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Your data matches 11 different statistics following compositions of up to 3 maps.
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Matching statistic: St000744
St000744: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> 1
[[1],[2]]
=> 1
[[1,2,3]]
=> 2
[[1,3],[2]]
=> 1
[[1,2],[3]]
=> 1
[[1],[2],[3]]
=> 2
[[1,2,3,4]]
=> 3
[[1,3,4],[2]]
=> 2
[[1,2,4],[3]]
=> 2
[[1,2,3],[4]]
=> 1
[[1,3],[2,4]]
=> 2
[[1,2],[3,4]]
=> 2
[[1,4],[2],[3]]
=> 1
[[1,3],[2],[4]]
=> 2
[[1,2],[3],[4]]
=> 2
[[1],[2],[3],[4]]
=> 3
[[1,2,3,4,5]]
=> 4
[[1,3,4,5],[2]]
=> 3
[[1,2,4,5],[3]]
=> 3
[[1,2,3,5],[4]]
=> 3
[[1,2,3,4],[5]]
=> 1
[[1,3,5],[2,4]]
=> 2
[[1,2,5],[3,4]]
=> 2
[[1,3,4],[2,5]]
=> 2
[[1,2,4],[3,5]]
=> 2
[[1,2,3],[4,5]]
=> 2
[[1,4,5],[2],[3]]
=> 2
[[1,3,5],[2],[4]]
=> 2
[[1,2,5],[3],[4]]
=> 2
[[1,3,4],[2],[5]]
=> 2
[[1,2,4],[3],[5]]
=> 2
[[1,2,3],[4],[5]]
=> 2
[[1,4],[2,5],[3]]
=> 2
[[1,3],[2,5],[4]]
=> 2
[[1,2],[3,5],[4]]
=> 2
[[1,3],[2,4],[5]]
=> 2
[[1,2],[3,4],[5]]
=> 2
[[1,5],[2],[3],[4]]
=> 1
[[1,4],[2],[3],[5]]
=> 3
[[1,3],[2],[4],[5]]
=> 3
[[1,2],[3],[4],[5]]
=> 3
[[1],[2],[3],[4],[5]]
=> 4
[[1,2,3,4,5,6]]
=> 5
[[1,3,4,5,6],[2]]
=> 4
[[1,2,4,5,6],[3]]
=> 4
[[1,2,3,5,6],[4]]
=> 4
[[1,2,3,4,6],[5]]
=> 4
[[1,2,3,4,5],[6]]
=> 1
[[1,3,5,6],[2,4]]
=> 3
[[1,2,5,6],[3,4]]
=> 3
Description
The length of the path to the largest entry in a standard Young tableau.
Matching statistic: St000264
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 36%●distinct values known / distinct values provided: 14%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 36%●distinct values known / distinct values provided: 14%
Values
[[1,2]]
=> [2] => [1] => ([],1)
=> ? ∊ {1,1}
[[1],[2]]
=> [2] => [1] => ([],1)
=> ? ∊ {1,1}
[[1,2,3]]
=> [3] => [1] => ([],1)
=> ? ∊ {1,1,2,2}
[[1,3],[2]]
=> [2,1] => [1,1] => ([(0,1)],2)
=> ? ∊ {1,1,2,2}
[[1,2],[3]]
=> [3] => [1] => ([],1)
=> ? ∊ {1,1,2,2}
[[1],[2],[3]]
=> [3] => [1] => ([],1)
=> ? ∊ {1,1,2,2}
[[1,2,3,4]]
=> [4] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,3,3}
[[1,3,4],[2]]
=> [2,2] => [2] => ([],2)
=> ? ∊ {1,1,2,2,2,2,2,2,3,3}
[[1,2,4],[3]]
=> [3,1] => [1,1] => ([(0,1)],2)
=> ? ∊ {1,1,2,2,2,2,2,2,3,3}
[[1,2,3],[4]]
=> [4] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,3,3}
[[1,3],[2,4]]
=> [2,2] => [2] => ([],2)
=> ? ∊ {1,1,2,2,2,2,2,2,3,3}
[[1,2],[3,4]]
=> [3,1] => [1,1] => ([(0,1)],2)
=> ? ∊ {1,1,2,2,2,2,2,2,3,3}
[[1,4],[2],[3]]
=> [3,1] => [1,1] => ([(0,1)],2)
=> ? ∊ {1,1,2,2,2,2,2,2,3,3}
[[1,3],[2],[4]]
=> [2,2] => [2] => ([],2)
=> ? ∊ {1,1,2,2,2,2,2,2,3,3}
[[1,2],[3],[4]]
=> [4] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,3,3}
[[1],[2],[3],[4]]
=> [4] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,3,3}
[[1,2,3,4,5]]
=> [5] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3,4,5],[2]]
=> [2,3] => [1,1] => ([(0,1)],2)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,4,5],[3]]
=> [3,2] => [1,1] => ([(0,1)],2)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,3,5],[4]]
=> [4,1] => [1,1] => ([(0,1)],2)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,3,4],[5]]
=> [5] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3,5],[2,4]]
=> [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,5],[3,4]]
=> [3,2] => [1,1] => ([(0,1)],2)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3,4],[2,5]]
=> [2,3] => [1,1] => ([(0,1)],2)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,4],[3,5]]
=> [3,2] => [1,1] => ([(0,1)],2)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,3],[4,5]]
=> [4,1] => [1,1] => ([(0,1)],2)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,4,5],[2],[3]]
=> [3,2] => [1,1] => ([(0,1)],2)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3,5],[2],[4]]
=> [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,5],[3],[4]]
=> [4,1] => [1,1] => ([(0,1)],2)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3,4],[2],[5]]
=> [2,3] => [1,1] => ([(0,1)],2)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,4],[3],[5]]
=> [3,2] => [1,1] => ([(0,1)],2)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,3],[4],[5]]
=> [5] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,4],[2,5],[3]]
=> [3,2] => [1,1] => ([(0,1)],2)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3],[2,5],[4]]
=> [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2],[3,5],[4]]
=> [4,1] => [1,1] => ([(0,1)],2)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3],[2,4],[5]]
=> [2,3] => [1,1] => ([(0,1)],2)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2],[3,4],[5]]
=> [3,2] => [1,1] => ([(0,1)],2)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,5],[2],[3],[4]]
=> [4,1] => [1,1] => ([(0,1)],2)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,4],[2],[3],[5]]
=> [3,2] => [1,1] => ([(0,1)],2)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3],[2],[4],[5]]
=> [2,3] => [1,1] => ([(0,1)],2)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2],[3],[4],[5]]
=> [5] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1],[2],[3],[4],[5]]
=> [5] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,3,4,5,6]]
=> [6] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,3,4,5,6],[2]]
=> [2,4] => [1,1] => ([(0,1)],2)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,4,5,6],[3]]
=> [3,3] => [2] => ([],2)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,5,6],[4]]
=> [4,2] => [1,1] => ([(0,1)],2)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,4,6],[5]]
=> [5,1] => [1,1] => ([(0,1)],2)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,4,5],[6]]
=> [6] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,3,5,6],[2,4]]
=> [2,2,2] => [3] => ([],3)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,5,6],[3,4]]
=> [3,3] => [2] => ([],2)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,3,4,6],[2,5]]
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,4,6],[3,5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4,6],[2],[5]]
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,4,6],[3],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4],[2,5,6]]
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,4],[3,5,6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,4,6],[2,5],[3]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,6],[2,4],[5]]
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,6],[3,4],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4],[2,6],[5]]
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,4],[3,6],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,4,6],[2],[3],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3],[2,4],[5,6]]
