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Your data matches 9 different statistics following compositions of up to 3 maps.
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Matching statistic: St000059
St000059: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> 0
[[1,2]]
=> 0
[[1],[2]]
=> 1
[[1,2,3]]
=> 0
[[1,3],[2]]
=> 1
[[1,2],[3]]
=> 2
[[1],[2],[3]]
=> 3
[[1,2,3,4]]
=> 0
[[1,3,4],[2]]
=> 1
[[1,2,4],[3]]
=> 2
[[1,2,3],[4]]
=> 3
[[1,3],[2,4]]
=> 2
[[1,2],[3,4]]
=> 4
[[1,4],[2],[3]]
=> 3
[[1,3],[2],[4]]
=> 4
[[1,2],[3],[4]]
=> 5
[[1],[2],[3],[4]]
=> 6
[[1,2,3,4,5]]
=> 0
[[1,3,4,5],[2]]
=> 1
[[1,2,4,5],[3]]
=> 2
[[1,2,3,5],[4]]
=> 3
[[1,2,3,4],[5]]
=> 4
[[1,3,5],[2,4]]
=> 2
[[1,2,5],[3,4]]
=> 4
[[1,3,4],[2,5]]
=> 3
[[1,2,4],[3,5]]
=> 5
[[1,2,3],[4,5]]
=> 6
[[1,4,5],[2],[3]]
=> 3
[[1,3,5],[2],[4]]
=> 4
[[1,2,5],[3],[4]]
=> 5
[[1,3,4],[2],[5]]
=> 5
[[1,2,4],[3],[5]]
=> 6
[[1,2,3],[4],[5]]
=> 7
[[1,4],[2,5],[3]]
=> 4
[[1,3],[2,5],[4]]
=> 5
[[1,2],[3,5],[4]]
=> 7
[[1,3],[2,4],[5]]
=> 6
[[1,2],[3,4],[5]]
=> 8
[[1,5],[2],[3],[4]]
=> 6
[[1,4],[2],[3],[5]]
=> 7
[[1,3],[2],[4],[5]]
=> 8
[[1,2],[3],[4],[5]]
=> 9
[[1],[2],[3],[4],[5]]
=> 10
[[1,2,3,4,5,6]]
=> 0
[[1,3,4,5,6],[2]]
=> 1
[[1,2,4,5,6],[3]]
=> 2
[[1,2,3,5,6],[4]]
=> 3
[[1,2,3,4,6],[5]]
=> 4
[[1,2,3,4,5],[6]]
=> 5
[[1,3,5,6],[2,4]]
=> 2
Description
The inversion number of a standard tableau as defined by Haglund and Stevens.
Their inversion number is the total number of inversion pairs for the tableau. An inversion pair is defined as a pair of cells (a,b), (x,y) such that the content of (x,y) is greater than the content of (a,b) and (x,y) is north of the inversion path of (a,b), where the inversion path is defined in detail in [1].
Matching statistic: St000456
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 66%
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 66%
Values
[[1]]
=> [1] => [1] => ([],1)
=> ? = 0
[[1,2]]
=> [2] => [2] => ([],2)
=> ? = 0
[[1],[2]]
=> [1,1] => [1,1] => ([(0,1)],2)
=> 1
[[1,2,3]]
=> [3] => [3] => ([],3)
=> ? ∊ {0,2}
[[1,3],[2]]
=> [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[[1,2],[3]]
=> [2,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,2}
[[1],[2],[3]]
=> [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,3,4]]
=> [4] => [4] => ([],4)
=> ? ∊ {0,2,3,3,5}
[[1,3,4],[2]]
=> [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[[1,2,4],[3]]
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,2,3,3,5}
[[1,2,3],[4]]
=> [3,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,2,3,3,5}
[[1,3],[2,4]]
=> [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[[1,2],[3,4]]
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,2,3,3,5}
[[1,4],[2],[3]]
=> [1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[1,3],[2],[4]]
=> [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[[1,2],[3],[4]]
=> [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,2,3,3,5}
[[1],[2],[3],[4]]
=> [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[[1,2,3,4,5]]
=> [5] => [5] => ([],5)
=> ? ∊ {0,2,4,4,4,5,5,6,6,7,7,7,9}
[[1,3,4,5],[2]]
=> [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[1,2,4,5],[3]]
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,2,4,4,4,5,5,6,6,7,7,7,9}
[[1,2,3,5],[4]]
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,2,4,4,4,5,5,6,6,7,7,7,9}
[[1,2,3,4],[5]]
=> [4,1] => [1,4] => ([(3,4)],5)
=> ? ∊ {0,2,4,4,4,5,5,6,6,7,7,7,9}
[[1,3,5],[2,4]]
=> [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,2,5],[3,4]]
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,2,4,4,4,5,5,6,6,7,7,7,9}
[[1,3,4],[2,5]]
=> [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[[1,2,4],[3,5]]
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,4,4,4,5,5,6,6,7,7,7,9}
[[1,2,3],[4,5]]
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,2,4,4,4,5,5,6,6,7,7,7,9}
[[1,4,5],[2],[3]]
=> [1,1,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,3,5],[2],[4]]
=> [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,2,5],[3],[4]]
=> [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,4,4,4,5,5,6,6,7,7,7,9}
[[1,3,4],[2],[5]]
=> [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[[1,2,4],[3],[5]]
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,4,4,4,5,5,6,6,7,7,7,9}
[[1,2,3],[4],[5]]
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,4,4,4,5,5,6,6,7,7,7,9}
[[1,4],[2,5],[3]]
=> [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[[1,3],[2,5],[4]]
=> [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,2],[3,5],[4]]
=> [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,4,4,4,5,5,6,6,7,7,7,9}
[[1,3],[2,4],[5]]
=> [1,2,1,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 8
[[1,2],[3,4],[5]]
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,4,4,4,5,5,6,6,7,7,7,9}
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 8
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,4,4,4,5,5,6,6,7,7,7,9}
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10
[[1,2,3,4,5,6]]
=> [6] => [6] => ([],6)
=> ? ∊ {0,2,3,5,5,5,5,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,14}
[[1,3,4,5,6],[2]]
=> [1,5] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,2,4,5,6],[3]]
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,2,3,5,5,5,5,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,14}
[[1,2,3,5,6],[4]]
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,2,3,5,5,5,5,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,14}
[[1,2,3,4,6],[5]]
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,2,3,5,5,5,5,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,14}
[[1,2,3,4,5],[6]]
=> [5,1] => [1,5] => ([(4,5)],6)
=> ? ∊ {0,2,3,5,5,5,5,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,14}
[[1,3,5,6],[2,4]]
=> [1,2,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2,5,6],[3,4]]
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,2,3,5,5,5,5,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,14}
[[1,3,4,6],[2,5]]
=> [1,3,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,2,4,6],[3,5]]
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,2,3,5,5,5,5,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,14}
[[1,2,3,6],[4,5]]
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,2,3,5,5,5,5,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,14}
[[1,3,4,5],[2,6]]
=> [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[[1,2,4,5],[3,6]]
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,2,3,5,5,5,5,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,14}
[[1,2,3,5],[4,6]]
=> [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,2,3,5,5,5,5,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,14}
[[1,2,3,4],[5,6]]
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,2,3,5,5,5,5,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,14}
[[1,4,5,6],[2],[3]]
=> [1,1,4] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,5,6],[2],[4]]
=> [1,2,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2,5,6],[3],[4]]
=> [2,1,3] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,2,3,5,5,5,5,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,14}
[[1,3,4,6],[2],[5]]
=> [1,3,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,2,4,6],[3],[5]]
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,2,3,5,5,5,5,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,14}
[[1,2,3,6],[4],[5]]
=> [3,1,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,2,3,5,5,5,5,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,14}
[[1,3,4,5],[2],[6]]
=> [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[[1,2,4,5],[3],[6]]
