=> [[1,2],[3,6],[4,7],[5,8],[9,10]]
=> [9,10,5,8,4,7,3,6,1,2] => ? = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> [[1,2],[3,6],[4,7],[5,9],[8,10]]
=> [8,10,5,9,4,7,3,6,1,2] => ? = 10 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> [[1,2],[3,6],[4,8],[5,9],[7,10]]
=> [7,10,5,9,4,8,3,6,1,2] => ? = 13 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> [[1,2],[3,7],[4,8],[5,9],[6,10]]
=> [6,10,5,9,4,8,3,7,1,2] => ? = 14 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [[1,2,5,7,9],[3,4,6,8,10]]
=> [[1,3],[2,4],[5,6],[7,8],[9,10]]
=> [9,10,7,8,5,6,2,4,1,3] => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [[1,2,5,7,8],[3,4,6,9,10]]
=> [[1,3],[2,4],[5,6],[7,9],[8,10]]
=> [8,10,7,9,5,6,2,4,1,3] => ? = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [[1,2,5,6,9],[3,4,7,8,10]]
=> [[1,3],[2,4],[5,7],[6,8],[9,10]]
=> [9,10,6,8,5,7,2,4,1,3] => ? = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [[1,2,5,6,8],[3,4,7,9,10]]
=> [[1,3],[2,4],[5,7],[6,9],[8,10]]
=> [8,10,6,9,5,7,2,4,1,3] => ? = 8 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> [[1,3],[2,4],[5,8],[6,9],[7,10]]
=> [7,10,6,9,5,8,2,4,1,3] => ? = 10 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [[1,2,4,7,9],[3,5,6,8,10]]
=> [[1,3],[2,5],[4,6],[7,8],[9,10]]
=> [9,10,7,8,4,6,2,5,1,3] => ? = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [[1,2,4,7,8],[3,5,6,9,10]]
=> [[1,3],[2,5],[4,6],[7,9],[8,10]]
=> [8,10,7,9,4,6,2,5,1,3] => ? = 8 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [[1,2,4,6,9],[3,5,7,8,10]]
=> [[1,3],[2,5],[4,7],[6,8],[9,10]]
=> [9,10,6,8,4,7,2,5,1,3] => ? = 8 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8],[3,5,7,9,10]]
=> [[1,3],[2,5],[4,7],[6,9],[8,10]]
=> [8,10,6,9,4,7,2,5,1,3] => ? = 16 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [[1,2,4,6,7],[3,5,8,9,10]]
=> [[1,3],[2,5],[4,8],[6,9],[7,10]]
=> [7,10,6,9,4,8,2,5,1,3] => ? = 20 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> [[1,3],[2,6],[4,7],[5,8],[9,10]]
=> [9,10,5,8,4,7,2,6,1,3] => ? = 10 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [[1,2,4,5,8],[3,6,7,9,10]]
=> [[1,3],[2,6],[4,7],[5,9],[8,10]]
=> [8,10,5,9,4,7,2,6,1,3] => ? = 20 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [[1,2,4,5,7],[3,6,8,9,10]]
=> [[1,3],[2,6],[4,8],[5,9],[7,10]]
=> [7,10,5,9,4,8,2,6,1,3] => ? = 26 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [[1,2,4,5,6],[3,7,8,9,10]]
=> [[1,3],[2,7],[4,8],[5,9],[6,10]]
=> [6,10,5,9,4,8,2,7,1,3] => ? = 28 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [[1,2,3,7,9],[4,5,6,8,10]]
=> [[1,4],[2,5],[3,6],[7,8],[9,10]]
=> [9,10,7,8,3,6,2,5,1,4] => ? = 5 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> [[1,4],[2,5],[3,6],[7,9],[8,10]]
=> [8,10,7,9,3,6,2,5,1,4] => ? = 10 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> [[1,4],[2,5],[3,7],[6,8],[9,10]]
=> [9,10,6,8,3,7,2,5,1,4] => ? = 10 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [[1,2,3,6,8],[4,5,7,9,10]]
=> [[1,4],[2,5],[3,7],[6,9],[8,10]]
=> [8,10,6,9,3,7,2,5,1,4] => ? = 20 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [[1,2,3,6,7],[4,5,8,9,10]]
=> [[1,4],[2,5],[3,8],[6,9],[7,10]]
=> [7,10,6,9,3,8,2,5,1,4] => ? = 25 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> [[1,4],[2,6],[3,7],[5,8],[9,10]]
=> [9,10,5,8,3,7,2,6,1,4] => ? = 13 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [[1,2,3,5,8],[4,6,7,9,10]]
=> [[1,4],[2,6],[3,7],[5,9],[8,10]]
=> [8,10,5,9,3,7,2,6,1,4] => ? = 26 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [[1,2,3,5,7],[4,6,8,9,10]]
=> [[1,4],[2,6],[3,8],[5,9],[7,10]]
=> [7,10,5,9,3,8,2,6,1,4] => ? = 34 - 1
Description
The number of cyclic alignments of a permutation.
