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Your data matches 47 different statistics following compositions of up to 3 maps.
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Matching statistic: St000110
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Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000110: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => 1
[[2,2]]
=> [1,2] => 1
[[1],[2]]
=> [2,1] => 2
[[1,3]]
=> [1,2] => 1
[[2,3]]
=> [1,2] => 1
[[3,3]]
=> [1,2] => 1
[[1],[3]]
=> [2,1] => 2
[[2],[3]]
=> [2,1] => 2
[[1,1,2]]
=> [1,2,3] => 1
[[1,2,2]]
=> [1,2,3] => 1
[[2,2,2]]
=> [1,2,3] => 1
[[1,1],[2]]
=> [3,1,2] => 3
[[1,2],[2]]
=> [2,1,3] => 2
[[1,4]]
=> [1,2] => 1
[[2,4]]
=> [1,2] => 1
[[3,4]]
=> [1,2] => 1
[[4,4]]
=> [1,2] => 1
[[1],[4]]
=> [2,1] => 2
[[2],[4]]
=> [2,1] => 2
[[3],[4]]
=> [2,1] => 2
[[1,1,3]]
=> [1,2,3] => 1
[[1,2,3]]
=> [1,2,3] => 1
[[1,3,3]]
=> [1,2,3] => 1
[[2,2,3]]
=> [1,2,3] => 1
[[2,3,3]]
=> [1,2,3] => 1
[[3,3,3]]
=> [1,2,3] => 1
[[1,1],[3]]
=> [3,1,2] => 3
[[1,2],[3]]
=> [3,1,2] => 3
[[1,3],[2]]
=> [2,1,3] => 2
[[1,3],[3]]
=> [2,1,3] => 2
[[2,2],[3]]
=> [3,1,2] => 3
[[2,3],[3]]
=> [2,1,3] => 2
[[1],[2],[3]]
=> [3,2,1] => 6
[[1,1,1,2]]
=> [1,2,3,4] => 1
[[1,1,2,2]]
=> [1,2,3,4] => 1
[[1,2,2,2]]
=> [1,2,3,4] => 1
[[2,2,2,2]]
=> [1,2,3,4] => 1
[[1,1,1],[2]]
=> [4,1,2,3] => 4
[[1,1,2],[2]]
=> [3,1,2,4] => 3
[[1,2,2],[2]]
=> [2,1,3,4] => 2
[[1,1],[2,2]]
=> [3,4,1,2] => 6
[[1,5]]
=> [1,2] => 1
[[2,5]]
=> [1,2] => 1
[[3,5]]
=> [1,2] => 1
[[4,5]]
=> [1,2] => 1
[[5,5]]
=> [1,2] => 1
[[1],[5]]
=> [2,1] => 2
[[2],[5]]
=> [2,1] => 2
[[3],[5]]
=> [2,1] => 2
[[4],[5]]
=> [2,1] => 2
Description
The number of permutations less than or equal to a permutation in left weak order.
This is the same as the number of permutations less than or equal to the given permutation in right weak order.
Matching statistic: St000100
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(load all 2 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => ([(0,1)],2)
=> 1
[[2,2]]
=> [1,2] => ([(0,1)],2)
=> 1
[[1],[2]]
=> [2,1] => ([],2)
=> 2
[[1,3]]
=> [1,2] => ([(0,1)],2)
=> 1
[[2,3]]
=> [1,2] => ([(0,1)],2)
=> 1
[[3,3]]
=> [1,2] => ([(0,1)],2)
=> 1
[[1],[3]]
=> [2,1] => ([],2)
=> 2
[[2],[3]]
=> [2,1] => ([],2)
=> 2
[[1,1,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,2,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[2,2,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,1],[2]]
=> [3,1,2] => ([(1,2)],3)
=> 3
[[1,2],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> 2
[[1,4]]
=> [1,2] => ([(0,1)],2)
=> 1
[[2,4]]
=> [1,2] => ([(0,1)],2)
=> 1
[[3,4]]
=> [1,2] => ([(0,1)],2)
=> 1
[[4,4]]
=> [1,2] => ([(0,1)],2)
=> 1
[[1],[4]]
=> [2,1] => ([],2)
=> 2
[[2],[4]]
=> [2,1] => ([],2)
=> 2
[[3],[4]]
=> [2,1] => ([],2)
=> 2
[[1,1,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[2,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[2,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[3,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,1],[3]]
=> [3,1,2] => ([(1,2)],3)
=> 3
[[1,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> 3
[[1,3],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> 2
[[1,3],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> 2
[[2,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> 3
[[2,3],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> 2
[[1],[2],[3]]
=> [3,2,1] => ([],3)
=> 6
[[1,1,1,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,2,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1,1],[2]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> 4
[[1,1,2],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 3
[[1,2,2],[2]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 2
[[1,1],[2,2]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> 6
[[1,5]]
=> [1,2] => ([(0,1)],2)
=> 1
[[2,5]]
=> [1,2] => ([(0,1)],2)
=> 1
[[3,5]]
=> [1,2] => ([(0,1)],2)
=> 1
[[4,5]]
=> [1,2] => ([(0,1)],2)
=> 1
[[5,5]]
=> [1,2] => ([(0,1)],2)
=> 1
[[1],[5]]
=> [2,1] => ([],2)
=> 2
[[2],[5]]
=> [2,1] => ([],2)
=> 2
[[3],[5]]
=> [2,1] => ([],2)
=> 2
[[4],[5]]
=> [2,1] => ([],2)
=> 2
Description
The number of linear extensions of a poset.
