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Your data matches 11 different statistics following compositions of up to 3 maps.
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Matching statistic: St001956
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
St001956: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 0
[1,0,1,0]
=> 0
[1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> 3
[1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> 3
[1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,0]
=> 4
[1,0,1,1,1,0,0,0]
=> 4
[1,1,0,0,1,0,1,0]
=> 4
[1,1,0,0,1,1,0,0]
=> 5
[1,1,0,1,0,0,1,0]
=> 5
[1,1,0,1,0,1,0,0]
=> 6
[1,1,0,1,1,0,0,0]
=> 3
[1,1,1,0,0,0,1,0]
=> 0
[1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> 6
[1,0,1,0,1,0,1,1,0,0]
=> 5
[1,0,1,0,1,1,0,0,1,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> 6
[1,0,1,1,0,1,0,0,1,0]
=> 6
[1,0,1,1,0,1,0,1,0,0]
=> 7
[1,0,1,1,0,1,1,0,0,0]
=> 7
[1,0,1,1,1,0,0,0,1,0]
=> 6
[1,0,1,1,1,0,0,1,0,0]
=> 7
[1,0,1,1,1,0,1,0,0,0]
=> 7
[1,0,1,1,1,1,0,0,0,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> 7
[1,1,0,0,1,0,1,1,0,0]
=> 6
[1,1,0,0,1,1,0,0,1,0]
=> 7
[1,1,0,0,1,1,0,1,0,0]
=> 8
[1,1,0,0,1,1,1,0,0,0]
=> 8
[1,1,0,1,0,0,1,0,1,0]
=> 8
[1,1,0,1,0,0,1,1,0,0]
=> 9
[1,1,0,1,0,1,0,0,1,0]
=> 9
[1,1,0,1,0,1,0,1,0,0]
=> 10
[1,1,0,1,0,1,1,0,0,0]
=> 7
[1,1,0,1,1,0,0,0,1,0]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> 5
[1,1,0,1,1,0,1,0,0,0]
=> 5
[1,1,0,1,1,1,0,0,0,0]
=> 6
Description
The comajor index for set-valued two-row standard Young tableaux.
The comajorindex is the sum $\sum_k (n+1-k)$ over all natural descents $k$.
Bijections via bicolored Motzkin paths (with two restrictions, see [1]) give the following for Dyck paths.
Let $j$ be smallest integer such that $2j$ is a down step.
Then $k$ is a natural descent if
* $k-2\ge j$ and positions $2(k-1)-1,2(k-1)$ are a valley i.e. [0,1], or
* $k-2\ge j$ and positions $2(k-1)-1,2(k-1)$ are a peak i.e. [1,0], or
* $k-1\ge j$ and positions $2(k-1),2k-1,2k$ form [0,1,1], or
* $k=j$ and positions $2k-1,2k$ are double down i.e. [0,0], or
* $k < j$ and positions $2k-1,2k$ are a valley i.e. [0,1].
Matching statistic: St000456
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 77%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 77%
Values
[1,0]
=> [1] => [1] => ([],1)
=> ? = 0
[1,0,1,0]
=> [1,2] => [1,2] => ([],2)
=> ? = 0
[1,1,0,0]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,2,2}
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,2,2}
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,2,2}
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,1,0,0,0]
=> [3,2,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,1,2,3,3,3,4,5,5}
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,1,2,3,3,3,4,5,5}
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,1,2,3,3,3,4,5,5}
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,1,2,3,3,3,4,5,5}
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,1,2,3,3,3,4,5,5}
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,1,2,3,3,3,4,5,5}
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {0,1,2,3,3,3,4,5,5}
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,1,2,3,3,3,4,5,5}
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,1,2,3,3,3,4,5,5}
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,6,7,7,7,7,7,7,7,8,9,9,10}
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,6,7,7,7,7,7,7,7,8,9,9,10}
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,6,7,7,7,7,7,7,7,8,9,9,10}
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,6,7,7,7,7,7,7,7,8,9,9,10}
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,6,7,7,7,7,7,7,7,8,9,9,10}
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,6,7,7,7,7,7,7,7,8,9,9,10}
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,6,7,7,7,7,7,7,7,8,9,9,10}
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,6,7,7,7,7,7,7,7,8,9,9,10}
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,6,7,7,7,7,7,7,7,8,9,9,10}
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,6,7,7,7,7,7,7,7,8,9,9,10}
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,6,7,7,7,7,7,7,7,8,9,9,10}
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,6,7,7,7,7,7,7,7,8,9,9,10}
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,6,7,7,7,7,7,7,7,8,9,9,10}
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,6,7,7,7,7,7,7,7,8,9,9,10}
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,6,7,7,7,7,7,7,7,8,9,9,10}
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,6,7,7,7,7,7,7,7,8,9,9,10}
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,6,7,7,7,7,7,7,7,8,9,9,10}
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,6,7,7,7,7,7,7,7,8,9,9,10}
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,6,7,7,7,7,7,7,7,8,9,9,10}
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,6,7,7,7,7,7,7,7,8,9,9,10}
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,6,7,7,7,7,7,7,7,8,9,9,10}
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,6,7,7,7,7,7,7,7,8,9,9,10}
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,6,7,7,7,7,7,7,7,8,9,9,10}
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,6,7,7,7,7,7,7,7,8,9,9,10}
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,6,7,7,7,7,7,7,7,8,9,9,10}
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,6,7,7,7,7,7,7,7,8,9,9,10}
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,6,7,7,7,7,7,7,7,8,9,9,10}
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 8
[1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 6
[1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,6,7,7,7,7,7,7,7,8,9,9,10}
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 8
[1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ? ∊ {0,1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,14,14,15}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [1,2,3,4,6,5] => ([(4,5)],6)
=> ? ∊ {0,1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,14,14,15}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ? ∊ {0,1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,14,14,15}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,14,14,15}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ? ∊ {0,1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,14,14,15}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ? ∊ {0,1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,14,14,15}
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ? ∊ {0,1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,14,14,15}
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => [1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,14,14,15}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,5,1] => [6,2,3,4,1,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,4,6,1] => [6,2,3,5,4,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,6,4,5,1] => [6,2,3,5,1,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,5,4,1] => [6,2,3,1,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,3,5,6,1] => [6,2,4,3,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,3,6,5,1] => [6,2,4,3,1,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,5,3,4,6,1] => [6,2,4,5,3,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 8
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,6,3,4,5,1] => [6,2,4,5,1,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,6,3,5,4,1] => [6,2,5,1,4,3] => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,5,4,3,6,1] => [6,2,5,3,4,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 8
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,6,4,3,5,1] => [6,2,5,3,1,4] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,6,4,5,3,1] => [6,2,1,5,3,4] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => [6,2,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,1] => [6,3,2,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,2,4,6,5,1] => [6,3,2,4,1,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [3,2,5,4,6,1] => [6,3,2,5,4,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 8
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,6,4,5,1] => [6,3,2,5,1,4] => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,2,6,5,4,1] => [6,3,2,1,4,5] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,2,3,5,6,1] => [6,3,4,2,5,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 8
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,2,3,6,5,1] => [6,3,4,2,1,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,2,3,4,6,1] => [6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 10
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,2,3,4,5,1] => [6,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 8
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,2,3,5,4,1] => [6,3,5,1,4,2] => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 8
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,4,3,6,1] => [6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 13
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [6,2,4,3,5,1] => [6,4,5,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 11
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [6,2,4,5,3,1] => [6,4,1,5,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 8
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => [6,5,1,3,4,2] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 8
Description
The monochromatic index of a connected graph.