The pair $(i,j)$ is a cyclic alignment of a permutation $\pi$ if $i, j, \pi(j), \pi(i)$ are cyclically ordered and all distinct, see Section 5 of [1]
Matching statistic: St001644
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001644: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 12%
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001644: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 12%
Values
[1,0]
=> [(1,2)]
=> [2,1] => ([(0,1)],2)
=> 1
[1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 1
[1,1,0,0]
=> [(1,4),(2,3)]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6)
=> 1
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 4
[1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => ([(0,7),(1,6),(2,5),(3,4)],8)
=> 1
[1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => ([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => ([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => ([(0,1),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => ([(0,1),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => ([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => ([(0,6),(0,7),(1,4),(1,5),(2,4),(2,5),(3,6),(3,7),(4,5),(6,7)],8)
=> ? = 4
[1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ([(0,1),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 4
[1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 8
[1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => ([(0,3),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 10
[1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => ([(0,1),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => ([(0,3),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 10
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => ([(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 13
[1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 14
[1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => ([(0,9),(1,8),(2,7),(3,6),(4,5)],10)
=> ? = 1
[1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,9,10,8,7] => ([(0,5),(1,4),(2,3),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 2
[1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [2,1,4,3,7,8,6,5,10,9] => ([(0,5),(1,4),(2,3),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 2
[1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,7,9,6,10,8,5] => ([(0,3),(1,2),(4,8),(4,9),(5,7),(5,9),(6,7),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4
[1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,8,9,10,7,6,5] => ([(0,3),(1,2),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [2,1,5,6,4,3,8,7,10,9] => ([(0,5),(1,4),(2,3),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 2
[1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> [2,1,5,6,4,3,9,10,8,7] => ([(0,1),(2,8),(2,9),(3,6),(3,7),(4,6),(4,7),(5,8),(5,9),(6,7),(8,9)],10)
=> ? = 4
[1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [2,1,5,7,4,8,6,3,10,9] => ([(0,3),(1,2),(4,8),(4,9),(5,7),(5,9),(6,7),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4
[1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,5,7,4,9,6,10,8,3] => ([(0,1),(2,8),(2,9),(3,7),(3,9),(4,5),(4,6),(4,9),(5,7),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 8
[1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,5,8,4,9,10,7,6,3] => ([(0,1),(2,5),(2,9),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 10
[1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,6,7,8,5,4,3,10,9] => ([(0,3),(1,2),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,6,7,9,5,4,10,8,3] => ([(0,1),(2,5),(2,9),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 10
[1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,6,8,9,5,10,7,4,3] => ([(0,1),(2,7),(2,8),(2,9),(3,6),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 13
[1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,7,8,9,10,6,5,4,3] => ([(0,1),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 14
[1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [3,4,2,1,6,5,8,7,10,9] => ([(0,5),(1,4),(2,3),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 2
[1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> [3,4,2,1,6,5,9,10,8,7] => ([(0,1),(2,8),(2,9),(3,6),(3,7),(4,6),(4,7),(5,8),(5,9),(6,7),(8,9)],10)
=> ? = 4
[1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [3,4,2,1,7,8,6,5,10,9] => ([(0,1),(2,8),(2,9),(3,6),(3,7),(4,6),(4,7),(5,8),(5,9),(6,7),(8,9)],10)
=> ? = 4
[1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [3,4,2,1,7,9,6,10,8,5] => ([(0,7),(0,8),(1,7),(1,8),(2,6),(2,9),(3,5),(3,9),(4,5),(4,6),(4,9),(5,9),(6,9),(7,8)],10)
=> ? = 8
[1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [3,4,2,1,8,9,10,7,6,5] => ([(0,5),(0,6),(1,5),(1,6),(2,7),(2,8),(2,9),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(7,8),(7,9),(8,9)],10)
=> ? = 10
[1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [3,5,2,6,4,1,8,7,10,9] => ([(0,3),(1,2),(4,8),(4,9),(5,7),(5,9),(6,7),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4