Matching statistic: St001855
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(load all 3 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001855: Signed permutations ⟶ ℤResult quality: 24% ●values known / values provided: 60%●distinct values known / distinct values provided: 24%
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001855: Signed permutations ⟶ ℤResult quality: 24% ●values known / values provided: 60%●distinct values known / distinct values provided: 24%
Values
[[1,2]]
=> [1,2] => [1,2] => [1,2] => 1
[[2,2]]
=> [1,2] => [1,2] => [1,2] => 1
[[1],[2]]
=> [2,1] => [2,1] => [2,1] => 2
[[1,3]]
=> [1,2] => [1,2] => [1,2] => 1
[[2,3]]
=> [1,2] => [1,2] => [1,2] => 1
[[3,3]]
=> [1,2] => [1,2] => [1,2] => 1
[[1],[3]]
=> [2,1] => [2,1] => [2,1] => 2
[[2],[3]]
=> [2,1] => [2,1] => [2,1] => 2
[[1,1,2]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1
[[1,2,2]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1
[[2,2,2]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1
[[1,1],[2]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2
[[1,2],[2]]
=> [2,1,3] => [2,3,1] => [2,3,1] => 3
[[1,4]]
=> [1,2] => [1,2] => [1,2] => 1
[[2,4]]
=> [1,2] => [1,2] => [1,2] => 1
[[3,4]]
=> [1,2] => [1,2] => [1,2] => 1
[[4,4]]
=> [1,2] => [1,2] => [1,2] => 1
[[1],[4]]
=> [2,1] => [2,1] => [2,1] => 2
[[2],[4]]
=> [2,1] => [2,1] => [2,1] => 2
[[3],[4]]
=> [2,1] => [2,1] => [2,1] => 2
[[1,1,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1
[[1,3,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1
[[2,2,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1
[[2,3,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1
[[3,3,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1
[[1,1],[3]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2
[[1,3],[2]]
=> [2,1,3] => [2,3,1] => [2,3,1] => 3
[[1,3],[3]]
=> [2,1,3] => [2,3,1] => [2,3,1] => 3
[[2,2],[3]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2
[[2,3],[3]]
=> [2,1,3] => [2,3,1] => [2,3,1] => 3
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 6
[[1,1,1,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[[1,1,1],[2]]
=> [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 2
[[1,1,2],[2]]
=> [3,1,2,4] => [1,3,4,2] => [1,3,4,2] => 3
[[1,2,2],[2]]
=> [2,1,3,4] => [2,3,4,1] => [2,3,4,1] => 4
[[1,1],[2,2]]
=> [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 6
[[1,5]]
=> [1,2] => [1,2] => [1,2] => 1
[[2,5]]
=> [1,2] => [1,2] => [1,2] => 1
[[3,5]]
=> [1,2] => [1,2] => [1,2] => 1
[[4,5]]
=> [1,2] => [1,2] => [1,2] => 1
[[5,5]]
=> [1,2] => [1,2] => [1,2] => 1
[[1],[5]]
=> [2,1] => [2,1] => [2,1] => 2
[[2],[5]]
=> [2,1] => [2,1] => [2,1] => 2
[[3],[5]]
=> [2,1] => [2,1] => [2,1] => 2
[[4],[5]]
=> [2,1] => [2,1] => [2,1] => 2
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {5,10}
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? ∊ {5,10}
[[1,2,2,3],[2]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {5,5,5,5,5,7,9,10,10,10,10,10,10,15,16,20,20,30}
[[1,2,3,3],[2]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {5,5,5,5,5,7,9,10,10,10,10,10,10,15,16,20,20,30}
[[1,3,3,3],[2]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {5,5,5,5,5,7,9,10,10,10,10,10,10,15,16,20,20,30}
[[1,3,3,3],[3]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {5,5,5,5,5,7,9,10,10,10,10,10,10,15,16,20,20,30}
[[2,3,3,3],[3]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {5,5,5,5,5,7,9,10,10,10,10,10,10,15,16,20,20,30}
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? ∊ {5,5,5,5,5,7,9,10,10,10,10,10,10,15,16,20,20,30}
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? ∊ {5,5,5,5,5,7,9,10,10,10,10,10,10,15,16,20,20,30}
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? ∊ {5,5,5,5,5,7,9,10,10,10,10,10,10,15,16,20,20,30}
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? ∊ {5,5,5,5,5,7,9,10,10,10,10,10,10,15,16,20,20,30}
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? ∊ {5,5,5,5,5,7,9,10,10,10,10,10,10,15,16,20,20,30}
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? ∊ {5,5,5,5,5,7,9,10,10,10,10,10,10,15,16,20,20,30}
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? ∊ {5,5,5,5,5,7,9,10,10,10,10,10,10,15,16,20,20,30}
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [2,3,5,4,1] => [2,3,5,4,1] => ? ∊ {5,5,5,5,5,7,9,10,10,10,10,10,10,15,16,20,20,30}
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [2,4,5,3,1] => [2,4,5,3,1] => ? ∊ {5,5,5,5,5,7,9,10,10,10,10,10,10,15,16,20,20,30}
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [3,4,5,2,1] => [3,4,5,2,1] => ? ∊ {5,5,5,5,5,7,9,10,10,10,10,10,10,15,16,20,20,30}
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [3,5,1,4,2] => [3,5,1,4,2] => ? ∊ {5,5,5,5,5,7,9,10,10,10,10,10,10,15,16,20,20,30}