This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path.
For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St001232
(load all 82 compositions to match this statistic)
(load all 82 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 59%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 59%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 0
[1,0,1,0]
=> [1,2] => [1,2] => [1,0,1,0]
=> 1
[1,1,0,0]
=> [2,1] => [2,1] => [1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> ? = 3
[1,0,1,1,0,0]
=> [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 0
[1,1,1,0,0,0]
=> [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> ? ∊ {1,4,4,5,5,6}
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> ? ∊ {1,4,4,5,5,6}
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> ? ∊ {1,4,4,5,5,6}
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 4
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> ? ∊ {1,4,4,5,5,6}
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> ? ∊ {1,4,4,5,5,6}
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> ? ∊ {1,4,4,5,5,6}
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {2,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> ? ∊ {2,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> ? ∊ {2,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {2,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> ? ∊ {2,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> 6
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> ? ∊ {2,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> ? ∊ {2,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {2,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {2,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {2,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {2,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {2,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {2,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> ? ∊ {2,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> ? ∊ {2,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {2,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> ? ∊ {2,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> ? ∊ {2,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> 6
[1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {2,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 4
[1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {2,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {2,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> ? ∊ {2,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> 4
[1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {2,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,3,3,3,3,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,2,3,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,3,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [2,3,4,5,1,6] => [1,1,0,1,0,1,0,1,0,0,1,0]
=> ? ∊ {1,2,3,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [2,3,4,6,1,5] => [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? ∊ {1,2,3,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,4,5] => [2,3,4,1,6,5] => [1,1,0,1,0,1,0,0,1,1,0,0]
=> ? ∊ {1,2,3,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => [2,3,4,1,5,6] => [1,1,0,1,0,1,0,0,1,0,1,0]
=> ? ∊ {1,2,3,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => [3,4,5,2,6,1] => [1,1,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {1,2,3,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => [2,3,5,1,4,6] => [1,1,0,1,0,1,1,0,0,0,1,0]
=> ? ∊ {1,2,3,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => [2,3,6,1,4,5] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? ∊ {1,2,3,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,3,5] => [2,3,1,5,6,4] => [1,1,0,1,0,0,1,1,0,1,0,0]
=> ? ∊ {1,2,3,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,3,4,6] => [2,3,1,5,4,6] => [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? ∊ {1,2,3,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,3,6,4] => [3,4,5,6,1,2] => [1,1,1,0,1,0,1,0,1,0,0,0]
=> ? ∊ {1,2,3,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,5,6,3,4] => [2,3,1,6,4,5] => [1,1,0,1,0,0,1,1,1,0,0,0]
=> ? ∊ {1,2,3,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,3,4,5] => [2,3,1,4,6,5] => [1,1,0,1,0,0,1,0,1,1,0,0]
=> ? ∊ {1,2,3,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => [2,3,1,4,5,6] => [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? ∊ {1,2,3,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => [3,4,2,5,6,1] => [1,1,1,0,1,0,0,1,0,1,0,0]
=> ? ∊ {1,2,3,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => [3,4,2,5,1,6] => [1,1,1,0,1,0,0,1,0,0,1,0]
=> ? ∊ {1,2,3,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,2,6] => [2,5,1,3,4,6] => [1,1,0,1,1,1,0,0,0,0,1,0]
=> 7
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 8
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,6,2,4,5] => [2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,5,2,3,6] => [2,1,5,3,4,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,4,5,6,2,3] => [2,1,6,3,4,5] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => [3,2,5,1,4,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
=> 5
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,1,4,5,6,3] => [3,2,6,1,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> 6
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,3,4,6] => [3,2,1,5,4,6] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,1,5,6,3,4] => [3,2,1,6,4,5] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [2,3,1,5,6,4] => [4,2,3,6,1,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> 4
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,1,6,4,5] => [4,2,3,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [2,3,4,1,6,5] => [5,2,3,4,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,1,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,1,6,4] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,6,3] => [4,2,6,1,3,5] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 6
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,4,5,1,6,3] => [5,2,6,1,3,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,1,3,4,6] => [1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,5,1,6,3,4] => [4,2,1,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> 4
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001879
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 41%
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 41%
Values
[1,0]
=> [[1]]
=> [1] => ([],1)
=> ? = 0
[1,0,1,0]
=> [[1,0],[0,1]]
=> [1,2] => ([(0,1)],2)
=> ? ∊ {0,1}
[1,1,0,0]
=> [[0,1],[1,0]]
=> [2,1] => ([],2)
=> ? ∊ {0,1}
[1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2
[1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> ? ∊ {0,1,2,3}
[1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ? ∊ {0,1,2,3}