[1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [3,5,2,6,4,1,9,10,8,7] => ([(0,7),(0,8),(1,7),(1,8),(2,6),(2,9),(3,5),(3,9),(4,5),(4,6),(4,9),(5,9),(6,9),(7,8)],10)
=> ? = 8
[1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [3,5,2,7,4,8,6,1,10,9] => ([(0,1),(2,8),(2,9),(3,7),(3,9),(4,5),(4,6),(4,9),(5,7),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 8
[1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [3,5,2,7,4,9,6,10,8,1] => ([(0,8),(0,9),(1,7),(1,9),(2,3),(2,4),(2,9),(3,5),(3,9),(4,6),(4,9),(5,7),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 16
[1,1,0,1,0,1,1,0,0,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> [3,5,2,8,4,9,10,7,6,1] => ([(0,5),(0,9),(1,4),(1,5),(1,9),(2,7),(2,8),(2,9),(3,7),(3,8),(3,9),(4,6),(4,9),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 20
[1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [3,6,2,7,8,5,4,1,10,9] => ([(0,1),(2,5),(2,9),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 10
[1,1,0,1,1,0,0,1,0,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9)]
=> [3,6,2,7,9,5,4,10,8,1] => ([(0,4),(0,9),(1,3),(1,9),(2,7),(2,8),(2,9),(3,5),(3,9),(4,6),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 20
[1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [3,6,2,8,9,5,10,7,4,1] => ([(0,2),(0,9),(1,7),(1,8),(1,9),(2,5),(2,9),(3,6),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 26
[1,1,0,1,1,1,0,0,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7)]
=> [3,7,2,8,9,10,6,5,4,1] => ([(0,1),(0,9),(1,5),(1,9),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 28
[1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [4,5,6,3,2,1,8,7,10,9] => ([(0,3),(1,2),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [4,5,6,3,2,1,9,10,8,7] => ([(0,5),(0,6),(1,5),(1,6),(2,7),(2,8),(2,9),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(7,8),(7,9),(8,9)],10)
=> ? = 10
[1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [4,5,7,3,2,8,6,1,10,9] => ([(0,1),(2,5),(2,9),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 10
[1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [4,5,7,3,2,9,6,10,8,1] => ([(0,5),(0,9),(1,4),(1,5),(1,9),(2,7),(2,8),(2,9),(3,7),(3,8),(3,9),(4,6),(4,9),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 20
[1,1,1,0,0,1,1,0,0,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8)]
=> [4,5,8,3,2,9,10,7,6,1] => ([(0,7),(0,8),(0,9),(1,5),(1,6),(1,9),(2,5),(2,6),(2,9),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,9),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 25
[1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [4,6,7,3,8,5,2,1,10,9] => ([(0,1),(2,7),(2,8),(2,9),(3,6),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 13
[1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [4,6,7,3,9,5,2,10,8,1] => ([(0,2),(0,9),(1,7),(1,8),(1,9),(2,5),(2,9),(3,6),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 26
Description
The dimension of a graph.
The dimension of a graph is the least integer $n$ such that there exists a representation of the graph in the Euclidean space of dimension $n$ with all vertices distinct and all edges having unit length. Edges are allowed to intersect, however.
Matching statistic: St000882
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000882: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 12%
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000882: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 12%
Values
[1,0]
=> [(1,2)]
=> [2,1] => [2,1] => 1
[1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => 1
[1,1,0,0]
=> [(1,4),(2,3)]
=> [3,4,2,1] => [4,1,3,2] => 2
[1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => 1
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [2,1,6,3,5,4] => 2
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [4,1,3,2,6,5] => 2
[1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [6,5,3,1,2,4] => ? = 4
[1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [6,1,2,4,5,3] => ? = 5
[1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => ? = 1
[1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => [2,1,4,3,8,5,7,6] => ? = 2
[1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [2,1,6,3,5,4,8,7] => ? = 2
[1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [2,1,8,7,5,3,4,6] => ? = 4
[1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => [2,1,8,3,4,6,7,5] => ? = 5
[1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [4,1,3,2,6,5,8,7] => ? = 2
[1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => [4,1,3,2,8,5,7,6] => ? = 4
[1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [6,5,3,1,2,4,8,7] => ? = 4
[1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [8,5,3,7,2,1,4,6] => ? = 8
[1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => [8,6,3,1,4,2,7,5] => ? = 10
[1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [6,1,2,4,5,3,8,7] => ? = 5
[1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [8,7,2,4,5,1,3,6] => ? = 10