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2],[3,4],[5,6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,4],[2,6],[3],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4,6,7],[2,5]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4,5,7],[2,6]]
=> [2,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,3,5,7],[4,6]]
=> [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4,6,7],[2],[5]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4,5,7],[2],[6]]
=> [2,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,3,5,7],[4],[6]]
=> [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4,7],[2,5,6]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4,6],[2,5,7]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4,5],[2,6,7]]
=> [2,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,3,5],[4,6,7]]
=> [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,6,7],[2,4],[5]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,5,7],[3,6],[4]]
=> [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4,7],[2,6],[5]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4,7],[2,5],[6]]
=> [2,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,3,7],[4,5],[6]]
=> [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4,6],[2,7],[5]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4,5],[2,7],[6]]
=> [2,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,3,5],[4,7],[6]]
=> [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4,6],[2,5],[7]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,6,7],[2],[4],[5]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,5,7],[3],[4],[6]]
=> [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4,7],[2],[5],[6]]
=> [2,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4,6],[2],[5],[7]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,6],[2,4,7],[5]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,5],[3,6,7],[4]]
=> [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4],[2,6,7],[5]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4],[2,5,7],[6]]
=> [2,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,3],[4,5,7],[6]]
=> [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4],[2,5,6],[7]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,7],[3,5],[4,6]]
=> [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,7],[2,4],[5,6]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,6],[2,4],[5,7]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4],[2,6],[5,7]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St000259
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 86%
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 86%
Values
[[1,2]]
=> [1,2] => [1,2] => ([],2)
=> ? = 1
[[1],[2]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,2}
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {1,2}
[[1,2],[3]]
=> [3,1,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,2,2,3}
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {1,2,2,3}
[[1,2,4],[3]]
=> [3,1,2,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {1,2,2,3}
[[1,2,3],[4]]
=> [4,1,2,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,3],[2,4]]
=> [2,4,1,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[[1,2],[3,4]]
=> [3,4,1,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 3
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,2,2,3}
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[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)
=> 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3,4}
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3,4}
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3,4}
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3,4}
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3,4}
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3,4}
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3,4}
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3,4}
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3,4}
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[[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)
=> ? ∊ {1,2,2,2,2,2,2,3,3,4}
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[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)
=> 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [2,3,1,4,5,6] => ([(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [2,3,4,1,5,6] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [3,4,1,2,5,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [3,4,5,1,2,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [2,4,5,1,3,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [2,3,5,1,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 3
[[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 3
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [3,4,2,1,5,6] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => [2,4,3,1,5,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[[1,3,4,6],[2],[5]]
=> [5,2,1,3,4,6] => [3,4,5,2,1,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => [2,4,5,3,1,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[[1,2,3,6],[4],[5]]
=> [5,4,1,2,3,6] => [2,3,5,4,1,6] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[[1,3,4,5],[2],[6]]
=> [6,2,1,3,4,5] => [3,4,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => [2,4,5,6,3,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,3,5],[4],[6]]
=> [6,4,1,2,3,5] => [2,3,5,6,4,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => [2,3,4,6,5,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 3
[[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 3
[[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 3
[[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 3
[[1,4,6],[2,5],[3]]
=> [3,2,5,1,4,6] => [4,5,1,3,2,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [3,5,1,4,2,6] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[[1,2,6],[3,5],[4]]
=> [4,3,5,1,2,6] => [2,5,1,4,3,6] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[[1,3,6],[2,4],[5]]
=> [5,2,4,1,3,6] => [3,4,1,5,2,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[[1,2,6],[3,4],[5]]
=> [5,3,4,1,2,6] => [2,4,1,5,3,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[[1,4,5],[2,6],[3]]
=> [3,2,6,1,4,5] => [4,5,6,1,3,2] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,5],[2,6],[4]]
=> [4,2,6,1,3,5] => [3,5,6,1,4,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[[1,2,5],[3,6],[4]]
=> [4,3,6,1,2,5] => [2,5,6,1,4,3] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 3
[[1,3,4],[2,6],[5]]
=> [5,2,6,1,3,4] => [3,4,6,1,5,2] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> 2
[[1,2,4],[3,6],[5]]
=> [5,3,6,1,2,4] => [2,4,6,1,5,3] => ([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[[1,2,3],[4,6],[5]]
=> [5,4,6,1,2,3] => [2,3,6,1,5,4] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 3