[1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> ? ∊ {0,1,2,3}
[1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => ([(1,2)],3)
=> ? ∊ {0,1,2,3}
[1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3
[1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ? ∊ {0,1,1,2,2,3,3,4,5,5,6}
[1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ? ∊ {0,1,1,2,2,3,3,4,5,5,6}
[1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {0,1,1,2,2,3,3,4,5,5,6}
[1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ? ∊ {0,1,1,2,2,3,3,4,5,5,6}
[1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ? ∊ {0,1,1,2,2,3,3,4,5,5,6}
[1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ? ∊ {0,1,1,2,2,3,3,4,5,5,6}
[1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {0,1,1,2,2,3,3,4,5,5,6}
[1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,1,1,2,2,3,3,4,5,5,6}
[1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ? ∊ {0,1,1,2,2,3,3,4,5,5,6}
[1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {0,1,1,2,2,3,3,4,5,5,6}
[1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? ∊ {0,1,1,2,2,3,3,4,5,5,6}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,0,1,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[1,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,0,0,1,1,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,0,1,0,0,1,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,0,1,0,1,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,0,1,1,0,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,1,0,0,0,1,0]
=> [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[1,1,1,0,1,0,0,1,0,0]
=> [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,1,0,1,0,0,0]
=> [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,1,1,0,0,0,0]
=> [[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,1,0,0,0,0,1,0]
=> [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,1,0,0,0,1,0,0]
=> [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,1,0,0,1,0,0,0]
=> [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,1,0,1,0,0,0,0]
=> [[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 7
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 7
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 7
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 7
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 8
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 7
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 7
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 7
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 8
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 7
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 7
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [[0,0,1,0,0,0],[1,0,-1,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 8
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [[0,0,0,1,0,0],[1,0,0,-1,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 7
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,2,3,5,4,6,7] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 7
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 7
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,2,3,6,4,5,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> 8
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 7
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,2,3,5,4,6,7] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 7
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 7
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,2,3,6,4,5,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> 8
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,2,5,3,4,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> 8
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,-1,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,2,3,6,4,5,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> 8
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,2,6,3,4,5,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> 9
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,3,2,4,5,6,7] => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 7
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,3,2,4,6,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 8
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Matching statistic: St000356
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 45%
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 45%
Values
[1,0]
=> [(1,2)]
=> [2,1] => [2,1] => 0
[1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => 1
[1,1,0,0]
=> [(1,4),(2,3)]
=> [4,3,2,1] => [4,3,2,1] => 0
[1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => 2
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [2,1,6,5,4,3] => 3
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [4,3,2,1,6,5] => 1
[1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [5,4,1,6,3,2] => 2
[1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => 0
[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] => 3
[1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,8,7,6,5] => [2,1,4,3,8,7,6,5] => 4
[1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => [2,1,6,5,4,3,8,7] => 4
[1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [2,1,7,6,3,8,5,4] => ? ∊ {1,2,3,4,5,6}
[1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,8,7,6,5,4,3] => [2,1,8,7,6,5,4,3] => 5
[1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => [4,3,2,1,6,5,8,7] => 2
[1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [4,3,2,1,8,7,6,5] => [4,3,2,1,8,7,6,5] => 3
[1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [5,4,1,6,3,2,8,7] => ? ∊ {1,2,3,4,5,6}
[1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => [7,6,1,5,4,8,3,2] => ? ∊ {1,2,3,4,5,6}
[1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [8,3,2,7,6,5,4,1] => [7,6,5,4,1,8,3,2] => ? ∊ {1,2,3,4,5,6}
[1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [6,5,4,3,2,1,8,7] => 1
[1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => [7,6,1,8,5,4,3,2] => ? ∊ {1,2,3,4,5,6}
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [6,5,2,1,8,7,4,3] => ? ∊ {1,2,3,4,5,6}
[1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 0
[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] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,10,9,8,7] => [2,1,4,3,6,5,10,9,8,7] => 5
[1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [2,1,4,3,8,7,6,5,10,9] => [2,1,4,3,8,7,6,5,10,9] => 5
[1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,10,7,6,9,8,5] => [2,1,4,3,9,8,5,10,7,6] => ? ∊ {2,2,3,3,3,3,3,4,4,4,5,6,6,6,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,10,9,8,7,6,5] => [2,1,4,3,10,9,8,7,6,5] => 6
[1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [2,1,6,5,4,3,8,7,10,9] => [2,1,6,5,4,3,8,7,10,9] => 5
[1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> [2,1,6,5,4,3,10,9,8,7] => [2,1,6,5,4,3,10,9,8,7] => 6
