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Your data matches 35 different statistics following compositions of up to 3 maps.
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Matching statistic: St000418
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St000418: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00118: Dyck paths —swap returns and last descent⟶ 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]
=> [1,1,0,0,1,0]
=> 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 4
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 5
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 8
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 4
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 10
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 4
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 10
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 5
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 13
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 4
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 14
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 5
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 16
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 8
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> 20
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> 8
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 20
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> 10
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 26
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> 8
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 28
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 10
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> 20
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> 10
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 25
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 10
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 5
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 26
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 13
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 34
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 8
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 37
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> 10
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 2
Description
The number of Dyck paths that are weakly below a Dyck path.
Matching statistic: St001832
(load all 3 compositions to match this statistic)
(load all 3 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
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St000421: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00118: Dyck paths —swap returns and last descent⟶ 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]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 7 = 8 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 9 = 10 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 9 = 10 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 12 = 13 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 13 = 14 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 15 = 16 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 7 = 8 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> 19 = 20 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> 7 = 8 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 19 = 20 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> 9 = 10 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 25 = 26 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> 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]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 27 = 28 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 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]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> 19 = 20 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> 9 = 10 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 24 = 25 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 9 = 10 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 25 = 26 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 12 = 13 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 33 = 34 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 7 = 8 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 36 = 37 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> 9 = 10 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 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] => ? ∊ {5,5,10,10,13,14}
[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] => ? ∊ {5,5,10,10,13,14}
[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] => ? ∊ {5,5,10,10,13,14}
[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] => ? ∊ {5,5,10,10,13,14}
[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] => ? ∊ {5,5,10,10,13,14}
[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] => ? ∊ {5,5,10,10,13,14}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
Description
The number of longest increasing subsequences of a permutation.
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: 12% ●values known / values provided: 23%●distinct values known / distinct values provided: 12%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 12% ●values known / values provided: 23%●distinct values known / distinct values provided: 12%
Values
[1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 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]
=> ? ∊ {4,5}
[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
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {4,5}
[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
[1,1,1,0,0,0]
=> [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
[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]
=> ? ∊ {4,4,4,5,5,8,10,10,13,14}
[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]
=> ? ∊ {4,4,4,5,5,8,10,10,13,14}
[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]
=> ? ∊ {4,4,4,5,5,8,10,10,13,14}
[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]
=> ? ∊ {4,4,4,5,5,8,10,10,13,14}
[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
[1,1,0,0,1,0,1,0]
=> [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]
=> ? ∊ {4,4,4,5,5,8,10,10,13,14}
[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]
=> ? ∊ {4,4,4,5,5,8,10,10,13,14}
[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]
=> ? ∊ {4,4,4,5,5,8,10,10,13,14}
[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]
=> ? ∊ {4,4,4,5,5,8,10,10,13,14}
[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
[1,1,1,0,0,0,1,0]
=> [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]
=> ? ∊ {4,4,4,5,5,8,10,10,13,14}