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,2,3,5,5,5,5,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,14}
[[1,2,3,5],[4],[6]]
=> [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,2,3,5,5,5,5,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,14}
[[1,2,3,4],[5],[6]]
=> [4,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,2,3,5,5,5,5,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,14}
[[1,3,5],[2,4,6]]
=> [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 8
[[1,2,5],[3,4,6]]
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,2,3,5,5,5,5,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,14}
[[1,3,4],[2,5,6]]
=> [1,3,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,2,4],[3,5,6]]
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,2,3,5,5,5,5,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,14}
[[1,2,3],[4,5,6]]
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,2,3,5,5,5,5,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,14}
[[1,4,6],[2,5],[3]]
=> [1,1,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[[1,3,6],[2,5],[4]]
=> [1,2,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2,6],[3,5],[4]]
=> [2,1,3] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,2,3,5,5,5,5,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,14}
[[1,3,6],[2,4],[5]]
=> [1,2,1,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[[1,2,6],[3,4],[5]]
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,2,3,5,5,5,5,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,14}
[[1,4,5],[2,6],[3]]
=> [1,1,3,1] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[[1,3,5],[2,6],[4]]
=> [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 8
[[1,2,5],[3,6],[4]]
=> [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,2,3,5,5,5,5,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,14}
[[1,3,4],[2,6],[5]]
=> [1,3,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,2,4],[3,6],[5]]
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,2,3,5,5,5,5,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,14}
[[1,2,3],[4,6],[5]]
=> [3,1,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,2,3,5,5,5,5,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,14}
[[1,3,5],[2,4],[6]]
=> [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 8
[[1,2,5],[3,4],[6]]
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,2,3,5,5,5,5,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,14}
[[1,3,4],[2,5],[6]]
=> [1,3,1,1] => [3,1,1,1] => ([(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
[[1,2,4],[3,5],[6]]
=> [2,2,1,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,2,3,5,5,5,5,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,14}
[[1,2,3],[4,5],[6]]
=> [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,2,3,5,5,5,5,6,6,7,7,7,7,7,7,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,12,12,12,12,14}
[[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,4,6],[2],[3],[5]]
=> [1,1,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[[1,3,6],[2],[4],[5]]
=> [1,2,1,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[[1,4,5],[2],[3],[6]]
=> [1,1,3,1] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[[1,3,5],[2],[4],[6]]
=> [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 8
[[1,3,4],[2],[5],[6]]
=> [1,3,1,1] => [3,1,1,1] => ([(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
[[1,4],[2,5],[3,6]]
=> [1,1,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[[1,3],[2,5],[4,6]]
=> [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 8
[[1,3],[2,4],[5,6]]
=> [1,2,1,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[[1,5],[2,6],[3],[4]]
=> [1,1,1,2,1] => [1,1,2,1,1] => ([(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)
=> 9
[[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
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 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 45%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 45%
Values
[[1]]
=> [1] => [1] => [1,0]
=> 0
[[1,2]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 1
[[1],[2]]
=> [2,1] => [2,1] => [1,1,0,0]
=> 0
[[1,2,3]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> ? = 3
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 0
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> ? ∊ {2,4,5,6}
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> ? ∊ {2,4,5,6}
[[1,2,4],[3]]
=> [3,1,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 3
[[1,2,3],[4]]
=> [4,1,2,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> ? ∊ {2,4,5,6}
[[1,3],[2,4]]
=> [2,4,1,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[[1,2],[3,4]]
=> [3,4,1,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> ? ∊ {2,4,5,6}
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 4
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 3
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,4,5,5,6,6,7,7,7,8,8,9,10}
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {2,4,5,5,6,6,7,7,7,8,8,9,10}
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {2,4,5,5,6,6,7,7,7,8,8,9,10}
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {2,4,5,5,6,6,7,7,7,8,8,9,10}
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,4,5,5,6,6,7,7,7,8,8,9,10}
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> ? ∊ {2,4,5,5,6,6,7,7,7,8,8,9,10}
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {2,4,5,5,6,6,7,7,7,8,8,9,10}
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {2,4,5,5,6,6,7,7,7,8,8,9,10}
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {2,4,5,5,6,6,7,7,7,8,8,9,10}
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> 5
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 4
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> ? ∊ {2,4,5,5,6,6,7,7,7,8,8,9,10}
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {2,4,5,5,6,6,7,7,7,8,8,9,10}
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> ? ∊ {2,4,5,5,6,6,7,7,7,8,8,9,10}
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 5
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> ? ∊ {2,4,5,5,6,6,7,7,7,8,8,9,10}
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0]
=> 6
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 6
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,3,3,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,3,3,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {2,3,3,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {2,3,3,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {2,3,3,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,3,3,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [2,1,4,3,5,6] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? ∊ {2,3,3,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,3,4,2,5,6] => [1,0,1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {2,3,3,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [2,1,3,5,4,6] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> ? ∊ {2,3,3,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [1,2,4,5,3,6] => [1,0,1,0,1,1,0,1,0,0,1,0]
=> ? ∊ {2,3,3,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [2,1,3,4,6,5] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,3,3,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {2,3,3,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {2,3,3,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {2,3,3,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? ∊ {2,3,3,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [2,4,1,3,5,6] => [1,1,0,1,1,0,0,0,1,0,1,0]
=> ? ∊ {2,3,3,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {2,3,3,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,4,6],[2],[5]]