[[1,3,5],[2,4],[6]]
=> [6,2,4,1,3,5] => [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 3
[[1,2,5],[3,4],[6]]
=> [6,3,4,1,2,5] => [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 4
[[1,3,4],[2,5],[6]]
=> [6,2,5,1,3,4] => [3,4,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[[1,2,4],[3,5],[6]]
=> [6,3,5,1,2,4] => [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 4
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [4,3,2,1,5,6] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => [4,5,3,2,1,6] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => [3,5,4,2,1,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[[1,2,6],[3],[4],[5]]
=> [5,4,3,1,2,6] => [2,5,4,3,1,6] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [5,4,3,2,1,6] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5}
[[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ([],7)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ([(5,6)],7)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[[1,2,4,5,6,7],[3]]
=> [3,1,2,4,5,6,7] => [2,3,1,4,5,6,7] => ([(4,6),(5,6)],7)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[[1,2,3,5,6,7],[4]]
=> [4,1,2,3,5,6,7] => [2,3,4,1,5,6,7] => ([(3,6),(4,6),(5,6)],7)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[[1,2,3,4,6,7],[5]]
=> [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[[1,2,3,4,5,7],[6]]
=> [6,1,2,3,4,5,7] => [2,3,4,5,6,1,7] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[[1,3,5,6,7],[2,4]]
=> [2,4,1,3,5,6,7] => [3,4,1,2,5,6,7] => ([(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St001330
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1] => ([],1)
=> 1
[[1],[2]]
=> [2,1] => [1] => ([],1)
=> 1
[[1,2,3]]
=> [1,2,3] => [1,2] => ([],2)
=> 1
[[1,3],[2]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 2
[[1,2],[3]]
=> [3,1,2] => [1,2] => ([],2)
=> 1
[[1],[2],[3]]
=> [3,2,1] => [2,1] => ([(0,1)],2)
=> 2
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3] => ([],3)
=> 1
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3] => ([(1,2)],3)
=> 2
[[1,2,4],[3]]
=> [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[1,2,3],[4]]
=> [4,1,2,3] => [1,2,3] => ([],3)
=> 1
[[1,3],[2,4]]
=> [2,4,1,3] => [2,1,3] => ([(1,2)],3)
=> 2
[[1,2],[3,4]]
=> [3,4,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,1,3] => ([(1,2)],3)
=> 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4] => ([],4)
=> 1
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4] => ([(2,3)],4)
=> 2
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,4] => ([],4)
=> 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 3
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 2
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3}
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3}
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 2
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3}
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3}
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 3
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3}
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3}
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5] => ([],5)
=> 1
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5] => ([(3,4)],5)
=> 2
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 2
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 2
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> 2
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> 2
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 2
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 2
[[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 3
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,3,4,6],[2],[5]]
=> [5,2,1,3,4,6] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,3,6],[4],[5]]
=> [5,4,1,2,3,6] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,4,6],[2,5],[3]]
=> [3,2,5,1,4,6] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,6],[3,5],[4]]
=> [4,3,5,1,2,6] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,3,6],[2,4],[5]]
=> [5,2,4,1,3,6] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,6],[3,4],[5]]
=> [5,3,4,1,2,6] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,3,5],[2,6],[4]]
=> [4,2,6,1,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,5],[3,6],[4]]
=> [4,3,6,1,2,5] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,3,4],[2,6],[5]]
=> [5,2,6,1,3,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,4],[3,6],[5]]
=> [5,3,6,1,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,3],[4,6],[5]]
=> [5,4,6,1,2,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,4],[3,5],[6]]
=> [6,3,5,1,2,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,6],[3],[4],[5]]
=> [5,4,3,1,2,6] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,5],[3],[4],[6]]
=> [6,4,3,1,2,5] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,4],[3],[5],[6]]
=> [6,5,3,1,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2],[3,6],[4],[5]]
=> [5,4,3,6,1,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000454
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1] => ([],1)
=> 0 = 1 - 1
[[1],[2]]
=> [2,1] => [1] => ([],1)
=> 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => [1,2] => ([],2)
=> 0 = 1 - 1
[[1,3],[2]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[[1,2],[3]]
=> [3,1,2] => [1,2] => ([],2)
=> 0 = 1 - 1
[[1],[2],[3]]
=> [3,2,1] => [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3] => ([(1,2)],3)
=> 1 = 2 - 1
[[1,2,4],[3]]
=> [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2} - 1
[[1,2,3],[4]]
=> [4,1,2,3] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[[1,3],[2,4]]
=> [2,4,1,3] => [2,1,3] => ([(1,2)],3)
=> 1 = 2 - 1
[[1,2],[3,4]]
=> [3,4,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2} - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,1,3] => ([(1,2)],3)
=> 1 = 2 - 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2} - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4] => ([(2,3)],4)
=> 1 = 2 - 1
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 3 - 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 1 = 2 - 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 1 = 2 - 1
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 3 - 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5] => ([],5)
=> 0 = 1 - 1
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5] => ([(3,4)],5)
=> 1 = 2 - 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4} - 1
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4} - 1
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0 = 1 - 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4} - 1
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 3 - 1