[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]
=> ? ∊ {4,4,4,5,5,8,10,10,13,14}
[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
[1,1,1,1,0,0,0,0]
=> [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
[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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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
[1,1,0,0,1,0,1,0,1,0]
=> [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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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
[1,1,1,0,0,0,1,0,1,0]
=> [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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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
[1,1,1,1,0,0,0,0,1,0]
=> [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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> ? ∊ {4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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
[1,1,1,1,1,0,0,0,0,0]
=> [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
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000123
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000123: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 29%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000123: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 29%
Values
[1,0]
=> [[1],[2]]
=> [2,1] => [2,1] => 0 = 1 - 1
[1,0,1,0]
=> [[1,3],[2,4]]
=> [2,4,1,3] => [4,2,1,3] => 1 = 2 - 1
[1,1,0,0]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [6,4,2,1,3,5] => 4 = 5 - 1
[1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [5,2,1,6,3,4] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [6,3,1,4,2,5] => 3 = 4 - 1
[1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [5,1,6,3,2,4] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> [2,4,6,8,1,3,5,7] => [8,6,4,2,1,3,5,7] => 9 = 10 - 1
[1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> [2,4,7,8,1,3,5,6] => [2,7,1,8,4,3,5,6] => ? ∊ {2,2,2,4,4,4,5,5,8,10,13,14} - 1
[1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> [2,5,6,8,1,3,4,7] => [8,5,2,1,6,3,4,7] => ? ∊ {2,2,2,4,4,4,5,5,8,10,13,14} - 1
[1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [2,5,7,8,1,3,4,6] => [7,2,1,8,5,3,4,6] => ? ∊ {2,2,2,4,4,4,5,5,8,10,13,14} - 1
[1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> [2,6,7,8,1,3,4,5] => [6,2,1,7,3,8,4,5] => ? ∊ {2,2,2,4,4,4,5,5,8,10,13,14} - 1
[1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> [3,4,6,8,1,2,5,7] => [8,6,3,1,4,2,5,7] => ? ∊ {2,2,2,4,4,4,5,5,8,10,13,14} - 1
[1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> [3,4,7,8,1,2,5,6] => [7,3,1,8,4,2,5,6] => ? ∊ {2,2,2,4,4,4,5,5,8,10,13,14} - 1
[1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> [3,5,6,8,1,2,4,7] => [8,5,1,6,3,2,4,7] => ? ∊ {2,2,2,4,4,4,5,5,8,10,13,14} - 1
[1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [3,5,7,8,1,2,4,6] => [7,1,8,5,3,2,4,6] => ? ∊ {2,2,2,4,4,4,5,5,8,10,13,14} - 1
[1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> [3,6,7,8,1,2,4,5] => [6,1,7,3,2,8,4,5] => ? ∊ {2,2,2,4,4,4,5,5,8,10,13,14} - 1
[1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> [4,5,6,8,1,2,3,7] => [8,4,1,5,2,6,3,7] => ? ∊ {2,2,2,4,4,4,5,5,8,10,13,14} - 1
[1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> [4,5,7,8,1,2,3,6] => [7,1,8,4,2,5,3,6] => ? ∊ {2,2,2,4,4,4,5,5,8,10,13,14} - 1
[1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> [4,6,7,8,1,2,3,5] => [6,1,7,2,8,4,3,5] => ? ∊ {2,2,2,4,4,4,5,5,8,10,13,14} - 1
[1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [5,1,6,2,7,3,8,4] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> [2,4,6,8,10,1,3,5,7,9] => [10,8,6,4,2,1,3,5,7,9] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> [2,4,6,9,10,1,3,5,7,8] => [9,10,2,1,6,4,3,5,7,8] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,6,9],[2,4,7,8,10]]
=> [2,4,7,8,10,1,3,5,6,9] => [7,10,2,1,8,4,3,5,6,9] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> [2,4,7,9,10,1,3,5,6,8] => [2,9,1,10,7,4,3,5,6,8] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> [2,4,8,9,10,1,3,5,6,7] => [2,8,1,9,4,3,10,5,6,7] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,0,1,1,0,0,1,0,1,0]
=> [[1,3,4,7,9],[2,5,6,8,10]]
=> [2,5,6,8,10,1,3,4,7,9] => [10,8,5,2,1,6,3,4,7,9] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> [2,5,6,9,10,1,3,4,7,8] => [9,5,2,1,10,6,3,4,7,8] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,0,1,1,0,1,0,0,1,0]
=> [[1,3,4,6,9],[2,5,7,8,10]]
=> [2,5,7,8,10,1,3,4,6,9] => [10,7,2,1,8,5,3,4,6,9] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> [2,5,7,9,10,1,3,4,6,8] => [9,2,1,10,7,5,3,4,6,8] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> [2,5,8,9,10,1,3,4,6,7] => [2,8,1,9,3,10,5,4,6,7] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,0,1,1,1,0,0,0,1,0]
=> [[1,3,4,5,9],[2,6,7,8,10]]
=> [2,6,7,8,10,1,3,4,5,9] => [10,6,2,1,7,3,8,4,5,9] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> [2,6,7,9,10,1,3,4,5,8] => [9,2,1,10,6,3,7,4,5,8] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> [2,6,8,9,10,1,3,4,5,7] => [8,2,1,9,3,10,6,4,5,7] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> [2,7,8,9,10,1,3,4,5,6] => [7,2,1,8,3,9,4,10,5,6] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,1,0,0,1,0,1,0,1,0]
=> [[1,2,5,7,9],[3,4,6,8,10]]
=> [3,4,6,8,10,1,2,5,7,9] => [10,8,6,3,1,4,2,5,7,9] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,1,0,0,1,0,1,1,0,0]
=> [[1,2,5,7,8],[3,4,6,9,10]]
=> [3,4,6,9,10,1,2,5,7,8] => [3,9,1,10,6,4,2,5,7,8] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,1,0,0,1,1,0,0,1,0]
=> [[1,2,5,6,9],[3,4,7,8,10]]
=> [3,4,7,8,10,1,2,5,6,9] => [10,7,3,1,8,4,2,5,6,9] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,1,0,0,1,1,0,1,0,0]
=> [[1,2,5,6,8],[3,4,7,9,10]]
=> [3,4,7,9,10,1,2,5,6,8] => [9,3,1,10,7,4,2,5,6,8] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> [3,4,8,9,10,1,2,5,6,7] => [8,3,1,9,4,2,10,5,6,7] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,1,0,1,0,0,1,0,1,0]
=> [[1,2,4,7,9],[3,5,6,8,10]]
=> [3,5,6,8,10,1,2,4,7,9] => [10,8,5,1,6,3,2,4,7,9] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,1,0,1,0,0,1,1,0,0]
=> [[1,2,4,7,8],[3,5,6,9,10]]
=> [3,5,6,9,10,1,2,4,7,8] => [9,5,1,10,6,3,2,4,7,8] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,1,0,1,0,1,0,0,1,0]
=> [[1,2,4,6,9],[3,5,7,8,10]]
=> [3,5,7,8,10,1,2,4,6,9] => [10,7,1,8,5,3,2,4,6,9] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8],[3,5,7,9,10]]
=> [3,5,7,9,10,1,2,4,6,8] => [9,1,10,7,5,3,2,4,6,8] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,1,0,1,0,1,1,0,0,0]
=> [[1,2,4,6,7],[3,5,8,9,10]]
=> [3,5,8,9,10,1,2,4,6,7] => [8,3,1,9,2,10,5,4,6,7] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> [3,6,7,8,10,1,2,4,5,9] => [10,6,1,7,3,2,8,4,5,9] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,1,0,1,1,0,0,1,0,0]