[1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [2,1,8,5,4,7,6,3,10,9] => [2,1,7,6,3,8,5,4,10,9] => ? ∊ {2,2,3,3,3,3,3,4,4,4,5,6,6,6,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,10,5,4,7,6,9,8,3] => [2,1,9,8,3,7,6,10,5,4] => ? ∊ {2,2,3,3,3,3,3,4,4,4,5,6,6,6,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,10,5,4,9,8,7,6,3] => [2,1,9,8,7,6,3,10,5,4] => ? ∊ {2,2,3,3,3,3,3,4,4,4,5,6,6,6,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,8,7,6,5,4,3,10,9] => [2,1,8,7,6,5,4,3,10,9] => 6
[1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,10,7,6,5,4,9,8,3] => [2,1,9,8,3,10,7,6,5,4] => ? ∊ {2,2,3,3,3,3,3,4,4,4,5,6,6,6,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,10,9,6,5,8,7,4,3] => [2,1,8,7,4,3,10,9,6,5] => ? ∊ {2,2,3,3,3,3,3,4,4,4,5,6,6,6,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,10,9,8,7,6,5,4,3] => [2,1,10,9,8,7,6,5,4,3] => 7
[1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [4,3,2,1,6,5,8,7,10,9] => [4,3,2,1,6,5,8,7,10,9] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> [4,3,2,1,6,5,10,9,8,7] => [4,3,2,1,6,5,10,9,8,7] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [4,3,2,1,8,7,6,5,10,9] => [4,3,2,1,8,7,6,5,10,9] => 4
[1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [4,3,2,1,10,7,6,9,8,5] => [4,3,2,1,9,8,5,10,7,6] => ? ∊ {2,2,3,3,3,3,3,4,4,4,5,6,6,6,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [4,3,2,1,10,9,8,7,6,5] => [4,3,2,1,10,9,8,7,6,5] => 5
[1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [6,3,2,5,4,1,8,7,10,9] => [5,4,1,6,3,2,8,7,10,9] => ? ∊ {2,2,3,3,3,3,3,4,4,4,5,6,6,6,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [6,3,2,5,4,1,10,9,8,7] => [5,4,1,6,3,2,10,9,8,7] => ? ∊ {2,2,3,3,3,3,3,4,4,4,5,6,6,6,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [8,3,2,5,4,7,6,1,10,9] => [7,6,1,5,4,8,3,2,10,9] => ? ∊ {2,2,3,3,3,3,3,4,4,4,5,6,6,6,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [10,3,2,5,4,7,6,9,8,1] => [9,8,1,5,4,7,6,10,3,2] => ? ∊ {2,2,3,3,3,3,3,4,4,4,5,6,6,6,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,0,1,1,0,0,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> [10,3,2,5,4,9,8,7,6,1] => [9,8,7,6,1,5,4,10,3,2] => ? ∊ {2,2,3,3,3,3,3,4,4,4,5,6,6,6,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [8,3,2,7,6,5,4,1,10,9] => [7,6,5,4,1,8,3,2,10,9] => ? ∊ {2,2,3,3,3,3,3,4,4,4,5,6,6,6,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,1,0,0,1,0,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9)]
=> [10,3,2,7,6,5,4,9,8,1] => [9,8,1,7,6,5,4,10,3,2] => ? ∊ {2,2,3,3,3,3,3,4,4,4,5,6,6,6,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [10,3,2,9,6,5,8,7,4,1] => [8,7,4,1,9,6,5,10,3,2] => ? ∊ {2,2,3,3,3,3,3,4,4,4,5,6,6,6,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,1,1,0,0,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7)]
=> [10,3,2,9,8,7,6,5,4,1] => [9,8,7,6,5,4,1,10,3,2] => ? ∊ {2,2,3,3,3,3,3,4,4,4,5,6,6,6,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [6,5,4,3,2,1,8,7,10,9] => [6,5,4,3,2,1,8,7,10,9] => 2
[1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [6,5,4,3,2,1,10,9,8,7] => [6,5,4,3,2,1,10,9,8,7] => 3
[1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [8,5,4,3,2,7,6,1,10,9] => [7,6,1,8,5,4,3,2,10,9] => ? ∊ {2,2,3,3,3,3,3,4,4,4,5,6,6,6,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [10,5,4,3,2,7,6,9,8,1] => [9,8,1,7,6,10,5,4,3,2] => ? ∊ {2,2,3,3,3,3,3,4,4,4,5,6,6,6,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,0,1,1,0,0,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8)]
=> [10,5,4,3,2,9,8,7,6,1] => [9,8,7,6,1,10,5,4,3,2] => ? ∊ {2,2,3,3,3,3,3,4,4,4,5,6,6,6,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [8,7,4,3,6,5,2,1,10,9] => [6,5,2,1,8,7,4,3,10,9] => ? ∊ {2,2,3,3,3,3,3,4,4,4,5,6,6,6,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [10,7,4,3,6,5,2,9,8,1] => [9,8,1,6,5,2,10,7,4,3] => ? ∊ {2,2,3,3,3,3,3,4,4,4,5,6,6,6,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,1,0,1,0,0,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [10,9,4,3,6,5,8,7,2,1] => [8,7,2,1,6,5,10,9,4,3] => ? ∊ {2,2,3,3,3,3,3,4,4,4,5,6,6,6,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> [10,9,4,3,8,7,6,5,2,1] => [8,7,6,5,2,1,10,9,4,3] => ? ∊ {2,2,3,3,3,3,3,4,4,4,5,6,6,6,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [8,7,6,5,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,10,9] => 1
[1,1,1,1,0,0,0,1,0,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9)]
=> [10,7,6,5,4,3,2,9,8,1] => [9,8,1,10,7,6,5,4,3,2] => ? ∊ {2,2,3,3,3,3,3,4,4,4,5,6,6,6,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,1,0,0,1,0,0,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8)]
=> [10,9,6,5,4,3,8,7,2,1] => [8,7,2,1,10,9,6,5,4,3] => ? ∊ {2,2,3,3,3,3,3,4,4,4,5,6,6,6,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> [10,9,8,5,4,7,6,3,2,1] => [7,6,3,2,1,10,9,8,5,4] => ? ∊ {2,2,3,3,3,3,3,4,4,4,5,6,6,6,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => [2,1,4,3,6,5,8,7,10,9,12,11] => 5
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> [2,1,4,3,6,5,8,7,12,11,10,9] => [2,1,4,3,6,5,8,7,12,11,10,9] => 6
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]
=> [2,1,4,3,6,5,10,9,8,7,12,11] => [2,1,4,3,6,5,10,9,8,7,12,11] => 6
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> [2,1,4,3,6,5,12,9,8,11,10,7] => [2,1,4,3,6,5,11,10,7,12,9,8] => ? ∊ {1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
=> [2,1,4,3,6,5,12,11,10,9,8,7] => [2,1,4,3,6,5,12,11,10,9,8,7] => 7
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> [2,1,4,3,8,7,6,5,10,9,12,11] => [2,1,4,3,8,7,6,5,10,9,12,11] => 6
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> [2,1,4,3,8,7,6,5,12,11,10,9] => [2,1,4,3,8,7,6,5,12,11,10,9] => 7
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9),(11,12)]
=> [2,1,4,3,10,7,6,9,8,5,12,11] => [2,1,4,3,9,8,5,10,7,6,12,11] => ? ∊ {1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
=> [2,1,4,3,12,7,6,9,8,11,10,5] => [2,1,4,3,11,10,5,9,8,12,7,6] => ? ∊ {1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,11),(9,10)]
=> [2,1,4,3,12,7,6,11,10,9,8,5] => [2,1,4,3,11,10,9,8,5,12,7,6] => ? ∊ {1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)]
=> [2,1,4,3,10,9,8,7,6,5,12,11] => [2,1,4,3,10,9,8,7,6,5,12,11] => 7
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)]
=> [2,1,4,3,12,9,8,7,6,11,10,5] => [2,1,4,3,11,10,5,12,9,8,7,6] => ? ∊ {1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10)]
=> [2,1,4,3,12,11,8,7,10,9,6,5] => [2,1,4,3,10,9,6,5,12,11,8,7] => ? ∊ {1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> [2,1,4,3,12,11,10,9,8,7,6,5] => [2,1,4,3,12,11,10,9,8,7,6,5] => 8
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
=> [2,1,6,5,4,3,8,7,10,9,12,11] => [2,1,6,5,4,3,8,7,10,9,12,11] => 6
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> [2,1,6,5,4,3,8,7,12,11,10,9] => [2,1,6,5,4,3,8,7,12,11,10,9] => 7