=> [5,2,1,3,4,6] => [2,1,5,3,4,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4
[[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => [1,3,5,2,4,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> ? ∊ {2,3,3,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3,6],[4],[5]]
=> [5,4,1,2,3,6] => [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {2,3,3,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,4,5],[2],[6]]
=> [6,2,1,3,4,5] => [2,1,3,6,4,5] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> ? ∊ {2,3,3,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => [1,3,2,6,4,5] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5
[[1,2,3,5],[4],[6]]
=> [6,4,1,2,3,5] => [1,2,4,6,3,5] => [1,0,1,0,1,1,0,1,1,0,0,0]
=> ? ∊ {2,3,3,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => [1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {2,3,3,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
[[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
=> ? ∊ {2,3,3,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [2,1,3,5,6,4] => [1,1,0,0,1,0,1,1,0,1,0,0]
=> ? ∊ {2,3,3,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [1,3,2,5,6,4] => [1,0,1,1,0,0,1,1,0,1,0,0]
=> ? ∊ {2,3,3,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [1,2,4,5,6,3] => [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {2,3,3,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,4,6],[2,5],[3]]
=> [3,2,5,1,4,6] => [3,2,1,5,4,6] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3
[[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [2,4,1,5,3,6] => [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {2,3,3,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,6],[3,5],[4]]
=> [4,3,5,1,2,6] => [1,4,3,5,2,6] => [1,0,1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {2,3,3,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,6],[2,4],[5]]
=> [5,2,4,1,3,6] => [2,1,5,4,3,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4
[[1,2,6],[3,4],[5]]
=> [5,3,4,1,2,6] => [1,3,5,4,2,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> ? ∊ {2,3,3,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,4,5],[2,6],[3]]
=> [3,2,6,1,4,5] => [3,2,1,4,6,5] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> ? ∊ {2,3,3,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,5],[2,6],[4]]
=> [4,2,6,1,3,5] => [2,4,1,3,6,5] => [1,1,0,1,1,0,0,0,1,1,0,0]
=> 6
[[1,2,5],[3,6],[4]]
=> [4,3,6,1,2,5] => [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5
[[1,3,4],[2,6],[5]]
=> [5,2,6,1,3,4] => [2,1,5,3,6,4] => [1,1,0,0,1,1,1,0,0,1,0,0]
=> 5
[[1,2,4],[3,6],[5]]
=> [5,3,6,1,2,4] => [1,3,5,2,6,4] => [1,0,1,1,0,1,1,0,0,1,0,0]
=> ? ∊ {2,3,3,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3],[4,6],[5]]
=> [5,4,6,1,2,3] => [1,2,5,4,6,3] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> ? ∊ {2,3,3,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,4],[3,5],[6]]
=> [6,3,5,1,2,4] => [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => [3,5,2,1,4,6] => [1,1,1,0,1,1,0,0,0,0,1,0]
=> 7
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => [2,5,4,1,3,6] => [1,1,0,1,1,1,0,0,0,0,1,0]
=> 7
[[1,2,6],[3],[4],[5]]
=> [5,4,3,1,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => [3,2,6,1,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> 6
[[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => [2,1,6,5,3,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [3,2,1,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => [1,4,3,6,5,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> 7
[[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [2,5,4,1,6,3] => [1,1,0,1,1,1,0,0,0,1,0,0]
=> 8
[[1,2],[3,6],[4],[5]]
=> [5,4,3,6,1,2] => [1,5,4,3,6,2] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> 6
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => [3,2,6,1,5,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> 6
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4
[[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => [4,6,3,2,1,5] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> 8
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => [3,6,5,2,1,4] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> 9
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000101
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
St000101: Semistandard tableaux ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 38%
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
St000101: Semistandard tableaux ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 38%
Values
[[1]]
=> [[1]]
=> [[1]]
=> 0
[[1,2]]
=> [[2,0],[1]]
=> [[1,2]]
=> 0
[[1],[2]]
=> [[1,1],[1]]
=> [[1],[2]]
=> 1
[[1,2,3]]
=> [[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> 0
[[1,3],[2]]
=> [[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> 2
[[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> 1
[[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> 3
[[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4]]
=> 0
[[1,3,4],[2]]
=> [[3,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,4],[2]]
=> 3
[[1,2,4],[3]]
=> [[3,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,4],[3]]
=> 2
[[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3],[4]]
=> 1
[[1,3],[2,4]]
=> [[2,2,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2,4]]
=> 4
[[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3,4]]
=> 2
[[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> [[1,4],[2],[3]]
=> 5
[[1,3],[2],[4]]
=> [[2,1,1,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2],[4]]
=> 4
[[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3],[4]]
=> 3
[[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> 6
[[1,2,3,4,5]]
=> [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4,5]]
=> 0
[[1,3,4,5],[2]]
=> [[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,4,5],[2]]
=> 4
[[1,2,4,5],[3]]
=> [[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,4,5],[3]]
=> 3
[[1,2,3,5],[4]]
=> [[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,5],[4]]
=> 2
[[1,2,3,4],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4],[5]]
=> 1
[[1,3,5],[2,4]]
=> [[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,5],[2,4]]
=> 6
[[1,2,5],[3,4]]
=> [[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,5],[3,4]]
=> 3
[[1,3,4],[2,5]]
=> [[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,4],[2,5]]
=> 5
[[1,2,4],[3,5]]
=> [[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,4],[3,5]]
=> 4
[[1,2,3],[4,5]]
=> [[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3],[4,5]]
=> 2
[[1,4,5],[2],[3]]
=> [[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> [[1,4,5],[2],[3]]
=> 7
[[1,3,5],[2],[4]]
=> [[3,1,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> [[1,3,5],[2],[4]]
=> 6
[[1,2,5],[3],[4]]
=> [[3,1,1,0,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> [[1,2,5],[3],[4]]
=> 5
[[1,3,4],[2],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,4],[2],[5]]
=> 5
[[1,2,4],[3],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,4],[3],[5]]
=> 4
[[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3],[4],[5]]
=> 3
[[1,4],[2,5],[3]]
=> [[2,2,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> [[1,4],[2,5],[3]]
=> 8
[[1,3],[2,5],[4]]
=> [[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2,5],[4]]
=> 6
[[1,2],[3,5],[4]]
=> [[2,2,1,0,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3,5],[4]]
=> 5
[[1,3],[2,4],[5]]
=> [[2,2,1,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2,4],[5]]
=> 7
[[1,2],[3,4],[5]]
=> [[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3,4],[5]]
=> 4