=> [[1,2,4,5,8],[3,6,7,9,10]]
=> [3,6,7,9,10,1,2,4,5,8] => [9,1,10,6,3,2,7,4,5,8] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,1,0,1,1,0,1,0,0,0]
=> [[1,2,4,5,7],[3,6,8,9,10]]
=> [3,6,8,9,10,1,2,4,5,7] => [8,1,9,3,2,10,6,4,5,7] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,1,0,1,1,1,0,0,0,0]
=> [[1,2,4,5,6],[3,7,8,9,10]]
=> [3,7,8,9,10,1,2,4,5,6] => [7,1,8,3,2,9,4,10,5,6] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,1,1,0,0,0,1,0,1,0]
=> [[1,2,3,7,9],[4,5,6,8,10]]
=> [4,5,6,8,10,1,2,3,7,9] => [10,8,4,1,5,2,6,3,7,9] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> [4,5,6,9,10,1,2,3,7,8] => [9,4,1,10,5,2,6,3,7,8] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> [4,5,7,8,10,1,2,3,6,9] => [10,7,1,8,4,2,5,3,6,9] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,1,1,0,0,1,0,1,0,0]
=> [[1,2,3,6,8],[4,5,7,9,10]]
=> [4,5,7,9,10,1,2,3,6,8] => [9,1,10,7,4,2,5,3,6,8] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,1,1,0,0,1,1,0,0,0]
=> [[1,2,3,6,7],[4,5,8,9,10]]
=> [4,5,8,9,10,1,2,3,6,7] => [8,1,9,4,2,10,5,3,6,7] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> [4,6,7,8,10,1,2,3,5,9] => [10,6,1,7,2,8,4,3,5,9] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,1,1,0,1,0,0,1,0,0]
=> [[1,2,3,5,8],[4,6,7,9,10]]
=> [4,6,7,9,10,1,2,3,5,8] => [9,1,10,6,2,7,4,3,5,8] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,1,1,0,1,0,1,0,0,0]
=> [[1,2,3,5,7],[4,6,8,9,10]]
=> [4,6,8,9,10,1,2,3,5,7] => [8,1,9,2,10,6,4,3,5,7] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,1,1,0,1,1,0,0,0,0]
=> [[1,2,3,5,6],[4,7,8,9,10]]
=> [4,7,8,9,10,1,2,3,5,6] => [7,1,8,2,9,4,3,10,5,6] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> [5,6,7,8,10,1,2,3,4,9] => [10,5,1,6,2,7,3,8,4,9] => ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
Description
The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map.
The Simion-Schmidt map takes a permutation and turns each occcurrence of [3,2,1] into an occurrence of [3,1,2], thus reducing the number of inversions of the permutation. This statistic records the difference in length of the permutation and its image.
Apparently, this statistic can be described as the number of occurrences of the mesh pattern ([3,2,1], {(0,3),(0,2)}). Equivalent mesh patterns are ([3,2,1], {(0,2),(1,2)}), ([3,2,1], {(0,3),(1,3)}) and ([3,2,1], {(1,2),(1,3)}).
Matching statistic: St000440
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000440: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 24%
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000440: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 24%
Values
[1,0]
=> [(1,2)]
=> [2,1] => [2,1] => 0 = 1 - 1
[1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[1,1,0,0]
=> [(1,4),(2,3)]
=> [3,4,2,1] => [4,1,3,2] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => 0 = 1 - 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] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [4,1,3,2,6,5] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [6,1,3,2,5,4] => 4 = 5 - 1
[1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [5,2,6,1,4,3] => 3 = 4 - 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] => [2,1,4,3,6,5,8,7] => 0 = 1 - 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] => ? ∊ {2,2,2,4,4,4,5,5,8,10,10,13,14} - 1
[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] => ? ∊ {2,2,2,4,4,4,5,5,8,10,10,13,14} - 1
[1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [2,1,8,3,5,4,7,6] => ? ∊ {2,2,2,4,4,4,5,5,8,10,10,13,14} - 1
[1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => [2,1,7,4,8,3,6,5] => ? ∊ {2,2,2,4,4,4,5,5,8,10,10,13,14} - 1
[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] => ? ∊ {2,2,2,4,4,4,5,5,8,10,10,13,14} - 1
[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] => ? ∊ {2,2,2,4,4,4,5,5,8,10,10,13,14} - 1
[1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [6,1,3,2,5,4,8,7] => ? ∊ {2,2,2,4,4,4,5,5,8,10,10,13,14} - 1
[1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [8,1,3,2,5,4,7,6] => ? ∊ {2,2,2,4,4,4,5,5,8,10,10,13,14} - 1
[1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => [7,4,8,1,3,2,6,5] => ? ∊ {2,2,2,4,4,4,5,5,8,10,10,13,14} - 1
[1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [5,2,6,1,4,3,8,7] => ? ∊ {2,2,2,4,4,4,5,5,8,10,10,13,14} - 1
[1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [5,2,8,1,4,3,7,6] => ? ∊ {2,2,2,4,4,4,5,5,8,10,10,13,14} - 1
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [8,1,4,3,7,2,6,5] => ? ∊ {2,2,2,4,4,4,5,5,8,10,10,13,14} - 1
[1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => [7,2,6,3,8,1,5,4] => ? ∊ {2,2,2,4,4,4,5,5,8,10,10,13,14} - 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] => [2,1,4,3,6,5,8,7,10,9] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,9,10,8,7] => [2,1,4,3,6,5,10,7,9,8] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [2,1,4,3,7,8,6,5,10,9] => [2,1,4,3,8,5,7,6,10,9] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,7,9,6,10,8,5] => [2,1,4,3,10,5,7,6,9,8] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,8,9,10,7,6,5] => [2,1,4,3,9,6,10,5,8,7] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[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] => [2,1,8,3,5,4,7,6,10,9] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[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] => [2,1,10,3,5,4,7,6,9,8] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[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] => [2,1,9,6,10,3,5,4,8,7] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[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] => [2,1,7,4,8,3,6,5,10,9] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[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] => [2,1,7,4,10,3,6,5,9,8] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[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] => [2,1,10,3,6,5,9,4,8,7] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[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] => [2,1,9,4,8,5,10,3,7,6] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[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] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[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] => [4,1,3,2,10,5,7,6,9,8] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[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] => [4,1,3,2,9,6,10,5,8,7] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[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] => [6,1,3,2,5,4,8,7,10,9] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[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] => [6,1,3,2,5,4,10,7,9,8] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[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] => [8,1,3,2,5,4,7,6,10,9] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[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] => [10,1,3,2,5,4,7,6,9,8] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[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] => [9,6,10,1,3,2,5,4,8,7] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[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] => [7,4,8,1,3,2,6,5,10,9] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[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] => [7,4,10,1,3,2,6,5,9,8] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[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] => [10,1,3,2,6,5,9,4,8,7] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[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] => [9,4,8,5,10,1,3,2,7,6] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[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] => [5,2,6,1,4,3,8,7,10,9] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[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] => [5,2,6,1,4,3,10,7,9,8] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[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] => [5,2,8,1,4,3,7,6,10,9] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[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] => [5,2,10,1,4,3,7,6,9,8] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[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] => [5,2,9,6,10,1,4,3,8,7] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[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] => [8,1,4,3,7,2,6,5,10,9] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[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] => [10,1,4,3,7,2,6,5,9,8] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[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] => [9,2,6,5,10,1,4,3,8,7] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[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] => [10,1,4,3,8,5,9,2,7,6] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
[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] => [7,2,6,3,8,1,5,4,10,9] => ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42} - 1
Description
The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation.
There is a bijection between permutations avoiding these two pattern and Schröder paths [1,2].
Matching statistic: St000681
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 18%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 18%
Values
[1,0]
=> [1,0]
=> [1,0]
=> []
=> ? = 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {1,2}
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {1,2}
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {1,2,2,4,5}
[1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {1,2,2,4,5}
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {1,2,2,4,5}
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {1,2,2,4,5}
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {1,2,2,4,5}
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,2,2,4,4,4,5,5,8,10,10,13,14}
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,2,2,4,4,4,5,5,8,10,10,13,14}
[1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,2,2,4,4,4,5,5,8,10,10,13,14}
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,2,2,4,4,4,5,5,8,10,10,13,14}
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,2,2,4,4,4,5,5,8,10,10,13,14}
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,2,2,4,4,4,5,5,8,10,10,13,14}
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,2,2,4,4,4,5,5,8,10,10,13,14}
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,2,2,4,4,4,5,5,8,10,10,13,14}
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,2,2,4,4,4,5,5,8,10,10,13,14}
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,2,2,4,4,4,5,5,8,10,10,13,14}
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,2,2,4,4,4,5,5,8,10,10,13,14}
[1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,2,2,4,4,4,5,5,8,10,10,13,14}
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,2,2,4,4,4,5,5,8,10,10,13,14}
[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]
=> 2
[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]
=> []
=> ? ∊ {2,2,2,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> []
=> ? ∊ {2,2,2,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> []
=> ? ∊ {2,2,2,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> []
=> ? ∊ {2,2,2,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> []
=> ? ∊ {2,2,2,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> []
=> ? ∊ {2,2,2,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> []
=> ? ∊ {2,2,2,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> []
=> ? ∊ {2,2,2,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> []
=> ? ∊ {2,2,2,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> []
=> ? ∊ {2,2,2,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> []
=> ? ∊ {2,2,2,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> []
=> ? ∊ {2,2,2,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> []
=> ? ∊ {2,2,2,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> []
=> ? ∊ {2,2,2,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> []
=> ? ∊ {2,2,2,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> []
=> ? ∊ {2,2,2,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> []
=> ? ∊ {2,2,2,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> []
=> ? ∊ {2,2,2,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> []
=> ? ∊ {2,2,2,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> []
=> ? ∊ {2,2,2,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> []
=> ? ∊ {2,2,2,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> []
=> ? ∊ {2,2,2,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> []
=> ? ∊ {2,2,2,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> []
=> ? ∊ {2,2,2,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> []
=> ? ∊ {2,2,2,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> []
=> ? ∊ {2,2,2,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> []
=> ? ∊ {2,2,2,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> []
=> ? ∊ {2,2,2,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> []
=> ? ∊ {2,2,2,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[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]
=> 4
[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]
=> 4
[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]
=> 4
[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]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 1
Description
The Grundy value of Chomp on Ferrers diagrams.