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4} - 1
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4} - 1
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4} - 1
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> 1 = 2 - 1
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4} - 1
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4} - 1
[[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4} - 1
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4} - 1
[[1,3,4,6],[2],[5]]
=> [5,2,1,3,4,6] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4} - 1
[[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4} - 1
[[1,2,3,6],[4],[5]]
=> [5,4,1,2,3,6] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[1,3,4,5],[2],[6]]
=> [6,2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> 1 = 2 - 1
[[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4} - 1
[[1,2,3,5],[4],[6]]
=> [6,4,1,2,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4} - 1
[[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4} - 1
[[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 3 - 1
[[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4} - 1
[[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4} - 1
[[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4} - 1
[[1,4,6],[2,5],[3]]
=> [3,2,5,1,4,6] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4} - 1
[[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4} - 1
[[1,2,6],[3,5],[4]]
=> [4,3,5,1,2,6] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4} - 1
[[1,3,6],[2,4],[5]]
=> [5,2,4,1,3,6] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4} - 1
[[1,2,6],[3,4],[5]]
=> [5,3,4,1,2,6] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4} - 1
[[1,4,5],[2,6],[3]]
=> [3,2,6,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[1,3,5],[2,6],[4]]
=> [4,2,6,1,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4} - 1
[[1,2,5],[3,6],[4]]
=> [4,3,6,1,2,5] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4} - 1
[[1,3,4],[2,6],[5]]
=> [5,2,6,1,3,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4} - 1
[[1,2,4],[3,6],[5]]
=> [5,3,6,1,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4} - 1
[[1,2,3],[4,6],[5]]
=> [5,4,6,1,2,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[1,3,5],[2,4],[6]]
=> [6,2,4,1,3,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4} - 1
[[1,2,5],[3,4],[6]]
=> [6,3,4,1,2,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 3 - 1
[[1,3,4],[2,5],[6]]
=> [6,2,5,1,3,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4} - 1
[[1,2,4],[3,5],[6]]
=> [6,3,5,1,2,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4} - 1
[[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4} - 1
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4} - 1
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4} - 1
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [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 = 5 - 1
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [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)
=> 4 = 5 - 1
[[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6] => ([],6)
=> 0 = 1 - 1
[[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [2,1,3,4,5,6] => ([(4,5)],6)
=> 1 = 2 - 1
[[1,2,3,4,6,7],[5]]
=> [5,1,2,3,4,6,7] => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
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: St001200
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 29%
Mp00252: Permutations —restriction⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 29%
Values
[[1,2]]
=> [1,2] => [1] => [1,0]
=> ? ∊ {1,1}
[[1],[2]]
=> [2,1] => [1] => [1,0]
=> ? ∊ {1,1}
[[1,2,3]]
=> [1,2,3] => [1,2] => [1,0,1,0]
=> 2
[[1,3],[2]]
=> [2,1,3] => [2,1] => [1,1,0,0]
=> ? ∊ {1,1}
[[1,2],[3]]
=> [3,1,2] => [1,2] => [1,0,1,0]
=> 2
[[1],[2],[3]]
=> [3,2,1] => [2,1] => [1,1,0,0]
=> ? ∊ {1,1}
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3] => [1,0,1,0,1,0]
=> 3
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[[1,2,4],[3]]
=> [3,1,2,4] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {1,1,2,2,2}
[[1,2,3],[4]]
=> [4,1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 3
[[1,3],[2,4]]
=> [2,4,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[[1,2],[3,4]]
=> [3,4,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {1,1,2,2,2}
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {1,1,2,2,2}
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {1,1,2,2,2}
[[1],[2],[3],[4]]
=> [4,3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {1,1,2,2,2}
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,2,3,4,4}
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,2,3,4,4}
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,2,3,4,4}
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,2,3,4,4}
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,2,3,4,4}
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,2,3,4,4}
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,2,3,4,4}
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,2,3,4,4}
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,2,3,4,4}
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,2,3,4,4}
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,2,2,2,3,4,4}
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 3
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 3
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 3
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> 3
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 3
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> 2
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> 2
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 3
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[[1,3,4,6],[2],[5]]
=> [5,2,1,3,4,6] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,6],[4],[5]]
=> [5,4,1,2,3,6] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,3,4,5],[2],[6]]
=> [6,2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 3
[[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[[1,2,3,5],[4],[6]]
=> [6,4,1,2,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> 3
[[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 3
[[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> 2
[[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> 2
[[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[[1,4,6],[2,5],[3]]
=> [3,2,5,1,4,6] => [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> 2
[[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[[1,2,6],[3,5],[4]]
=> [4,3,5,1,2,6] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[[1,3,6],[2,4],[5]]
=> [5,2,4,1,3,6] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,6],[3,4],[5]]