[[1,5],[2],[3],[4]]
=> [[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1,5],[2],[3],[4]]
=> 9
[[1,4],[2],[3],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> [[1,4],[2],[3],[5]]
=> 8
[[1,3],[2],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2],[4],[5]]
=> 7
[[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3],[4],[5]]
=> 6
[[1],[2],[3],[4],[5]]
=> [[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4],[5]]
=> 10
[[1,2,3,4,5,6]]
=> [[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4,5,6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,4,5,6],[2]]
=> [[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,4,5,6],[2]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,4,5,6],[3]]
=> [[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,4,5,6],[3]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3,5,6],[4]]
=> [[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,5,6],[4]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3,4,6],[5]]
=> [[5,1,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4,6],[5]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3,4,5],[6]]
=> [[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4,5],[6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,5,6],[2,4]]
=> [[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,5,6],[2,4]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[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]]
=> [[1,2,5,6],[3,4]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,4,6],[2,5]]
=> [[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,4,6],[2,5]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,4,6],[3,5]]
=> [[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,4,6],[3,5]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3,6],[4,5]]
=> [[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,6],[4,5]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,4,5],[2,6]]
=> [[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,4,5],[2,6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,4,5],[3,6]]
=> [[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,4,5],[3,6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3,5],[4,6]]
=> [[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,5],[4,6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3,4],[5,6]]
=> [[4,2,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4],[5,6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[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]]
=> [[1,4,5,6],[2],[3]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,5,6],[2],[4]]
=> [[4,1,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> [[1,3,5,6],[2],[4]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,5,6],[3],[4]]
=> [[4,1,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> [[1,2,5,6],[3],[4]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,4,6],[2],[5]]
=> [[4,1,1,0,0,0],[3,1,1,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,4,6],[2],[5]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,4,6],[3],[5]]
=> [[4,1,1,0,0,0],[3,1,1,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,4,6],[3],[5]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3,6],[4],[5]]
=> [[4,1,1,0,0,0],[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,6],[4],[5]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,4,5],[2],[6]]
=> [[4,1,1,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,4,5],[2],[6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,4,5],[3],[6]]
=> [[4,1,1,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,4,5],[3],[6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3,5],[4],[6]]
=> [[4,1,1,0,0,0],[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,5],[4],[6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3,4],[5],[6]]
=> [[4,1,1,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4],[5],[6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,5],[2,4,6]]
=> [[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,5],[2,4,6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,5],[3,4,6]]
=> [[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,5],[3,4,6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,4],[2,5,6]]
=> [[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,4],[2,5,6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,4],[3,5,6]]
=> [[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,4],[3,5,6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[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]]
=> [[1,2,3],[4,5,6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,4,6],[2,5],[3]]
=> [[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> [[1,4,6],[2,5],[3]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[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]]
=> [[1,3,6],[2,5],[4]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,6],[3,5],[4]]
=> [[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> [[1,2,6],[3,5],[4]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,6],[2,4],[5]]
=> [[3,2,1,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,6],[2,4],[5]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,6],[3,4],[5]]
=> [[3,2,1,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,6],[3,4],[5]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,4,5],[2,6],[3]]
=> [[3,2,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> [[1,4,5],[2,6],[3]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,5],[2,6],[4]]
=> [[3,2,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> [[1,3,5],[2,6],[4]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,5],[3,6],[4]]
=> [[3,2,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> [[1,2,5],[3,6],[4]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,4],[2,6],[5]]
=> [[3,2,1,0,0,0],[3,1,1,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,4],[2,6],[5]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,4],[3,6],[5]]
=> [[3,2,1,0,0,0],[3,1,1,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,4],[3,6],[5]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3],[4,6],[5]]
=> [[3,2,1,0,0,0],[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3],[4,6],[5]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,5],[2,4],[6]]
=> [[3,2,1,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,5],[2,4],[6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,5],[3,4],[6]]
=> [[3,2,1,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,5],[3,4],[6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,4],[2,5],[6]]
=> [[3,2,1,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,4],[2,5],[6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,4],[3,5],[6]]
=> [[3,2,1,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,4],[3,5],[6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3],[4,5],[6]]
=> [[3,2,1,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3],[4,5],[6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[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]]
=> [[1,5,6],[2],[3],[4]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,4,6],[2],[3],[5]]
=> [[3,1,1,1,0,0],[2,1,1,1,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> [[1,4,6],[2],[3],[5]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,6],[2],[4],[5]]
=> [[3,1,1,1,0,0],[2,1,1,1,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> [[1,3,6],[2],[4],[5]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,6],[3],[4],[5]]
=> [[3,1,1,1,0,0],[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> [[1,2,6],[3],[4],[5]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
Description
The cocharge of a semistandard tableau.