Players take turns and choose a cell of the diagram, cutting off all cells below and to the right of this cell in English notation. The player who is left with the single cell partition looses. The traditional version is played on chocolate bars, see [1].
This statistic is the Grundy value of the partition, that is, the smallest non-negative integer which does not occur as value of a partition obtained by a single move.
Matching statistic: St000713
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000713: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 14%●distinct values known / distinct values provided: 12%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000713: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 14%●distinct values known / distinct values provided: 12%
Values
[1,0]
=> [2,1] => [2]
=> []
=> ? = 1
[1,0,1,0]
=> [3,1,2] => [3]
=> []
=> ? ∊ {1,2}
[1,1,0,0]
=> [2,3,1] => [3]
=> []
=> ? ∊ {1,2}
[1,0,1,0,1,0]
=> [4,1,2,3] => [4]
=> []
=> ? ∊ {1,2,2,4,5}
[1,0,1,1,0,0]
=> [3,1,4,2] => [4]
=> []
=> ? ∊ {1,2,2,4,5}
[1,1,0,0,1,0]
=> [2,4,1,3] => [4]
=> []
=> ? ∊ {1,2,2,4,5}
[1,1,0,1,0,0]
=> [4,3,1,2] => [4]
=> []
=> ? ∊ {1,2,2,4,5}
[1,1,1,0,0,0]
=> [2,3,4,1] => [4]
=> []
=> ? ∊ {1,2,2,4,5}
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5]
=> []
=> ? ∊ {1,2,2,2,4,4,4,5,5,8,10,13,14}
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5]
=> []
=> ? ∊ {1,2,2,2,4,4,4,5,5,8,10,13,14}
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5]
=> []
=> ? ∊ {1,2,2,2,4,4,4,5,5,8,10,13,14}
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5]
=> []
=> ? ∊ {1,2,2,2,4,4,4,5,5,8,10,13,14}
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5]
=> []
=> ? ∊ {1,2,2,2,4,4,4,5,5,8,10,13,14}
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5]
=> []
=> ? ∊ {1,2,2,2,4,4,4,5,5,8,10,13,14}
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5]
=> []
=> ? ∊ {1,2,2,2,4,4,4,5,5,8,10,13,14}
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5]
=> []
=> ? ∊ {1,2,2,2,4,4,4,5,5,8,10,13,14}
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [3,2]
=> [2]
=> 10
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5]
=> []
=> ? ∊ {1,2,2,2,4,4,4,5,5,8,10,13,14}
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5]
=> []
=> ? ∊ {1,2,2,2,4,4,4,5,5,8,10,13,14}
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5]
=> []
=> ? ∊ {1,2,2,2,4,4,4,5,5,8,10,13,14}
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5]
=> []
=> ? ∊ {1,2,2,2,4,4,4,5,5,8,10,13,14}
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5]
=> []
=> ? ∊ {1,2,2,2,4,4,4,5,5,8,10,13,14}
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6]
=> []
=> ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,13,13,14,14,16,25,26,26,28,28,34,37,37,41,42}
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6]
=> []
=> ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,13,13,14,14,16,25,26,26,28,28,34,37,37,41,42}
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [6]
=> []
=> ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,13,13,14,14,16,25,26,26,28,28,34,37,37,41,42}
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6]
=> []
=> ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,13,13,14,14,16,25,26,26,28,28,34,37,37,41,42}
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6]
=> []
=> ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,13,13,14,14,16,25,26,26,28,28,34,37,37,41,42}
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6]
=> []
=> ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,13,13,14,14,16,25,26,26,28,28,34,37,37,41,42}
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6]
=> []
=> ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,13,13,14,14,16,25,26,26,28,28,34,37,37,41,42}
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6]
=> []
=> ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,13,13,14,14,16,25,26,26,28,28,34,37,37,41,42}
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [4,2]
=> [2]
=> 10
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6]
=> []
=> ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,13,13,14,14,16,25,26,26,28,28,34,37,37,41,42}
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6]
=> []
=> ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,13,13,14,14,16,25,26,26,28,28,34,37,37,41,42}
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [6]
=> []
=> ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,13,13,14,14,16,25,26,26,28,28,34,37,37,41,42}
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6]
=> []
=> ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,13,13,14,14,16,25,26,26,28,28,34,37,37,41,42}
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6]
=> []
=> ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,13,13,14,14,16,25,26,26,28,28,34,37,37,41,42}
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [6]
=> []
=> ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,13,13,14,14,16,25,26,26,28,28,34,37,37,41,42}
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6]
=> []
=> ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,13,13,14,14,16,25,26,26,28,28,34,37,37,41,42}
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [6]