=> [5,3,4,1,2,6] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,4,5],[2,6],[3]]
=> [3,2,6,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[[1,3,4],[2,6],[5]]
=> [5,2,6,1,3,4] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,4],[3,6],[5]]
=> [5,3,6,1,2,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,3],[4,6],[5]]
=> [5,4,6,1,2,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,6],[3],[4],[5]]
=> [5,4,3,1,2,6] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,4],[3],[5],[6]]
=> [6,5,3,1,2,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2],[3,6],[4],[5]]
=> [5,4,3,6,1,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6}
Description
The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St000260
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 43%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 43%
Values
[[1,2]]
=> [1,2] => [1,2] => ([],2)
=> ? = 1
[[1],[2]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,2,2}
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {1,2,2}
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {1,2,2}
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,2,2,2,2,2,3,3}
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,3,3}
[[1,2,4],[3]]
=> [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,3,3}
[[1,2,3],[4]]
=> [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,3,3}
[[1,3],[2,4]]
=> [2,4,1,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {1,2,2,2,2,2,3,3}
[[1,2],[3,4]]
=> [3,4,1,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,3,3}
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,3,3}
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,3,3}
[[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)
=> 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[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)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[[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)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[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)
=> 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => ([(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => ([(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => [3,2,6,1,4,5] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 2
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [2,4,1,6,5,3] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 2
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [3,5,2,1,6,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [2,5,4,1,6,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => [3,2,6,1,5,4] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> 2
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [2,4,6,1,5,3] => ([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => [4,6,3,2,1,5] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => [3,6,5,2,1,4] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => [2,6,5,4,1,3] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[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)
=> 1
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => [2,4,1,7,3,5,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 3
[[1,3,6],[2,5],[4,7]]
=> [4,7,2,5,1,3,6] => [2,4,1,5,7,3,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 3
[[1,4,5],[2,7],[3],[6]]
=> [6,3,2,7,1,4,5] => [3,2,6,1,4,7,5] => ([(0,6),(1,4),(2,3),(2,5),(3,5),(4,6),(5,6)],7)
=> 2
[[1,3,5],[2,7],[4],[6]]
=> [6,4,2,7,1,3,5] => [2,4,6,1,3,7,5] => ([(0,6),(1,4),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> 2
[[1,3,6],[2,5],[4],[7]]
=> [7,4,2,5,1,3,6] => [2,4,1,7,5,3,6] => ([(0,6),(1,4),(2,5),(2,6),(3,4),(3,5),(5,6)],7)
=> 3
[[1,3,5],[2,6],[4],[7]]
=> [7,4,2,6,1,3,5] => [2,4,1,7,3,6,5] => ([(0,5),(1,4),(1,5),(2,3),(2,6),(3,6),(4,6)],7)
=> 3
[[1,5,6],[2],[3],[4],[7]]
=> [7,4,3,2,1,5,6] => [4,3,7,2,1,5,6] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 2
[[1,4,6],[2],[3],[5],[7]]
=> [7,5,3,2,1,4,6] => [3,5,7,2,1,4,6] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 2
[[1,3,6],[2],[4],[5],[7]]
=> [7,5,4,2,1,3,6] => [2,5,7,4,1,3,6] => ([(0,6),(1,5),(2,3),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[[1,4,5],[2],[3],[6],[7]]
=> [7,6,3,2,1,4,5] => [3,2,7,6,1,4,5] => ([(0,1),(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[[1,3,5],[2],[4],[6],[7]]
=> [7,6,4,2,1,3,5] => [2,4,7,6,1,3,5] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 2
[[1,4],[2,6],[3,7],[5]]
=> [5,3,7,2,6,1,4] => [3,5,2,1,7,6,4] => ([(0,4),(0,5),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5)],7)
=> 2
[[1,3],[2,6],[4,7],[5]]
=> [5,4,7,2,6,1,3] => [2,5,4,1,7,6,3] => ([(0,5),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 2
[[1,4],[2,5],[3,7],[6]]
=> [6,3,7,2,5,1,4] => [3,2,6,1,7,5,4] => ([(0,4),(0,5),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5)],7)
=> 2
[[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => [2,4,6,1,7,5,3] => ([(0,5),(1,4),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6)],7)
=> 2
[[1,3],[2,5],[4,6],[7]]
=> [7,4,6,2,5,1,3] => [2,4,1,7,6,5,3] => ([(0,2),(1,2),(1,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => [4,6,3,2,1,7,5] => ([(0,1),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[[1,4],[2,7],[3],[5],[6]]
=> [6,5,3,2,7,1,4] => [3,6,5,2,1,7,4] => ([(0,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[[1,3],[2,7],[4],[5],[6]]
=> [6,5,4,2,7,1,3] => [2,6,5,4,1,7,3] => ([(0,6),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 2
[[1,5],[2,6],[3],[4],[7]]
=> [7,4,3,2,6,1,5] => [4,3,7,2,1,6,5] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 2
[[1,4],[2,6],[3],[5],[7]]
=> [7,5,3,2,6,1,4] => [3,5,7,2,1,6,4] => ([(0,5),(0,6),(1,3),(1,4),(2,3),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> 2
[[1,3],[2,6],[4],[5],[7]]
=> [7,5,4,2,6,1,3] => [2,5,7,4,1,6,3] => ([(0,6),(1,4),(1,5),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 2
[[1,4],[2,5],[3],[6],[7]]
=> [7,6,3,2,5,1,4] => [3,2,7,6,1,5,4] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 2
[[1,3],[2,5],[4],[6],[7]]
=> [7,6,4,2,5,1,3] => [2,4,7,6,1,5,3] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 2
[[1,6],[2],[3],[4],[5],[7]]
=> [7,5,4,3,2,1,6] => [5,7,4,3,2,1,6] => ([(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)
=> 2
[[1,5],[2],[3],[4],[6],[7]]
=> [7,6,4,3,2,1,5] => [4,7,6,3,2,1,5] => ([(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)
=> 2
[[1,4],[2],[3],[5],[6],[7]]
=> [7,6,5,3,2,1,4] => [3,7,6,5,2,1,4] => ([(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)
=> 2
[[1,3],[2],[4],[5],[6],[7]]
=> [7,6,5,4,2,1,3] => [2,7,6,5,4,1,3] => ([(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)
=> 2