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [4,5,1,3,2] => [4,5,1,3,2] => ? ∊ {5,5,5,5,5,7,9,10,10,10,10,10,10,15,16,20,20,30}
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [4,5,2,3,1] => [4,5,2,3,1] => ? ∊ {5,5,5,5,5,7,9,10,10,10,10,10,10,15,16,20,20,30}
[[1,1,1,1,1,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,2,3,4,5,6,6,10,15,20}
[[1,1,1,1,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,2,3,4,5,6,6,10,15,20}
[[1,1,1,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,2,3,4,5,6,6,10,15,20}
[[1,1,2,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,2,3,4,5,6,6,10,15,20}
[[1,2,2,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,2,3,4,5,6,6,10,15,20}
[[2,2,2,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,2,3,4,5,6,6,10,15,20}
[[1,1,1,1,1],[2]]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? ∊ {1,1,1,1,1,1,2,3,4,5,6,6,10,15,20}
[[1,1,1,1,2],[2]]
=> [5,1,2,3,4,6] => [1,2,3,5,6,4] => [1,2,3,5,6,4] => ? ∊ {1,1,1,1,1,1,2,3,4,5,6,6,10,15,20}
[[1,1,1,2,2],[2]]
=> [4,1,2,3,5,6] => [1,2,4,5,6,3] => [1,2,4,5,6,3] => ? ∊ {1,1,1,1,1,1,2,3,4,5,6,6,10,15,20}
[[1,1,2,2,2],[2]]
=> [3,1,2,4,5,6] => [1,3,4,5,6,2] => [1,3,4,5,6,2] => ? ∊ {1,1,1,1,1,1,2,3,4,5,6,6,10,15,20}
[[1,2,2,2,2],[2]]
=> [2,1,3,4,5,6] => [2,3,4,5,6,1] => [2,3,4,5,6,1] => ? ∊ {1,1,1,1,1,1,2,3,4,5,6,6,10,15,20}
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [1,2,5,6,3,4] => [1,2,5,6,3,4] => ? ∊ {1,1,1,1,1,1,2,3,4,5,6,6,10,15,20}
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [1,4,5,6,2,3] => [1,4,5,6,2,3] => ? ∊ {1,1,1,1,1,1,2,3,4,5,6,6,10,15,20}
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [3,4,5,6,1,2] => [3,4,5,6,1,2] => ? ∊ {1,1,1,1,1,1,2,3,4,5,6,6,10,15,20}
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [4,5,6,1,2,3] => [4,5,6,1,2,3] => ? ∊ {1,1,1,1,1,1,2,3,4,5,6,6,10,15,20}
[[1,2,2,4],[2]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,14,15,15,15,15,15,16,16,16,16,16,20,20,20,20,20,20,20,20,20,20,25,30,30,30,30,30,30,40,60}
[[1,2,3,4],[2]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,14,15,15,15,15,15,16,16,16,16,16,20,20,20,20,20,20,20,20,20,20,25,30,30,30,30,30,30,40,60}
[[1,2,4,4],[2]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,14,15,15,15,15,15,16,16,16,16,16,20,20,20,20,20,20,20,20,20,20,25,30,30,30,30,30,30,40,60}
[[1,3,3,4],[2]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,14,15,15,15,15,15,16,16,16,16,16,20,20,20,20,20,20,20,20,20,20,25,30,30,30,30,30,30,40,60}
[[1,3,4,4],[2]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,14,15,15,15,15,15,16,16,16,16,16,20,20,20,20,20,20,20,20,20,20,25,30,30,30,30,30,30,40,60}
[[1,4,4,4],[2]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,14,15,15,15,15,15,16,16,16,16,16,20,20,20,20,20,20,20,20,20,20,25,30,30,30,30,30,30,40,60}
[[1,3,3,4],[3]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,14,15,15,15,15,15,16,16,16,16,16,20,20,20,20,20,20,20,20,20,20,25,30,30,30,30,30,30,40,60}
[[1,3,4,4],[3]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,14,15,15,15,15,15,16,16,16,16,16,20,20,20,20,20,20,20,20,20,20,25,30,30,30,30,30,30,40,60}
[[1,4,4,4],[3]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,14,15,15,15,15,15,16,16,16,16,16,20,20,20,20,20,20,20,20,20,20,25,30,30,30,30,30,30,40,60}
[[1,4,4,4],[4]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,14,15,15,15,15,15,16,16,16,16,16,20,20,20,20,20,20,20,20,20,20,25,30,30,30,30,30,30,40,60}
[[2,3,3,4],[3]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,14,15,15,15,15,15,16,16,16,16,16,20,20,20,20,20,20,20,20,20,20,25,30,30,30,30,30,30,40,60}
[[2,3,4,4],[3]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,14,15,15,15,15,15,16,16,16,16,16,20,20,20,20,20,20,20,20,20,20,25,30,30,30,30,30,30,40,60}
[[2,4,4,4],[3]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,14,15,15,15,15,15,16,16,16,16,16,20,20,20,20,20,20,20,20,20,20,25,30,30,30,30,30,30,40,60}
[[2,4,4,4],[4]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,14,15,15,15,15,15,16,16,16,16,16,20,20,20,20,20,20,20,20,20,20,25,30,30,30,30,30,30,40,60}
[[3,4,4,4],[4]]
=> [2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,14,15,15,15,15,15,16,16,16,16,16,20,20,20,20,20,20,20,20,20,20,25,30,30,30,30,30,30,40,60}
Description
The number of signed permutations less than or equal to a signed permutation in left weak order.
Matching statistic: St001877
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 8% ●values known / values provided: 36%●distinct values known / distinct values provided: 8%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 8% ●values known / values provided: 36%●distinct values known / distinct values provided: 8%
Values