=> []
=> ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,13,13,14,14,16,25,26,26,28,28,34,37,37,41,42}
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6]
=> []
=> ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,13,13,14,14,16,25,26,26,28,28,34,37,37,41,42}
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6]
=> []
=> ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,13,13,14,14,16,25,26,26,28,28,34,37,37,41,42}
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [6]
=> []
=> ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,13,13,14,14,16,25,26,26,28,28,34,37,37,41,42}
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6]
=> []
=> ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,13,13,14,14,16,25,26,26,28,28,34,37,37,41,42}
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,2]
=> [2]
=> 10
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [3,3]
=> [3]
=> 20
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,2]
=> [2]
=> 10
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [6]
=> []
=> ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,13,13,14,14,16,25,26,26,28,28,34,37,37,41,42}
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [6]
=> []
=> ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,13,13,14,14,16,25,26,26,28,28,34,37,37,41,42}
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [3,3]
=> [3]
=> 20
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6]
=> []
=> ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,13,13,14,14,16,25,26,26,28,28,34,37,37,41,42}
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [6]
=> []
=> ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,13,13,14,14,16,25,26,26,28,28,34,37,37,41,42}
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [6]
=> []
=> ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,13,13,14,14,16,25,26,26,28,28,34,37,37,41,42}
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [6]
=> []
=> ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,13,13,14,14,16,25,26,26,28,28,34,37,37,41,42}
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [4,2]
=> [2]
=> 10
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [6]
=> []
=> ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,13,13,14,14,16,25,26,26,28,28,34,37,37,41,42}
[1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [6]
=> []
=> ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,13,13,14,14,16,25,26,26,28,28,34,37,37,41,42}
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [3,3]
=> [3]
=> 20
[1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [4,2]
=> [2]
=> 10
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [6]
=> []
=> ? ∊ {1,2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,13,13,14,14,16,25,26,26,28,28,34,37,37,41,42}
Description
The dimension of the irreducible representation of Sp(4) labelled by an integer partition.
Consider the symplectic group $Sp(2n)$. Then the integer partition $(\mu_1,\dots,\mu_k)$ of length at most $n$ corresponds to the weight vector $(\mu_1-\mu_2,\dots,\mu_{k-2}-\mu_{k-1},\mu_n,0,\dots,0)$.
For example, the integer partition $(2)$ labels the symmetric square of the vector representation, whereas the integer partition $(1,1)$ labels the second fundamental representation.
Matching statistic: St000307
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 12%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 12%
Values
[1,0]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[1,0,1,0]
=> [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1
[1,1,0,0]
=> [2,3,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,0]
=> [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? ∊ {4,5}
[1,1,0,0,1,0]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,1,0,0]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? ∊ {4,5}
[1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5,3,2,1,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? ∊ {2,2,2,4,4,4,5,5,8,10,10,13,14}
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5,4,2,1,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? ∊ {2,2,2,4,4,4,5,5,8,10,10,13,14}
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? ∊ {2,2,2,4,4,4,5,5,8,10,10,13,14}
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5,2,1,3,4] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? ∊ {2,2,2,4,4,4,5,5,8,10,10,13,14}
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,4,4,4,5,5,8,10,10,13,14}
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5,3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? ∊ {2,2,2,4,4,4,5,5,8,10,10,13,14}
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? ∊ {2,2,2,4,4,4,5,5,8,10,10,13,14}
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,2,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? ∊ {2,2,2,4,4,4,5,5,8,10,10,13,14}
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5,2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? ∊ {2,2,2,4,4,4,5,5,8,10,10,13,14}
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5,4,1,2,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {2,2,2,4,4,4,5,5,8,10,10,13,14}
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? ∊ {2,2,2,4,4,4,5,5,8,10,10,13,14}