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> [2,1,6,5,4,3,10,9,8,7,12,11] => [2,1,6,5,4,3,10,9,8,7,12,11] => 7
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,9),(10,11)]
=> [2,1,6,5,4,3,12,9,8,11,10,7] => [2,1,6,5,4,3,11,10,7,12,9,8] => ? ∊ {1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
=> [2,1,6,5,4,3,12,11,10,9,8,7] => [2,1,6,5,4,3,12,11,10,9,8,7] => 8
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]
=> [2,1,8,5,4,7,6,3,10,9,12,11] => [2,1,7,6,3,8,5,4,10,9,12,11] => ? ∊ {1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
=> [2,1,8,5,4,7,6,3,12,11,10,9] => [2,1,7,6,3,8,5,4,12,11,10,9] => ? ∊ {1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)]
=> [2,1,10,5,4,7,6,9,8,3,12,11] => [2,1,9,8,3,7,6,10,5,4,12,11] => ? ∊ {1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
=> [2,1,12,5,4,7,6,9,8,11,10,3] => [2,1,11,10,3,7,6,9,8,12,5,4] => ? ∊ {1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,11),(9,10)]
=> [2,1,12,5,4,7,6,11,10,9,8,3] => [2,1,11,10,9,8,3,7,6,12,5,4] => ? ∊ {1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)]
=> [2,1,10,5,4,9,8,7,6,3,12,11] => [2,1,9,8,7,6,3,10,5,4,12,11] => ? ∊ {1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,9),(7,8),(10,11)]
=> [2,1,12,5,4,9,8,7,6,11,10,3] => [2,1,11,10,3,9,8,7,6,12,5,4] => ? ∊ {1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [(1,2),(3,12),(4,5),(6,11),(7,8),(9,10)]
=> [2,1,12,5,4,11,8,7,10,9,6,3] => [2,1,10,9,6,3,11,8,7,12,5,4] => ? ∊ {1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,5),(6,11),(7,10),(8,9)]
=> [2,1,12,5,4,11,10,9,8,7,6,3] => [2,1,11,10,9,8,7,6,3,12,5,4] => ? ∊ {1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10),(11,12)]
=> [2,1,8,7,6,5,4,3,10,9,12,11] => [2,1,8,7,6,5,4,3,10,9,12,11] => 7
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> [2,1,8,7,6,5,4,3,12,11,10,9] => [2,1,8,7,6,5,4,3,12,11,10,9] => 8
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9),(11,12)]
=> [2,1,10,7,6,5,4,9,8,3,12,11] => [2,1,9,8,3,10,7,6,5,4,12,11] => ? ∊ {1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
=> [2,1,12,7,6,5,4,9,8,11,10,3] => [2,1,11,10,3,9,8,12,7,6,5,4] => ? ∊ {1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7),(11,12)]
=> [2,1,10,9,8,7,6,5,4,3,12,11] => [2,1,10,9,8,7,6,5,4,3,12,11] => 8
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> [2,1,12,11,10,9,8,7,6,5,4,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 9
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)]
=> [4,3,2,1,6,5,8,7,10,9,12,11] => [4,3,2,1,6,5,8,7,10,9,12,11] => 4
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)]
=> [4,3,2,1,6,5,8,7,12,11,10,9] => [4,3,2,1,6,5,8,7,12,11,10,9] => 5
Description
The number of occurrences of the pattern 13-2.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $13\!\!-\!\!2$.
Matching statistic: St001880
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 23%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 23%
Values
[1,0]
=> [1,0]
=> [1] => ([],1)
=> ? = 0
[1,0,1,0]
=> [1,0,1,0]
=> [1,2] => ([(0,1)],2)
=> ? ∊ {0,1}
[1,1,0,0]
=> [1,1,0,0]
=> [2,1] => ([],2)
=> ? ∊ {0,1}
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> ? ∊ {0,1,2,2}
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ? ∊ {0,1,2,2}
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> ? ∊ {0,1,2,2}
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => ([(1,2)],3)
=> ? ∊ {0,1,2,2}
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ? ∊ {0,1,1,2,2,3,3,3,5,5,6}
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ? ∊ {0,1,1,2,2,3,3,3,5,5,6}
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {0,1,1,2,2,3,3,3,5,5,6}
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ? ∊ {0,1,1,2,2,3,3,3,5,5,6}
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ? ∊ {0,1,1,2,2,3,3,3,5,5,6}
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ? ∊ {0,1,1,2,2,3,3,3,5,5,6}
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {0,1,1,2,2,3,3,3,5,5,6}
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,1,1,2,2,3,3,3,5,5,6}
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ? ∊ {0,1,1,2,2,3,3,3,5,5,6}
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {0,1,1,2,2,3,3,3,5,5,6}
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? ∊ {0,1,1,2,2,3,3,3,5,5,6}
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 5
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 5
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 5
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 5
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 4
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 5
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 4
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 5
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 4
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 4
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 7
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 7
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 7
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,4,5,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> 6
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 7
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 7
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,4,5,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> 6
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,3,4,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> 6
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,4,5,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> 6
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,3,4,5,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> 5
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 7
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 7
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 7
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 7
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,3,4,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> 6
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,3,4,5,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> 5
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,4,2,3,5,6,7] => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
=> 6
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,3,4,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> 6
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,3,4,5,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> 5
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,2,3,4,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> 5
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St000186
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000186: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 18%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000186: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 18%
Values
[1,0]
=> []
=> []
=> ?