Matching statistic: St000102
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
St000102: Semistandard tableaux ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 38%
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
St000102: Semistandard tableaux ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 38%
Values
[[1]]
=> [[1]]
=> [[1]]
=> 0
[[1,2]]
=> [[2,0],[1]]
=> [[1,2]]
=> 1
[[1],[2]]
=> [[1,1],[1]]
=> [[1],[2]]
=> 0
[[1,2,3]]
=> [[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> 3
[[1,3],[2]]
=> [[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> 1
[[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> 2
[[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> 0
[[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4]]
=> 6
[[1,3,4],[2]]
=> [[3,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,4],[2]]
=> 3
[[1,2,4],[3]]
=> [[3,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,4],[3]]
=> 4
[[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3],[4]]
=> 5
[[1,3],[2,4]]
=> [[2,2,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2,4]]
=> 2
[[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3,4]]
=> 4
[[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> [[1,4],[2],[3]]
=> 1
[[1,3],[2],[4]]
=> [[2,1,1,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2],[4]]
=> 2
[[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3],[4]]
=> 3
[[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> 0
[[1,2,3,4,5]]
=> [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4,5]]
=> 10
[[1,3,4,5],[2]]
=> [[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,4,5],[2]]
=> 6
[[1,2,4,5],[3]]
=> [[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,4,5],[3]]
=> 7
[[1,2,3,5],[4]]
=> [[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,5],[4]]
=> 8
[[1,2,3,4],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4],[5]]
=> 9
[[1,3,5],[2,4]]
=> [[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,5],[2,4]]
=> 4
[[1,2,5],[3,4]]
=> [[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,5],[3,4]]
=> 7
[[1,3,4],[2,5]]
=> [[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,4],[2,5]]
=> 5
[[1,2,4],[3,5]]
=> [[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,4],[3,5]]
=> 6
[[1,2,3],[4,5]]
=> [[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3],[4,5]]
=> 8
[[1,4,5],[2],[3]]
=> [[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> [[1,4,5],[2],[3]]
=> 3
[[1,3,5],[2],[4]]
=> [[3,1,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> [[1,3,5],[2],[4]]
=> 4
[[1,2,5],[3],[4]]
=> [[3,1,1,0,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> [[1,2,5],[3],[4]]
=> 5
[[1,3,4],[2],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,4],[2],[5]]
=> 5
[[1,2,4],[3],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,4],[3],[5]]
=> 6
[[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3],[4],[5]]
=> 7
[[1,4],[2,5],[3]]
=> [[2,2,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> [[1,4],[2,5],[3]]
=> 2
[[1,3],[2,5],[4]]
=> [[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2,5],[4]]
=> 4
[[1,2],[3,5],[4]]
=> [[2,2,1,0,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3,5],[4]]
=> 5
[[1,3],[2,4],[5]]
=> [[2,2,1,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2,4],[5]]
=> 3
[[1,2],[3,4],[5]]
=> [[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3,4],[5]]
=> 6
[[1,5],[2],[3],[4]]
=> [[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1,5],[2],[3],[4]]
=> 1
[[1,4],[2],[3],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> [[1,4],[2],[3],[5]]
=> 2
[[1,3],[2],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2],[4],[5]]
=> 3
[[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3],[4],[5]]
=> 4
[[1],[2],[3],[4],[5]]
=> [[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4],[5]]
=> 0
[[1,2,3,4,5,6]]
=> [[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4,5,6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,4,5,6],[2]]
=> [[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,4,5,6],[2]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,4,5,6],[3]]
=> [[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,4,5,6],[3]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3,5,6],[4]]
=> [[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,5,6],[4]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3,4,6],[5]]
=> [[5,1,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4,6],[5]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3,4,5],[6]]
=> [[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4,5],[6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,5,6],[2,4]]
=> [[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,5,6],[2,4]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[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]]
=> [[1,2,5,6],[3,4]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,4,6],[2,5]]
=> [[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,4,6],[2,5]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,4,6],[3,5]]
=> [[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,4,6],[3,5]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3,6],[4,5]]
=> [[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,6],[4,5]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,4,5],[2,6]]
=> [[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,4,5],[2,6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,4,5],[3,6]]
=> [[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,4,5],[3,6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3,5],[4,6]]
=> [[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,5],[4,6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3,4],[5,6]]
=> [[4,2,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4],[5,6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[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]]
=> [[1,4,5,6],[2],[3]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,5,6],[2],[4]]
=> [[4,1,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> [[1,3,5,6],[2],[4]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,5,6],[3],[4]]
=> [[4,1,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> [[1,2,5,6],[3],[4]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,4,6],[2],[5]]
=> [[4,1,1,0,0,0],[3,1,1,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,4,6],[2],[5]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,4,6],[3],[5]]
=> [[4,1,1,0,0,0],[3,1,1,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,4,6],[3],[5]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3,6],[4],[5]]
=> [[4,1,1,0,0,0],[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,6],[4],[5]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,4,5],[2],[6]]
=> [[4,1,1,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,4,5],[2],[6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,4,5],[3],[6]]
=> [[4,1,1,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,4,5],[3],[6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3,5],[4],[6]]
=> [[4,1,1,0,0,0],[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,5],[4],[6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3,4],[5],[6]]
=> [[4,1,1,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4],[5],[6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,5],[2,4,6]]
=> [[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,5],[2,4,6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,5],[3,4,6]]
=> [[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,5],[3,4,6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,4],[2,5,6]]
=> [[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,4],[2,5,6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,4],[3,5,6]]
=> [[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,4],[3,5,6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[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]]
=> [[1,2,3],[4,5,6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,4,6],[2,5],[3]]
=> [[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> [[1,4,6],[2,5],[3]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[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]]
=> [[1,3,6],[2,5],[4]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,6],[3,5],[4]]
=> [[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> [[1,2,6],[3,5],[4]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,6],[2,4],[5]]
=> [[3,2,1,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,6],[2,4],[5]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,6],[3,4],[5]]
=> [[3,2,1,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,6],[3,4],[5]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,4,5],[2,6],[3]]
=> [[3,2,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> [[1,4,5],[2,6],[3]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,5],[2,6],[4]]
=> [[3,2,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> [[1,3,5],[2,6],[4]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,5],[3,6],[4]]
=> [[3,2,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> [[1,2,5],[3,6],[4]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,4],[2,6],[5]]
=> [[3,2,1,0,0,0],[3,1,1,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,4],[2,6],[5]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,4],[3,6],[5]]
=> [[3,2,1,0,0,0],[3,1,1,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,4],[3,6],[5]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3],[4,6],[5]]
=> [[3,2,1,0,0,0],[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3],[4,6],[5]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,5],[2,4],[6]]
=> [[3,2,1,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,5],[2,4],[6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,5],[3,4],[6]]
=> [[3,2,1,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,5],[3,4],[6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,4],[2,5],[6]]
=> [[3,2,1,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,4],[2,5],[6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,4],[3,5],[6]]
=> [[3,2,1,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,4],[3,5],[6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3],[4,5],[6]]
=> [[3,2,1,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3],[4,5],[6]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[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]]
=> [[1,5,6],[2],[3],[4]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,4,6],[2],[3],[5]]
=> [[3,1,1,1,0,0],[2,1,1,1,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> [[1,4,6],[2],[3],[5]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,6],[2],[4],[5]]
=> [[3,1,1,1,0,0],[2,1,1,1,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> [[1,3,6],[2],[4],[5]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,6],[3],[4],[5]]
=> [[3,1,1,1,0,0],[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> [[1,2,6],[3],[4],[5]]
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
Description
The charge of a semistandard tableau.