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [8,1,7,4,2,6,3,5] => ? = 13
[1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => [8,1,2,3,5,6,7,4] => ? = 14
[1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => ? = 1
[1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,9,10,8,7] => [2,1,4,3,6,5,10,7,9,8] => ? = 2
[1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [2,1,4,3,7,8,6,5,10,9] => [2,1,4,3,8,5,7,6,10,9] => ? = 2
[1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,7,9,6,10,8,5] => [2,1,4,3,10,9,7,5,6,8] => ? = 4
[1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,8,9,10,7,6,5] => [2,1,4,3,10,5,6,8,9,7] => ? = 5
[1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [2,1,5,6,4,3,8,7,10,9] => [2,1,6,3,5,4,8,7,10,9] => ? = 2
[1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> [2,1,5,6,4,3,9,10,8,7] => [2,1,6,3,5,4,10,7,9,8] => ? = 4
[1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [2,1,5,7,4,8,6,3,10,9] => [2,1,8,7,5,3,4,6,10,9] => ? = 4
[1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,5,7,4,9,6,10,8,3] => [2,1,10,7,5,9,4,3,6,8] => ? = 8
[1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,5,8,4,9,10,7,6,3] => [2,1,10,8,5,3,6,4,9,7] => ? = 10
[1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,6,7,8,5,4,3,10,9] => [2,1,8,3,4,6,7,5,10,9] => ? = 5
[1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,6,7,9,5,4,10,8,3] => [2,1,10,9,4,6,7,3,5,8] => ? = 10
[1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,6,8,9,5,10,7,4,3] => [2,1,10,3,9,6,4,8,5,7] => ? = 13
[1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,7,8,9,10,6,5,4,3] => [2,1,10,3,4,5,7,8,9,6] => ? = 14
[1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [3,4,2,1,6,5,8,7,10,9] => [4,1,3,2,6,5,8,7,10,9] => ? = 2
[1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> [3,4,2,1,6,5,9,10,8,7] => [4,1,3,2,6,5,10,7,9,8] => ? = 4
[1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [3,4,2,1,7,8,6,5,10,9] => [4,1,3,2,8,5,7,6,10,9] => ? = 4
[1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [3,4,2,1,7,9,6,10,8,5] => [4,1,3,2,10,9,7,5,6,8] => ? = 8
[1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [3,4,2,1,8,9,10,7,6,5] => [4,1,3,2,10,5,6,8,9,7] => ? = 10
[1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [3,5,2,6,4,1,8,7,10,9] => [6,5,3,1,2,4,8,7,10,9] => ? = 4
[1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [3,5,2,6,4,1,9,10,8,7] => [6,5,3,1,2,4,10,7,9,8] => ? = 8
[1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [3,5,2,7,4,8,6,1,10,9] => [8,5,3,7,2,1,4,6,10,9] => ? = 8
[1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [3,5,2,7,4,9,6,10,8,1] => [10,5,3,7,2,9,4,1,6,8] => ? = 16
[1,1,0,1,0,1,1,0,0,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> [3,5,2,8,4,9,10,7,6,1] => [10,5,3,8,2,1,6,4,9,7] => ? = 20
[1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [3,6,2,7,8,5,4,1,10,9] => [8,6,3,1,4,2,7,5,10,9] => ? = 10
[1,1,0,1,1,0,0,1,0,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9)]
=> [3,6,2,7,9,5,4,10,8,1] => [10,6,3,9,4,2,7,1,5,8] => ? = 20
[1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [3,6,2,8,9,5,10,7,4,1] => [10,6,3,1,9,2,4,8,5,7] => ? = 26
[1,1,0,1,1,1,0,0,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7)]
=> [3,7,2,8,9,10,6,5,4,1] => [10,7,3,1,4,5,2,8,9,6] => ? = 28
[1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [4,5,6,3,2,1,8,7,10,9] => [6,1,2,4,5,3,8,7,10,9] => ? = 5
[1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [4,5,6,3,2,1,9,10,8,7] => [6,1,2,4,5,3,10,7,9,8] => ? = 10
[1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [4,5,7,3,2,8,6,1,10,9] => [8,7,2,4,5,1,3,6,10,9] => ? = 10
[1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [4,5,7,3,2,9,6,10,8,1] => [10,7,2,4,5,9,3,1,6,8] => ? = 20
[1,1,1,0,0,1,1,0,0,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8)]
=> [4,5,8,3,2,9,10,7,6,1] => [10,8,2,4,5,1,6,3,9,7] => ? = 25
[1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [4,6,7,3,8,5,2,1,10,9] => [8,1,7,4,2,6,3,5,10,9] => ? = 13
Description
The number of connected components of short braid edges in the graph of braid moves of a permutation.
Given a permutation $\pi$, let $\operatorname{Red}(\pi)$ denote the set of reduced words for $\pi$ in terms of simple transpositions $s_i = (i,i+1)$. We now say that two reduced words are connected by a short braid move if they are obtained from each other by a modification of the form $s_i s_j \leftrightarrow s_j s_i$ for $|i-j| > 1$ as a consecutive subword of a reduced word.
For example, the two reduced words $s_1s_3s_2$ and $s_3s_1s_2$ for
$$(1243) = (12)(34)(23) = (34)(12)(23)$$
share an edge because they are obtained from each other by interchanging $s_1s_3 \leftrightarrow s_3s_1$.
This statistic counts the number connected components of such short braid moves among all reduced words.
The following 1 statistic also match your data. Click on any of them to see the details.
St001330The hat guessing number of a graph.
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