[[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)
=> 1
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St001712
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 43%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 43%
Values
[[1,2]]
=> [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[[1],[2]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 1
[[1,2,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> 1
[[1,3],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2
[[1,2],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2
[[1],[2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 1
[[1,2,3,4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? ∊ {3,3}
[[1,3,4],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 2
[[1,2,4],[3]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 2
[[1,2,3],[4]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 2
[[1,3],[2,4]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 1
[[1,2],[3,4]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? ∊ {3,3}
[[1,2,3,4,5]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> ? ∊ {1,1,3,3,3,3,3,3,4,4}
[[1,3,4,5],[2]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? ∊ {1,1,3,3,3,3,3,3,4,4}
[[1,2,4,5],[3]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? ∊ {1,1,3,3,3,3,3,3,4,4}
[[1,2,3,5],[4]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? ∊ {1,1,3,3,3,3,3,3,4,4}
[[1,2,3,4],[5]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? ∊ {1,1,3,3,3,3,3,3,4,4}
[[1,3,5],[2,4]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2
[[1,2,5],[3,4]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2
[[1,3,4],[2,5]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2
[[1,2,4],[3,5]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2
[[1,2,3],[4,5]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2
[[1,4,5],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2
[[1,3,5],[2],[4]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2
[[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2
[[1,3,4],[2],[5]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2
[[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2
[[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2
[[1,4],[2,5],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2
[[1,3],[2,5],[4]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2
[[1,2],[3,5],[4]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2
[[1,3],[2,4],[5]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2
[[1,2],[3,4],[5]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? ∊ {1,1,3,3,3,3,3,3,4,4}
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? ∊ {1,1,3,3,3,3,3,3,4,4}
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? ∊ {1,1,3,3,3,3,3,3,4,4}
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? ∊ {1,1,3,3,3,3,3,3,4,4}
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? ∊ {1,1,3,3,3,3,3,3,4,4}
[[1,2,3,4,5,6]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,6,13],[7,8,9,10,11,12,14]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,3,4,5,6],[2]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,4,5,6],[3]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,5,6],[4]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,4,6],[5]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,4,5],[6]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,3,5,6],[2,4]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,5,6],[3,4]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,3,4,6],[2,5]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,4,6],[3,5]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,6],[4,5]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,3,4,5],[2,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,4,5],[3,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,5],[4,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,4],[5,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,4,5,6],[2],[3]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,3,5,6],[2],[4]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,5,6],[3],[4]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,3,4,6],[2],[5]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,4,6],[3],[5]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,6],[4],[5]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,3,4,5],[2],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,4,5],[3],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,5],[4],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,4],[5],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,3,5],[2,4,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,5],[3,4,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,3,4],[2,5,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,4],[3,5,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,3],[4,5,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,4,6],[2,5],[3]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 3
[[1,3,6],[2,5],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 3
[[1,2,6],[3,5],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 3
[[1,3,6],[2,4],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 3
[[1,2,6],[3,4],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 3
[[1,4,5],[2,6],[3]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 3
[[1,3,5],[2,6],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 3
[[1,2,5],[3,6],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 3
[[1,3,4],[2,6],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 3
[[1,2,4],[3,6],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 3
[[1,2,3],[4,6],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 3
[[1,3,5],[2,4],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 3
[[1,2,5],[3,4],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 3
[[1,3,4],[2,5],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 3
[[1,2,4],[3,5],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 3
[[1,2,3],[4,5],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 3
[[1,5,6],[2],[3],[4]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,4,6],[2],[3],[5]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,3,6],[2],[4],[5]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,6],[3],[4],[5]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,4,5],[2],[3],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,3,5],[2],[4],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,5],[3],[4],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,3,4],[2],[5],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
Description
The number of natural descents of a standard Young tableau.
A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation.