Matching statistic: St000450
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000450: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 38%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000450: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 38%
Values
[[1]]
=> [1] => ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
[[1,2]]
=> [2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[[1],[2]]
=> [1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1,2,3]]
=> [3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,3],[2]]
=> [1,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,2],[3]]
=> [2,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[1],[2],[3]]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[1,2,3,4]]
=> [4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[[1,3,4],[2]]
=> [1,3] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,2,3],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[[1,3],[2,4]]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,4],[2],[3]]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[[1,3],[2],[4]]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[[1,2],[3],[4]]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6 = 5 + 1
[[1],[2],[3],[4]]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7 = 6 + 1
[[1,2,3,4,5]]
=> [5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[[1,3,4,5],[2]]
=> [1,4] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
[[1,3,5],[2,4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
[[1,2,5],[3,4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,3,4],[2,5]]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[[1,2,4],[3,5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7 = 6 + 1
[[1,2,3],[4,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,4,5],[2],[3]]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,3,5],[2],[4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
[[1,2,5],[3],[4]]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[[1,3,4],[2],[5]]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[[1,2,4],[3],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7 = 6 + 1
[[1,2,3],[4],[5]]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> 8 = 7 + 1
[[1,4],[2,5],[3]]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> 8 = 7 + 1
[[1,3],[2,5],[4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
[[1,2],[3,5],[4]]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[[1,3],[2,4],[5]]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> 9 = 8 + 1
[[1,2],[3,4],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7 = 6 + 1
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7 = 6 + 1
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> 8 = 7 + 1
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> 9 = 8 + 1
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> 10 = 9 + 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => ([(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),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 11 = 10 + 1
[[1,2,3,4,5,6]]
=> [6] => ([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,3,4,5,6],[2]]
=> [1,5] => ([(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,4,5,6],[3]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,3,5,6],[4]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,3,4,6],[5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,3,4,5],[6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,3,5,6],[2,4]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,5,6],[3,4]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,3,4,6],[2,5]]
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,4,6],[3,5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,3,6],[4,5]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,3,4,5],[2,6]]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,4,5],[3,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,3,5],[4,6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,3,4],[5,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,4,5,6],[2],[3]]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,3,5,6],[2],[4]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,5,6],[3],[4]]
=> [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,3,4,6],[2],[5]]
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,4,6],[3],[5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,3,6],[4],[5]]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,3,4,5],[2],[6]]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,4,5],[3],[6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,3,5],[4],[6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,3,4],[5],[6]]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,3,5],[2,4,6]]
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,5],[3,4,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,3,4],[2,5,6]]
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,4],[3,5,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,3],[4,5,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,4,6],[2,5],[3]]
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(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)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,3,6],[2,5],[4]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,6],[3,5],[4]]
=> [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,3,6],[2,4],[5]]
=> [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,6],[3,4],[5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,4,5],[2,6],[3]]
=> [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(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)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,3,5],[2,6],[4]]
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,5],[3,6],[4]]
=> [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,3,4],[2,6],[5]]
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,4],[3,6],[5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,3],[4,6],[5]]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,3,5],[2,4],[6]]
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,5],[3,4],[6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,3,4],[2,5],[6]]
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,4],[3,5],[6]]
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,3],[4,5],[6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,4,6],[2],[3],[5]]
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(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)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,3,6],[2],[4],[5]]
=> [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,6],[3],[4],[5]]
=> [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
Description
The number of edges minus the number of vertices plus 2 of a graph.
When G is connected and planar, this is also the number of its faces.
When $G=(V,E)$ is a connected graph, this is its $k$-monochromatic index for $k>2$: for $2\leq k\leq |V|$, the $k$-monochromatic index of $G$ is the maximum number of edge colors allowed such that for each set $S$ of $k$ vertices, there exists a monochromatic tree in $G$ which contains all vertices from $S$. It is shown in [1] that for $k>2$, this is given by this statistic.