=> ? = 0
[1,0,1,0]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[1,1,0,0]
=> []
=> []
=> ?
=> ? = 0
[1,0,1,0,1,0]
=> [2,1]
=> [[1,3],[2]]
=> [[2,1,0],[1,1],[1]]
=> 3
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> [[1,1],[1]]
=> 2
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 2
[1,1,0,1,0,0]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[1,1,1,0,0,0]
=> []
=> []
=> ?
=> ? = 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {0,1,5,5,6}
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {0,1,5,5,6}
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? ∊ {0,1,5,5,6}
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> 4
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 3
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {0,1,5,5,6}
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> 4
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,3,4],[2]]
=> [[3,1,0,0],[2,1,0],[1,1],[1]]
=> 4
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,3],[2]]
=> [[2,1,0],[1,1],[1]]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [[1,1],[1]]
=> 2
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> [[3,0,0],[2,0],[1]]
=> 3
[1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 2
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[1,1,1,1,0,0,0,0]
=> []
=> []
=> ?
=> ? ∊ {0,1,5,5,6}
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,3,6,10],[2,5,9],[4,8],[7]]
=> [[4,3,2,1,0,0,0,0,0,0],[3,3,2,1,0,0,0,0,0],[3,2,2,1,0,0,0,0],[3,2,1,1,0,0,0],[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {0,2,3,3,3,3,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> [[3,3,2,1,0,0,0,0,0],[3,2,2,1,0,0,0,0],[3,2,1,1,0,0,0],[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {0,2,3,3,3,3,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> [[4,2,2,1,0,0,0,0,0],[3,2,2,1,0,0,0,0],[2,2,2,1,0,0,0],[2,2,1,1,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {0,2,3,3,3,3,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [[3,2,2,1,0,0,0,0],[2,2,2,1,0,0,0],[2,2,1,1,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {0,2,3,3,3,3,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [[2,2,2,1,0,0,0],[2,2,1,1,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {0,2,3,3,3,3,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> [[4,3,1,1,0,0,0,0,0],[3,3,1,1,0,0,0,0],[3,2,1,1,0,0,0],[3,1,1,1,0,0],[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? ∊ {0,2,3,3,3,3,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [[3,3,1,1,0,0,0,0],[3,2,1,1,0,0,0],[3,1,1,1,0,0],[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? ∊ {0,2,3,3,3,3,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [[4,2,1,1,0,0,0,0],[3,2,1,1,0,0,0],[2,2,1,1,0,0],[2,1,1,1,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? ∊ {0,2,3,3,3,3,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [[3,2,1,1,0,0,0],[2,2,1,1,0,0],[2,1,1,1,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? ∊ {0,2,3,3,3,3,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [[2,2,1,1,0,0],[2,1,1,1,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? ∊ {0,2,3,3,3,3,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [[4,1,1,1,0,0,0],[3,1,1,1,0,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? ∊ {0,2,3,3,3,3,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [[3,1,1,1,0,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? ∊ {0,2,3,3,3,3,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? ∊ {0,2,3,3,3,3,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> [[4,3,2,0,0,0,0,0,0],[3,3,2,0,0,0,0,0],[3,2,2,0,0,0,0],[3,2,1,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {0,2,3,3,3,3,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [[3,3,2,0,0,0,0,0],[3,2,2,0,0,0,0],[3,2,1,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {0,2,3,3,3,3,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [[4,2,2,0,0,0,0,0],[3,2,2,0,0,0,0],[2,2,2,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {0,2,3,3,3,3,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [[3,2,2,0,0,0,0],[2,2,2,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {0,2,3,3,3,3,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[2,2,2,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {0,2,3,3,3,3,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [[4,3,1,0,0,0,0,0],[3,3,1,0,0,0,0],[3,2,1,0,0,0],[3,1,1,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? ∊ {0,2,3,3,3,3,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [[3,3,1,0,0,0,0],[3,2,1,0,0,0],[3,1,1,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? ∊ {0,2,3,3,3,3,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [[4,2,1,0,0,0,0],[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {0,2,3,3,3,3,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {0,2,3,3,3,3,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {0,2,3,3,3,3,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [[4,1,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? ∊ {0,2,3,3,3,3,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? ∊ {0,2,3,3,3,3,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> [[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? ∊ {0,2,3,3,3,3,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? ∊ {0,2,3,3,3,3,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> [[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {0,2,3,3,3,3,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {0,2,3,3,3,3,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> 4
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,3,4,5],[2]]
=> [[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? ∊ {0,2,3,3,3,3,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,3,4],[2]]
=> [[3,1,0,0],[2,1,0],[1,1],[1]]
=> 4
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,3],[2]]
=> [[2,1,0],[1,1],[1]]
=> 3
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [[1,1],[1]]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> 4
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[1,2,3]]
=> [[3,0,0],[2,0],[1]]
=> 3
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ?