Matching statistic: St001060
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 43%
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 43%
Values
[[1,2]]
=> [2] => [2] => ([],2)
=> ? ∊ {1,1}
[[1],[2]]
=> [2] => [2] => ([],2)
=> ? ∊ {1,1}
[[1,2,3]]
=> [3] => [3] => ([],3)
=> ? ∊ {1,1,2,2}
[[1,3],[2]]
=> [2,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,2,2}
[[1,2],[3]]
=> [3] => [3] => ([],3)
=> ? ∊ {1,1,2,2}
[[1],[2],[3]]
=> [3] => [3] => ([],3)
=> ? ∊ {1,1,2,2}
[[1,2,3,4]]
=> [4] => [4] => ([],4)
=> ? ∊ {1,1,2,2,2,3,3}
[[1,3,4],[2]]
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[[1,2,4],[3]]
=> [3,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,2,2,2,3,3}
[[1,2,3],[4]]
=> [4] => [4] => ([],4)
=> ? ∊ {1,1,2,2,2,3,3}
[[1,3],[2,4]]
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[[1,2],[3,4]]
=> [3,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,2,2,2,3,3}
[[1,4],[2],[3]]
=> [3,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,2,2,2,3,3}
[[1,3],[2],[4]]
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[[1,2],[3],[4]]
=> [4] => [4] => ([],4)
=> ? ∊ {1,1,2,2,2,3,3}
[[1],[2],[3],[4]]
=> [4] => [4] => ([],4)
=> ? ∊ {1,1,2,2,2,3,3}
[[1,2,3,4,5]]
=> [5] => [5] => ([],5)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4}
[[1,3,4,5],[2]]
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[[1,2,4,5],[3]]
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4}
[[1,2,3,5],[4]]
=> [4,1] => [1,4] => ([(3,4)],5)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4}
[[1,2,3,4],[5]]
=> [5] => [5] => ([],5)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4}
[[1,3,5],[2,4]]
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,2,5],[3,4]]
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4}
[[1,3,4],[2,5]]
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[[1,2,4],[3,5]]
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4}
[[1,2,3],[4,5]]
=> [4,1] => [1,4] => ([(3,4)],5)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4}
[[1,4,5],[2],[3]]
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4}
[[1,3,5],[2],[4]]
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,2,5],[3],[4]]
=> [4,1] => [1,4] => ([(3,4)],5)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4}
[[1,3,4],[2],[5]]
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[[1,2,4],[3],[5]]
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4}
[[1,2,3],[4],[5]]
=> [5] => [5] => ([],5)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4}
[[1,4],[2,5],[3]]
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4}
[[1,3],[2,5],[4]]
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,2],[3,5],[4]]
=> [4,1] => [1,4] => ([(3,4)],5)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4}
[[1,3],[2,4],[5]]
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[[1,2],[3,4],[5]]
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4}
[[1,5],[2],[3],[4]]
=> [4,1] => [1,4] => ([(3,4)],5)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4}
[[1,4],[2],[3],[5]]
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4}
[[1,3],[2],[4],[5]]
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[[1,2],[3],[4],[5]]
=> [5] => [5] => ([],5)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4}
[[1],[2],[3],[4],[5]]
=> [5] => [5] => ([],5)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4}
[[1,2,3,4,5,6]]
=> [6] => [6] => ([],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5}
[[1,3,4,5,6],[2]]
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[1,2,4,5,6],[3]]
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5}
[[1,2,3,5,6],[4]]
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5}
[[1,2,3,4,6],[5]]
=> [5,1] => [1,5] => ([(4,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5}
[[1,2,3,4,5],[6]]
=> [6] => [6] => ([],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5}
[[1,3,5,6],[2,4]]
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,5,6],[3,4]]
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5}
[[1,3,4,6],[2,5]]
=> [2,3,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5}
[[1,2,4,6],[3,5]]
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,3,6],[4,5]]
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5}
[[1,3,4,5],[2,6]]
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[1,2,4,5],[3,6]]
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5}
[[1,2,3,5],[4,6]]
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5}
[[1,2,3,4],[5,6]]
=> [5,1] => [1,5] => ([(4,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5}
[[1,4,5,6],[2],[3]]
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5}
[[1,3,5,6],[2],[4]]
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,5,6],[3],[4]]
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5}
[[1,3,4,6],[2],[5]]
=> [2,3,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5}
[[1,2,4,6],[3],[5]]
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,3,6],[4],[5]]
=> [5,1] => [1,5] => ([(4,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5}
[[1,3,4,5],[2],[6]]
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[1,2,4,5],[3],[6]]
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5}
[[1,2,3,5],[4],[6]]
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5}
[[1,2,3,4],[5],[6]]
=> [6] => [6] => ([],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5}
[[1,3,5],[2,4,6]]
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,5],[3,4,6]]
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,5,5}
[[1,2,4],[3,5,6]]
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,4,6],[2,5],[3]]
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,6],[2,5],[4]]
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,6],[3,4],[5]]
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,5],[2,6],[4]]
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,4],[3,6],[5]]
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,5],[2,4],[6]]
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,4],[2,5],[6]]
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[1,4,6],[2],[3],[5]]
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,5],[2],[4],[6]]