Matching statistic: St001645
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 24%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 24%
Values
[[1]]
=> [1] => ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,2]]
=> [2] => ([],2)
=> ([],1)
=> 1 = 0 + 1
[[1],[2]]
=> [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[[1,2,3]]
=> [3] => ([],3)
=> ([],1)
=> 1 = 0 + 1
[[1,3],[2]]
=> [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 3 + 1
[[1,2],[3]]
=> [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
[[1],[2],[3]]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[1,2,3,4]]
=> [4] => ([],4)
=> ([],1)
=> 1 = 0 + 1
[[1,3,4],[2]]
=> [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,3,4,4,5,6} + 1
[[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,3,4,4,5,6} + 1
[[1,2,3],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
[[1,3],[2,4]]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,4,4,5,6} + 1
[[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,3,4,4,5,6} + 1
[[1,4],[2],[3]]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,4,4,5,6} + 1
[[1,3],[2],[4]]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,4,4,5,6} + 1
[[1,2],[3],[4]]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[1],[2],[3],[4]]
=> [1,1,1,1] => ([(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
[[1,2,3,4,5]]
=> [5] => ([],5)
=> ([],1)
=> 1 = 0 + 1
[[1,3,4,5],[2]]
=> [1,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,3,3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,3,3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,3,3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
[[1,3,5],[2,4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,2,5],[3,4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,3,3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,3,4],[2,5]]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,2,4],[3,5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,2,3],[4,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {2,3,3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,4,5],[2],[3]]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,3,5],[2],[4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,2,5],[3],[4]]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,3,4],[2],[5]]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,2,4],[3],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,2,3],[4],[5]]
=> [3,1,1] => ([(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
[[1,4],[2,5],[3]]
=> [1,1,2,1] => ([(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)
=> ? ∊ {2,3,3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,3],[2,5],[4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,2],[3,5],[4]]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,3],[2,4],[5]]
=> [1,2,1,1] => ([(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)
=> ? ∊ {2,3,3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,2],[3,4],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => ([(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)
=> ? ∊ {2,3,3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => ([(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)
=> ? ∊ {2,3,3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => ([(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)
=> ? ∊ {2,3,3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => ([(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],[2],[3],[4],[5]]
=> [1,1,1,1,1] => ([(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
[[1,2,3,4,5,6]]
=> [6] => ([],6)
=> ([],1)
=> 1 = 0 + 1
[[1,3,4,5,6],[2]]
=> [1,5] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,4,5,6],[3]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,3,5,6],[4]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,3,4,6],[5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,3,4,5],[6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 2 = 1 + 1
[[1,3,5,6],[2,4]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,5,6],[3,4]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,3,4,6],[2,5]]
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,4,6],[3,5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,3,6],[4,5]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,3,4,5],[2,6]]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,4,5],[3,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,3,5],[4,6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,3,4],[5,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,4,5,6],[2],[3]]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,3,5,6],[2],[4]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,5,6],[3],[4]]
=> [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,3,4,6],[2],[5]]
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,4,6],[3],[5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,3,6],[4],[5]]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,3,4,5],[2],[6]]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,4,5],[3],[6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,3,5],[4],[6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,3,4],[5],[6]]
=> [4,1,1] => ([(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
[[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => ([(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
[[1,3],[2,4],[5],[6]]
=> [1,2,1,1,1] => ([(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
[[1,3],[2],[4],[5],[6]]
=> [1,2,1,1,1] => ([(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
[[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => ([(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],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => ([(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
[[1,2,3,4,5,6,7]]
=> [7] => ([],7)
=> ([],1)
=> 1 = 0 + 1
[[1,2,3,4,5,6],[7]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> 2 = 1 + 1
[[1,2,3,4,5],[6],[7]]
=> [5,1,1] => ([(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
[[1,2,3,4],[5],[6],[7]]
=> [4,1,1,1] => ([(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
[[1,3,4],[2,5],[6],[7]]
=> [1,3,1,1,1] => ([(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
[[1,2,4],[3,5],[6],[7]]
=> [2,2,1,1,1] => ([(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
[[1,3,4],[2],[5],[6],[7]]
=> [1,3,1,1,1] => ([(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
[[1,2,4],[3],[5],[6],[7]]
=> [2,2,1,1,1] => ([(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
[[1,2,3],[4],[5],[6],[7]]
=> [3,1,1,1,1] => ([(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
[[1,4],[2,5],[3],[6],[7]]
=> [1,1,2,1,1,1] => ([(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
[[1,3],[2,4],[5],[6],[7]]
=> [1,2,1,1,1,1] => ([(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
[[1,2],[3,4],[5],[6],[7]]
=> [2,2,1,1,1] => ([(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
[[1,4],[2],[3],[5],[6],[7]]
=> [1,1,2,1,1,1] => ([(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
[[1,3],[2],[4],[5],[6],[7]]
=> [1,2,1,1,1,1] => ([(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
[[1,2],[3],[4],[5],[6],[7]]
=> [2,1,1,1,1,1] => ([(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],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => ([(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
[[1,2,3,4],[5],[6],[7],[8]]
=> [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
Description
The pebbling number of a connected graph.