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5,2,3,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? ∊ {2,2,2,4,4,4,5,5,8,10,10,13,14}
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,4,4,4,5,5,8,10,10,13,14}
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6,4,3,2,1,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [6,5,3,2,1,4] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6,4,5,3,2,1] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6,3,2,1,4,5] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6,5,4,2,1,3] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6,4,2,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(1,19),(2,9),(2,12),(2,19),(3,8),(3,12),(3,19),(4,6),(4,8),(4,10),(4,19),(5,6),(5,9),(5,11),(5,19),(6,13),(6,17),(6,18),(8,16),(8,17),(9,16),(9,18),(10,13),(10,17),(11,13),(11,18),(12,16),(13,15),(14,7),(15,7),(16,14),(17,14),(17,15),(18,14),(18,15),(19,16),(19,17),(19,18)],20)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6,5,3,4,2,1] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [5,3,6,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6,3,4,2,1,5] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,16),(2,8),(2,10),(2,12),(3,8),(3,9),(3,11),(4,9),(4,10),(4,13),(5,1),(5,11),(5,12),(5,13),(6,17),(8,19),(9,14),(9,19),(10,15),(10,19),(11,14),(11,16),(11,19),(12,15),(12,16),(12,19),(13,6),(13,14),(13,15),(14,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,7),(18,7),(19,18)],20)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6,5,2,1,3,4] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [6,4,5,2,1,3] => ([(0,2),(0,3),(0,4),(0,5),(1,14),(1,16),(2,7),(2,8),(2,9),(3,7),(3,10),(3,12),(4,8),(4,11),(4,12),(5,9),(5,10),(5,11),(7,17),(8,15),(8,17),(9,1),(9,15),(9,17),(10,13),(10,17),(11,13),(11,15),(12,13),(12,15),(12,17),(13,14),(13,16),(14,6),(15,14),(15,16),(16,6),(17,16)],18)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6,2,1,3,4,5] => ([(0,3),(0,4),(0,5),(1,14),(2,1),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(6,14),(7,13),(7,14),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8),(14,8)],15)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [6,5,4,3,1,2] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6,4,3,1,2,5] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [6,5,3,1,2,4] => ([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6,4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,13),(2,8),(2,9),(2,13),(3,7),(3,9),(3,13),(4,6),(4,7),(4,8),(6,15),(7,12),(7,15),(8,11),(8,12),(8,15),(9,11),(9,12),(10,5),(11,10),(11,14),(12,10),(12,14),(13,11),(13,15),(14,5),(15,14)],16)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6,3,1,2,4,5] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [6,5,4,2,3,1] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6,4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,15),(2,10),(2,12),(2,16),(3,11),(3,13),(3,15),(3,16),(4,8),(4,10),(4,11),(4,15),(5,8),(5,9),(5,13),(5,16),(6,18),(6,19),(8,14),(8,17),(8,20),(9,17),(9,21),(10,20),(10,21),(11,14),(11,20),(12,21),(13,6),(13,14),(13,17),(14,19),(15,17),(15,20),(15,21),(16,6),(16,20),(16,21),(17,18),(17,19),(18,7),(19,7),(20,18),(20,19),(21,18)],22)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,2,6,5,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,18),(1,21),(2,11),(2,12),(2,19),(3,9),(3,13),(3,19),(4,10),(4,13),(4,19),(5,10),(5,12),(5,14),(5,19),(6,1),(6,9),(6,11),(6,14),(6,19),(8,17),(8,20),(9,18),(9,21),(10,16),(10,21),(11,15),(11,18),(11,21),(12,15),(12,16),(13,21),(14,8),(14,15),(14,16),(14,18),(15,17),(15,20),(16,17),(16,20),(17,7),(18,17),(18,20),(19,16),(19,18),(19,21),(20,7),(21,20)],22)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [5,3,1,6,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(1,16),(1,17),(1,24),(2,11),(2,15),(2,17),(2,24),(3,9),(3,13),(3,15),(3,24),(4,10),(4,14),(4,16),(4,24),(5,7),(5,9),(5,11),(5,14),(5,24),(6,7),(6,10),(6,12),(6,13),(6,24),(7,21),(7,22),(7,25),(9,21),(9,25),(10,22),(10,25),(11,19),(11,21),(11,25),(12,20),(12,22),(12,25),(13,19),(13,22),(13,25),(14,20),(14,21),(14,25),(15,19),(15,25),(16,20),(16,25),(17,19),(17,20),(18,8),(19,18),(19,23),(20,18),(20,23),(21,18),(21,23),(22,18),(22,23),(23,8),(24,19),(24,20),(24,21),(24,22),(25,23)],26)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,2,6,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,19),(1,20),(2,15),(2,16),(2,19),(2,20),(3,9),(3,12),(3,13),(3,19),(4,8),(4,11),(4,13),(4,15),(4,20),(5,7),(5,11),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,16),(6,17),(7,21),(7,25),(7,26),(7,27),(8,21),(8,24),(8,26),(9,24),(9,25),(9,26),(11,18),(11,21),(11,25),(12,18),(12,25),(12,27),(13,18),(13,26),(13,27),(14,21),(14,27),(15,24),(15,25),(15,27),(16,24),(16,26),(17,24),(17,26),(17,27),(18,23),(19,24),(19,27),(20,21),(20,25),(20,26),(20,27),(21,22),(21,23),(22,10),(23,10),(24,22),(25,22),(25,23),(26,22),(26,23),(27,22),(27,23)],28)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [6,5,2,3,1,4] => ([(0,1),(0,2),(0,3),(0,5),(1,11),(1,14),(2,10),(2,13),(2,14),(3,10),(3,12),(3,14),(4,7),(4,8),(4,9),(5,4),(5,11),(5,12),(5,13),(7,17),(8,17),(8,18),(9,17),(9,18