=> ? ∊ {0,2,3,3,3,3,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
=> [[5,4,3,2,1,0,0,0,0,0,0,0,0,0,0],[4,4,3,2,1,0,0,0,0,0,0,0,0,0],[4,3,3,2,1,0,0,0,0,0,0,0,0],[4,3,2,2,1,0,0,0,0,0,0,0],[4,3,2,1,1,0,0,0,0,0,0],[4,3,2,1,0,0,0,0,0,0],[3,3,2,1,0,0,0,0,0],[3,2,2,1,0,0,0,0],[3,2,1,1,0,0,0],[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {0,1,2,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,3,6,10],[2,5,9,14],[4,8,13],[7,12],[11]]
=> [[4,4,3,2,1,0,0,0,0,0,0,0,0,0],[4,3,3,2,1,0,0,0,0,0,0,0,0],[4,3,2,2,1,0,0,0,0,0,0,0],[4,3,2,1,1,0,0,0,0,0,0],[4,3,2,1,0,0,0,0,0,0],[3,3,2,1,0,0,0,0,0],[3,2,2,1,0,0,0,0],[3,2,1,1,0,0,0],[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {0,1,2,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,3,6,13,14],[2,5,9],[4,8,12],[7,11],[10]]
=> [[5,3,3,2,1,0,0,0,0,0,0,0,0,0],[4,3,3,2,1,0,0,0,0,0,0,0,0],[3,3,3,2,1,0,0,0,0,0,0,0],[3,3,2,2,1,0,0,0,0,0,0],[3,3,2,1,1,0,0,0,0,0],[3,3,2,1,0,0,0,0,0],[3,2,2,1,0,0,0,0],[3,2,1,1,0,0,0],[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {0,1,2,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [[1,3,6,13],[2,5,9],[4,8,12],[7,11],[10]]
=> [[4,3,3,2,1,0,0,0,0,0,0,0,0],[3,3,3,2,1,0,0,0,0,0,0,0],[3,3,2,2,1,0,0,0,0,0,0],[3,3,2,1,1,0,0,0,0,0],[3,3,2,1,0,0,0,0,0],[3,2,2,1,0,0,0,0],[3,2,1,1,0,0,0],[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {0,1,2,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8,12],[7,11],[10]]
=> [[3,3,3,2,1,0,0,0,0,0,0,0],[3,3,2,2,1,0,0,0,0,0,0],[3,3,2,1,1,0,0,0,0,0],[3,3,2,1,0,0,0,0,0],[3,2,2,1,0,0,0,0],[3,2,1,1,0,0,0],[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {0,1,2,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,3,8,9,14],[2,5,12,13],[4,7],[6,11],[10]]
=> [[5,4,2,2,1,0,0,0,0,0,0,0,0,0],[4,4,2,2,1,0,0,0,0,0,0,0,0],[4,3,2,2,1,0,0,0,0,0,0,0],[4,2,2,2,1,0,0,0,0,0,0],[4,2,2,1,1,0,0,0,0,0],[4,2,2,1,0,0,0,0,0],[3,2,2,1,0,0,0,0],[2,2,2,1,0,0,0],[2,2,1,1,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {0,1,2,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,3,8,9],[2,5,12,13],[4,7],[6,11],[10]]
=> [[4,4,2,2,1,0,0,0,0,0,0,0,0],[4,3,2,2,1,0,0,0,0,0,0,0],[4,2,2,2,1,0,0,0,0,0,0],[4,2,2,1,1,0,0,0,0,0],[4,2,2,1,0,0,0,0,0],[3,2,2,1,0,0,0,0],[2,2,2,1,0,0,0],[2,2,1,1,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {0,1,2,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,3,8,12,13],[2,5,11],[4,7],[6,10],[9]]
=> [[5,3,2,2,1,0,0,0,0,0,0,0,0],[4,3,2,2,1,0,0,0,0,0,0,0],[3,3,2,2,1,0,0,0,0,0,0],[3,2,2,2,1,0,0,0,0,0],[3,2,2,1,1,0,0,0,0],[3,2,2,1,0,0,0,0],[2,2,2,1,0,0,0],[2,2,1,1,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {0,1,2,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
=> [[4,3,2,2,1,0,0,0,0,0,0,0],[3,3,2,2,1,0,0,0,0,0,0],[3,2,2,2,1,0,0,0,0,0],[3,2,2,1,1,0,0,0,0],[3,2,2,1,0,0,0,0],[2,2,2,1,0,0,0],[2,2,1,1,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {0,1,2,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
=> [[3,3,2,2,1,0,0,0,0,0,0],[3,2,2,2,1,0,0,0,0,0],[3,2,2,1,1,0,0,0,0],[3,2,2,1,0,0,0,0],[2,2,2,1,0,0,0],[2,2,1,1,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {0,1,2,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[1,3,10,11,12],[2,5],[4,7],[6,9],[8]]
=> [[5,2,2,2,1,0,0,0,0,0,0,0],[4,2,2,2,1,0,0,0,0,0,0],[3,2,2,2,1,0,0,0,0,0],[2,2,2,2,1,0,0,0,0],[2,2,2,1,1,0,0,0],[2,2,2,1,0,0,0],[2,2,1,1,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {0,1,2,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> 4
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> 4
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 3
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> 4
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [[1,3,4],[2]]
=> [[3,1,0,0],[2,1,0],[1,1],[1]]
=> 4
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [[1,3],[2]]
=> [[2,1,0],[1,1],[1]]
=> 3
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [[1,1],[1]]
=> 2
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> [[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> 4
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> [[1,2,3]]
=> [[3,0,0],[2,0],[1]]
=> 3
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 2
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> 4
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> 4
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 3
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> 4
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [3,1]
=> [[1,3,4],[2]]
=> [[3,1,0,0],[2,1,0],[1,1],[1]]
=> 4
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,1]
=> [[1,3],[2]]
=> [[2,1,0],[1,1],[1]]
=> 3
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [[1,1],[1]]
=> 2
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [4]
=> [[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> 4
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [3]
=> [[1,2,3]]
=> [[3,0,0],[2,0],[1]]
=> 3
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1]
=> [[1]]
=> [[1]]
=> 1
Description
The sum of the first row in a Gelfand-Tsetlin pattern.