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,4],[2],[5],[6]]
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[1,3],[2,5],[4,6]]
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2],[3,4],[5,6]]
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,4],[2,6],[3],[5]]
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3],[2,5],[4],[6]]
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3],[2,4],[5],[6]]
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[1,3],[2],[4],[5],[6]]
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St000845
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000845: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 43%
Mp00209: Permutations —pattern poset⟶ Posets
St000845: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 43%
Values
[[1,2]]
=> [1,2] => ([(0,1)],2)
=> 1
[[1],[2]]
=> [2,1] => ([(0,1)],2)
=> 1
[[1,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,3],[2]]
=> [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1,2],[3]]
=> [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1],[2],[3]]
=> [3,2,1] => ([(0,2),(2,1)],3)
=> 1
[[1,2,3,4]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,3,4],[2]]
=> [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[[1,2,4],[3]]
=> [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3
[[1,2,3],[4]]
=> [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[[1,3],[2,4]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 2
[[1,2],[3,4]]
=> [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2
[[1,4],[2],[3]]
=> [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[[1,3],[2],[4]]
=> [4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3
[[1,2],[3],[4]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4}
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => ([(0,2),(0,3),(0,4),(1,8),(1,9),(2,1),(2,10),(2,11),(3,6),(3,7),(3,11),(4,6),(4,7),(4,10),(6,14),(7,12),(7,14),(8,13),(8,15),(9,13),(9,15),(10,8),(10,12),(10,14),(11,9),(11,12),(11,14),(12,13),(12,15),(13,5),(14,15),(15,5)],16)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(1,20),(1,21),(2,13),(2,14),(3,8),(3,13),(3,15),(4,9),(4,10),(4,13),(4,15),(5,7),(5,8),(5,9),(5,14),(6,1),(6,7),(6,10),(6,14),(6,15),(7,12),(7,16),(7,18),(7,20),(8,18),(8,21),(9,16),(9,18),(9,21),(10,16),(10,20),(10,21),(12,17),(12,19),(13,21),(14,20),(14,21),(15,18),(15,20),(15,21),(16,17),(16,19),(17,11),(18,17),(18,19),(19,11),(20,17),(20,19),(21,19)],22)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,18),(1,21),(2,11),(2,12),(2,19),(3,9),(3,13),(3,19),(4,10),(4,13),(4,19),(5,10),(5,12),(5,14),(5,19),(6,1),(6,9),(6,11),(6,14),(6,19),(8,17),(8,20),(9,18),(9,21),(10,16),(10,21),(11,15),(11,18),(11,21),(12,15),(12,16),(13,21),(14,8),(14,15),(14,16),(14,18),(15,17),(15,20),(16,17),(16,20),(17,7),(18,17),(18,20),(19,16),(19,18),(19,21),(20,7),(21,20)],22)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => ([(0,3),(0,4),(0,5),(1,14),(2,6),(2,8),(2,14),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,13),(6,15),(8,13),(8,15),(9,12),(9,14),(10,8),(10,12),(11,6),(11,12),(11,14),(12,13),(12,15),(13,7),(14,15),(15,7)],16)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => ([(0,2),(0,3),(0,4),(0,6),(1,15),(1,17),(2,12),(2,13),(3,7),(3,12),(4,8),(4,12),(4,13),(5,1),(5,10),(5,11),(5,14),(6,5),(6,7),(6,8),(6,13),(7,10),(7,16),(8,11),(8,14),(8,16),(10,15),(10,17),(11,15),(11,17),(12,16),(13,14),(13,16),(14,15),(14,17),(15,9),(16,17),(17,9)],18)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => ([(0,3),(0,4),(0,5),(0,6),(1,11),(1,18),(2,12),(2,17),(2,18),(3,7),(3,14),(4,1),(4,10),(4,13),(4,14),(5,2),(5,9),(5,13),(5,14),(6,7),(6,9),(6,10),(7,17),(9,15),(9,17),(10,15),(10,17),(10,18),(11,16),(11,19),(12,16),(12,19),(13,11),(13,12),(13,15),(13,18),(14,17),(14,18),(15,16),(15,19),(16,8),(17,19),(18,16),(18,19),(19,8)],20)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => ([(0,2),(0,3),(0,4),(0,6),(1,15),(1,17),(2,7),(2,14),(3,9),(3,14),(4,9),(4,10),(4,14),(5,1),(5,11),(5,12),(5,16),(6,5),(6,7),(6,10),(6,14),(7,11),(7,16),(9,13),(10,12),(10,13),(10,16),(11,15),(11,17),(12,15),(12,17),(13,17),(14,13),(14,16),(15,8),(16,15),(16,17),(17,8)],18)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ([(0,4),(0,5),(1,7),(2,9),(2,11),(3,2),(3,10),(4,3),(4,6),(5,1),(5,6),(6,7),(6,10),(7,11),(9,8),(10,9),(10,11),(11,8)],12)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => ([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,3,4,6],[2],[5]]
=> [5,2,1,3,4,6] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(1,19),(2,9),(2,12),(2,19),(3,8),(3,12),(3,19),(4,6),(4,8),(4,10),(4,19),(5,6),(5,9),(5,11),(5,19),(6,13),(6,17),(6,18),(8,16),(8,17),(9,16),(9,18),(10,13),(10,17),(11,13),(11,18),(12,16),(13,15),(14,7),(15,7),(16,14),(17,14),(17,15),(18,14),(18,15),(19,16),(19,17),(19,18)],20)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,6],[4],[5]]
=> [5,4,1,2,3,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,3,4,5],[2],[6]]
=> [6,2,1,3,4,5] => ([(0,3),(0,4),(0,5),(1,14),(2,1),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(6,14),(7,13),(7,14),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8),(14,8)],15)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,5],[4],[6]]
=> [6,4,1,2,3,5] => ([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(1,16),(1,17),(1,24),(2,11),(2,15),(2,17),(2,24),(3,9),(3,13),(3,15),(3,24),(4,10),(4,14),(4,16),(4,24),(5,7),(5,9),(5,11),(5,14),(5,24),(6,7),(6,10),(6,12),(6,13),(6,24),(7,21),(7,22),(7,25),(9,21),(9,25),(10,22),(10,25),(11,19),(11,21),(11,25),(12,20),(12,22),(12,25),(13,19),(13,22),(13,25),(14,20),(14,21),(14,25),(15,19),(15,25),(16,20),(16,25),(17,19),(17,20),(18,8),(19,18),(19,23),(20,18),(20,23),(21,18),(21,23),(22,18),(22,23),(23,8),(24,19),(24,20),(24,21),(24,22),(25,23)],26)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5}
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
Description
The maximal number of elements covered by an element in a poset.
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St000846The maximal number of elements covering an element of a poset.
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