),(10,15),(11,7),(11,16),(12,8),(12,15),(12,16),(13,9),(13,15),(13,16),(14,15),(14,16),(15,18),(16,17),(16,18),(17,6),(18,6)],19)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [6,4,5,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,15),(2,7),(2,9),(2,15),(3,6),(3,7),(3,15),(4,6),(4,8),(4,15),(6,14),(7,11),(7,14),(8,12),(8,14),(9,11),(9,12),(10,5),(11,10),(11,13),(12,10),(12,13),(13,5),(14,13),(15,11),(15,12),(15,14)],16)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [5,2,4,6,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,16),(1,21),(2,8),(2,16),(2,17),(2,18),(2,21),(3,9),(3,10),(3,17),(3,18),(4,12),(4,13),(4,17),(4,21),(5,11),(5,13),(5,16),(5,18),(6,8),(6,10),(6,11),(6,12),(6,21),(8,15),(8,19),(8,20),(8,25),(9,24),(10,19),(10,20),(10,24),(11,14),(11,20),(11,25),(12,14),(12,15),(12,19),(13,14),(13,25),(14,23),(15,22),(15,23),(16,24),(16,25),(17,19),(17,24),(17,25),(18,20),(18,24),(18,25),(19,22),(19,23),(20,22),(20,23),(21,15),(21,24),(21,25),(22,7),(23,7),(24,22),(25,22),(25,23)],26)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6,2,3,1,4,5] => ([(0,1),(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,8),(2,13),(3,10),(3,12),(3,15),(4,2),(4,11),(4,12),(4,15),(5,6),(5,10),(5,11),(6,16),(7,17),(8,17),(8,18),(10,14),(10,16),(11,8),(11,14),(11,16),(12,7),(12,13),(12,14),(13,17),(13,18),(14,17),(14,18),(15,13),(15,16),(16,18),(17,9),(18,9)],19)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [6,5,4,1,2,3] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [6,4,1,2,3,5] => ([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [6,5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,7),(1,13),(2,6),(2,12),(3,1),(3,9),(3,12),(4,6),(4,9),(4,12),(6,10),(7,8),(7,11),(8,5),(9,7),(9,10),(9,13),(10,11),(11,5),(12,10),(12,13),(13,8),(13,11)],14)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [5,3,6,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,10),(1,15),(1,16),(2,11),(2,15),(2,16),(3,13),(3,15),(3,16),(4,12),(4,15),(4,16),(5,8),(5,9),(5,14),(5,20),(6,5),(6,10),(6,11),(6,12),(6,13),(8,17),(8,19),(9,17),(9,19),(10,18),(10,20),(11,14),(11,18),(11,20),(12,8),(12,18),(12,20),(13,9),(13,18),(13,20),(14,17),(14,19),(15,14),(15,20),(16,14),(16,18),(17,7),(18,19),(19,7),(20,17),(20,19)],21)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [6,3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(1,11),(1,12),(2,8),(2,9),(2,12),(3,6),(3,7),(3,9),(3,11),(4,6),(4,7),(4,8),(4,10),(6,14),(6,19),(7,13),(7,15),(7,19),(8,13),(8,19),(9,13),(9,16),(9,19),(10,14),(10,15),(10,19),(11,14),(11,15),(11,16),(12,16),(12,19),(13,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,5),(18,5),(19,17),(19,18)],20)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [6,5,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [5,2,3,6,4,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,19),(1,23),(2,8),(2,16),(2,17),(3,10),(3,11),(3,17),(4,12),(4,13),(4,16),(4,17),(5,1),(5,9),(5,11),(5,12),(5,16),(6,8),(6,9),(6,10),(6,13),(8,18),(8,20),(9,14),(9,15),(9,20),(9,23),(10,14),(10,18),(11,14),(11,19),(11,23),(12,15),(12,19),(12,23),(13,15),(13,18),(13,20),(14,21),(14,22),(15,21),(15,22),(16,19),(16,20),(16,23),(17,18),(17,23),(18,21),(19,22),(20,21),(20,22),(21,7),(22,7),(23,21),(23,22)],24)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
[1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [5,2,6,3,4,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,18),(1,23),(2,13),(2,14),(2,19),(3,11),(3,12),(3,19),(4,9),(4,10),(4,14),(4,19),(5,1),(5,9),(5,12),(5,15),(5,19),(6,10),(6,11),(6,13),(6,15),(7,21),(7,22),(9,17),(9,18),(9,23),(10,16),(10,17),(10,20),(11,20),(11,23),(12,18),(12,23),(13,16),(13,20),(14,16),(14,23),(15,7),(15,17),(15,20),(15,23),(16,22),(17,21),(17,22),(18,21),(19,18),(19,20),(19,23),(20,21),(20,22),(21,8),(22,8),(23,21),(23,22)],24)
=> ? ∊ {2,2,2,2,4,4,4,4,4,4,5,5,5,8,8,8,8,10,10,10,10,10,10,13,13,14,14,16,20,20,20,25,26,26,28,28,34,37,37,41,42}
Description
The number of rowmotion orbits of a poset.
Rowmotion is an operation on order ideals in a poset $P$. It sends an order ideal $I$ to the order ideal generated by the minimal antichain of $P \setminus I$.
The following 25 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000462The major index minus the number of excedences of a permutation. St000472The sum of the ascent bottoms of a permutation. St000538The number of even inversions of a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000961The shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St001171The 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 $K[x]/(x^n)$. St001377The major index minus the number of inversions of a permutation. St001535The number of cyclic alignments of a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000100The number of linear extensions of a poset. St000454The largest eigenvalue of a graph if it is integral. St000633The size of the automorphism group of a poset. St001644The dimension of a graph. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001330The hat guessing number of a graph. St000455The second largest eigenvalue of a graph if it is integral.
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