Matching statistic: St001645
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 32%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 32%
Values
[1,0]
=> [1] => ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,0,1,0]
=> [1,2] => ([],2)
=> ([],1)
=> 1 = 0 + 1
[1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> ([],1)
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {2,3} + 1
[1,1,0,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {2,3} + 1
[1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> ([],1)
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {1,2,3,3,4,4,4,5,5,6} + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {1,2,3,3,4,4,4,5,5,6} + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {1,2,3,3,4,4,4,5,5,6} + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,2,3,3,4,4,4,5,5,6} + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {1,2,3,3,4,4,4,5,5,6} + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? ∊ {1,2,3,3,4,4,4,5,5,6} + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {1,2,3,3,4,4,4,5,5,6} + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,2,3,3,4,4,4,5,5,6} + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,2,3,3,4,4,4,5,5,6} + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,2,3,3,4,4,4,5,5,6} + 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10} + 1
[1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {1,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15} + 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ([],7)
=> ([],1)
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [7,2,3,4,5,6,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [7,3,4,5,6,2,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [7,3,4,6,5,2,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [7,3,5,4,6,2,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [7,3,6,4,5,2,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [7,3,6,5,4,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [7,4,3,5,6,2,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [7,4,3,6,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [7,5,3,4,6,2,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [7,6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [7,6,3,5,4,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [7,5,4,3,6,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [7,6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [7,6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
Description
The pebbling number of a connected graph.
Matching statistic: St000769
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000769: Integer compositions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 32%
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000769: Integer compositions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 32%
Values
[1,0]
=> 10 => 10 => [1,2] => 0
[1,0,1,0]
=> 1010 => 0110 => [2,1,2] => 1
[1,1,0,0]
=> 1100 => 1010 => [1,2,2] => 0
[1,0,1,0,1,0]
=> 101010 => 100110 => [1,3,1,2] => 2
[1,0,1,1,0,0]
=> 101100 => 110010 => [1,1,3,2] => 3
[1,1,0,0,1,0]
=> 110010 => 010110 => [2,2,1,2] => 2
[1,1,0,1,0,0]
=> 110100 => 011010 => [2,1,2,2] => 1
[1,1,1,0,0,0]
=> 111000 => 101010 => [1,2,2,2] => 0
[1,0,1,0,1,0,1,0]
=> 10101010 => 01100110 => [2,1,3,1,2] => 4
[1,0,1,0,1,1,0,0]
=> 10101100 => 01110010 => [2,1,1,3,2] => 5
[1,0,1,1,0,0,1,0]
=> 10110010 => 10100110 => [1,2,3,1,2] => 3
[1,0,1,1,0,1,0,0]
=> 10110100 => 10110010 => [1,2,1,3,2] => 6
[1,0,1,1,1,0,0,0]
=> 10111000 => 11010010 => [1,1,2,3,2] => 4
[1,1,0,0,1,0,1,0]
=> 11001010 => 00110110 => [3,1,2,1,2] => 4
[1,1,0,0,1,1,0,0]
=> 11001100 => 00111010 => [3,1,1,2,2] => 1
[1,1,0,1,0,0,1,0]
=> 11010010 => 10010110 => [1,3,2,1,2] => 5
[1,1,0,1,0,1,0,0]
=> 11010100 => 10011010 => [1,3,1,2,2] => 2
[1,1,0,1,1,0,0,0]
=> 11011000 => 11001010 => [1,1,3,2,2] => 3
[1,1,1,0,0,0,1,0]
=> 11100010 => 01010110 => [2,2,2,1,2] => 3
[1,1,1,0,0,1,0,0]
=> 11100100 => 01011010 => [2,2,1,2,2] => 2
[1,1,1,0,1,0,0,0]
=> 11101000 => 01101010 => [2,1,2,2,2] => 1
[1,1,1,1,0,0,0,0]
=> 11110000 => 10101010 => [1,2,2,2,2] => 0
[1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,0,1,1,0,0,0]
=> 1101011000 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,1,0,0,1,0,0]
=> 1101100100 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,0,1,1,0,0,0]
=> 1110011000 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,1,0,0,1,0,0]
=> 1110100100 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,1,1,0,0,0,0]
=> 1110110000 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,1,0,0,0,1,0,0]
=> 1111000100 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,1,0,0,1,0,0,0]
=> 1111001000 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,1,0,1,0,0,0,0]
=> 1111010000 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,1,1,0,0,0,0,0]
=> 1111100000 => ? => ? => ? ∊ {0,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => ? => ? => ? ∊ {0,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => ? => ? => ? ∊ {0,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> 101010110010 => ? => ? => ? ∊ {0,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? => ? => ? ∊ {0,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> 101010111000 => ? => ? => ? ∊ {0,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> 101011001010 => ? => ? => ? ∊ {0,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,1,0,0,1,1,0,0]
=> 101011001100 => ? => ? => ? ∊ {0,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
[1,0,1,0,1,1,0,1,0,0,1,0]
=> 101011010010 => ? => ? => ? ∊ {0,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,13,13,13,14,14,15}
Description
The major index of a composition regarded as a word.
This is the sum of the positions of the descents of the composition.
For the statistic which interprets the composition as a descent set, see [[St000008]].
Matching statistic: St001582
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001582: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 14%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001582: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 14%
Values
[1,0]
=> []
=> []
=> [1] => ? = 0
[1,0,1,0]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[1,1,0,0]
=> []
=> []
=> [1] => ? = 0
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[1,1,1,0,0,0]
=> []
=> []
=> [1] => ? = 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? ∊ {0,1,3,3,4,4,4,5,5,6}
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? ∊ {0,1,3,3,4,4,4,5,5,6}
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => ? ∊ {0,1,3,3,4,4,4,5,5,6}
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? ∊ {0,1,3,3,4,4,4,5,5,6}
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? ∊ {0,1,3,3,4,4,4,5,5,6}
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ? ∊ {0,1,3,3,4,4,4,5,5,6}
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? ∊ {0,1,3,3,4,4,4,5,5,6}
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? ∊ {0,1,3,3,4,4,4,5,5,6}
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? ∊ {0,1,3,3,4,4,4,5,5,6}
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[1,1,1,1,0,0,0,0]
=> []
=> []
=> [1] => ? ∊ {0,1,3,3,4,4,4,5,5,6}
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? ∊ {0,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,9,9,10}
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
Description
The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.
The following 1 statistic also match your data. Click on any of them to see the details.
St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
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