Matching statistic: St000454
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 24%
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 24%
Values
[[1]]
=> [1] => [1] => ([],1)
=> 0
[[1,2]]
=> [1,2] => [2,1] => ([(0,1)],2)
=> 1
[[1],[2]]
=> [2,1] => [1,2] => ([],2)
=> 0
[[1,2,3]]
=> [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
=> ? = 3
[[1,3],[2]]
=> [2,1,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> 1
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => ([],3)
=> 0
[[1,2,3,4]]
=> [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {2,3,4,4,5,6}
[[1,3,4],[2]]
=> [2,1,3,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,4,4,5,6}
[[1,2,4],[3]]
=> [3,1,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,4,4,5,6}
[[1,2,3],[4]]
=> [4,1,2,3] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {2,3,4,4,5,6}
[[1,3],[2,4]]
=> [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {2,3,4,4,5,6}
[[1,2],[3,4]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,4,4,5,6}
[[1,4],[2],[3]]
=> [3,2,1,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[1,3],[2],[4]]
=> [4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,2,4,3] => ([(2,3)],4)
=> 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => ([],4)
=> 0
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10}
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10}
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10}
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10}
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10}
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10}
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10}
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10}
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10}
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10}
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10}
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10}
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10}
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10}
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10}
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10}
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10}
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10}
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10}
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [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)
=> 4
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,2,3,5,4] => ([(3,4)],5)
=> 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,3,4,5] => ([],5)
=> 0
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [3,2,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [4,2,3,5,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [5,2,3,4,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [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)
=> ? ∊ {3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [3,5,2,4,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [4,5,2,3,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? ∊ {3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [3,6,2,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [4,6,2,3,5,1] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [5,6,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? ∊ {3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [4,1,3,5,6,2] => ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [5,1,3,4,6,2] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [6,1,3,4,5,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [4,3,2,5,6,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [5,3,2,4,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => [5,4,2,3,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,4,6],[2],[5]]
=> [5,2,1,3,4,6] => [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)
=> ? ∊ {3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => [6,4,2,3,5,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3,6],[4],[5]]
=> [5,4,1,2,3,6] => [6,5,2,3,4,1] => ([(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)
=> ? ∊ {3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,4,5],[2],[6]]
=> [6,2,1,3,4,5] => [1,4,3,5,6,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => [1,5,3,4,6,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,2,3,5],[4],[6]]
=> [6,4,1,2,3,5] => [1,6,3,4,5,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [3,5,1,4,6,2] => ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15}
[[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [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)
=> 5
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => [1,2,6,5,4,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> 2
[[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [1,2,3,4,6,5] => ([(4,5)],6)
=> 1
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([],6)
=> 0
[[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => [1,2,4,5,6,7,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[[1,2,3,5],[4],[6],[7]]
=> [7,6,4,1,2,3,5] => [1,2,7,4,5,6,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => [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)
=> 6
[[1,6],[2],[3],[4],[5],[7]]
=> [7,5,4,3,2,1,6] => [1,7,6,5,4,3,2] => ([(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)
=> 5
[[1,5],[2],[3],[4],[6],[7]]
=> [7,6,4,3,2,1,5] => [1,2,7,6,5,4,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[[1,4],[2],[3],[5],[6],[7]]
=> [7,6,5,3,2,1,4] => [1,2,3,7,6,5,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[[1,3],[2],[4],[5],[6],[7]]
=> [7,6,5,4,2,1,3] => [1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
=> 2
[[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => [1,2,3,4,5,7,6] => ([(5,6)],7)
=> 1
[[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([],7)
=> 0
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St000777
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 21%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 21%
Values
[[1]]
=> [1] => [1] => ([],1)
=> 1 = 0 + 1
[[1,2]]
=> [1,2] => [1,2] => ([],2)
=> ? = 0 + 1
[[1],[2]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,2,3} + 1
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,2,3} + 1
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,2,3} + 1
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,2,2,3,4,4,5,6} + 1
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,2,2,3,4,4,5,6} + 1
[[1,2,4],[3]]
=> [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,2,2,3,4,4,5,6} + 1
[[1,2,3],[4]]
=> [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,2,2,3,4,4,5,6} + 1
[[1,3],[2,4]]
=> [2,4,1,3] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,2,2,3,4,4,5,6} + 1
[[1,2],[3,4]]
=> [3,4,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,2,2,3,4,4,5,6} + 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,2,2,3,4,4,5,6} + 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 4 = 3 + 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,2,2,3,4,4,5,6} + 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,2,2,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,2,2,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,2,2,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,9,10} + 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [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)
=> 2 = 1 + 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ? ∊ {0,2,2,3,3,3,3,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(4,5)],6)
=> ? ∊ {0,2,2,3,3,3,3,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => ([(4,5)],6)
=> ? ∊ {0,2,2,3,3,3,3,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ? ∊ {0,2,2,3,3,3,3,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ? ∊ {0,2,2,3,3,3,3,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => ([(4,5)],6)
=> ? ∊ {0,2,2,3,3,3,3,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> ? ∊ {0,2,2,3,3,3,3,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,4,2,3,5,6] => ([(3,5),(4,5)],6)
=> ? ∊ {0,2,2,3,3,3,3,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> ? ∊ {0,2,2,3,3,3,3,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ? ∊ {0,2,2,3,3,3,3,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ? ∊ {0,2,2,3,3,3,3,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> ? ∊ {0,2,2,3,3,3,3,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ? ∊ {0,2,2,3,3,3,3,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,2,2,3,3,3,3,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ? ∊ {0,2,2,3,3,3,3,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,14,15} + 1
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [4,1,5,2,6,3] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => [4,1,2,6,5,3] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 6 = 5 + 1
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => [5,4,3,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => [5,4,1,6,3,2] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => [5,1,6,4,3,2] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [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)
=> 2 = 1 + 1
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => [4,1,2,6,3,7,5] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 7 = 6 + 1
[[1,4,6],[2],[3],[5],[7]]
=> [7,5,3,2,1,4,6] => [5,4,1,6,2,7,3] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[[1,3,6],[2],[4],[5],[7]]
=> [7,5,4,2,1,3,6] => [5,1,6,4,2,7,3] => ([(0,6),(1,5),(2,3),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[[1,4,5],[2],[3],[6],[7]]
=> [7,6,3,2,1,4,5] => [5,4,1,2,7,6,3] => ([(0,1),(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[[1,3,5],[2],[4],[6],[7]]
=> [7,6,4,2,1,3,5] => [5,1,6,2,7,4,3] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[[1,3,4],[2],[5],[6],[7]]
=> [7,6,5,2,1,3,4] => [5,1,2,7,6,4,3] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[[1,6],[2],[3],[4],[5],[7]]
=> [7,5,4,3,2,1,6] => [6,5,4,3,1,7,2] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 4 + 1
[[1,5],[2],[3],[4],[6],[7]]
=> [7,6,4,3,2,1,5] => [6,5,4,1,7,3,2] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 4 + 1
[[1,4],[2],[3],[5],[6],[7]]
=> [7,6,5,3,2,1,4] => [6,5,1,7,4,3,2] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 4 + 1
[[1,3],[2],[4],[5],[6],[7]]
=> [7,6,5,4,2,1,3] => [6,1,7,5,4,3,2] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 4 + 1
[[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [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)
=> 2 = 1 + 1
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
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