[[1,2]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[2,2]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1],[2]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[1,3]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[2,3]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[3,3]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1],[3]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[2],[3]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[1,1,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,2,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[2,2,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,1],[2]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[[1,2],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 2 - 1
[[1,4]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[2,4]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[3,4]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[4,4]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[2],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[3],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[1,1,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[2,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[2,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[3,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,1],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[[1,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[[1,3],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 2 - 1
[[1,3],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 2 - 1
[[2,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[[2,3],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 2 - 1
[[1],[2],[3]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 6 - 1
[[1,1,1,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[1,1,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[1,2,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[2,2,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[1,1,1],[2]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {4,6} - 1
[[1,1,2],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 3 - 1
[[1,2,2],[2]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1 = 2 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(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)
=> ? ∊ {4,6} - 1
[[1,5]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[2,5]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[3,5]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[4,5]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[5,5]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[2],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[3],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[4],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[1,1,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,2,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,3,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1],[2],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? ∊ {6,6,6} - 1
[[1],[3],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? ∊ {6,6,6} - 1
[[2],[3],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? ∊ {6,6,6} - 1
[[1,1,1],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {4,4,4,4,5,6,6,6,6,6,8,12} - 1
[[1,1,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {4,4,4,4,5,6,6,6,6,6,8,12} - 1
[[1,2,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {4,4,4,4,5,6,6,6,6,6,8,12} - 1
[[2,2,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {4,4,4,4,5,6,6,6,6,6,8,12} - 1
[[1,1],[2,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(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)
=> ? ∊ {4,4,4,4,5,6,6,6,6,6,8,12} - 1
[[1,1],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(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)
=> ? ∊ {4,4,4,4,5,6,6,6,6,6,8,12} - 1
[[1,2],[2,3]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? ∊ {4,4,4,4,5,6,6,6,6,6,8,12} - 1
[[1,2],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(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)
=> ? ∊ {4,4,4,4,5,6,6,6,6,6,8,12} - 1
[[2,2],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(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)
=> ? ∊ {4,4,4,4,5,6,6,6,6,6,8,12} - 1
[[1,1],[2],[3]]
=> [4,3,1,2] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? ∊ {4,4,4,4,5,6,6,6,6,6,8,12} - 1
[[1,2],[2],[3]]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? ∊ {4,4,4,4,5,6,6,6,6,6,8,12} - 1
[[1,3],[2],[3]]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ? ∊ {4,4,4,4,5,6,6,6,6,6,8,12} - 1
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],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)
=> ? ∊ {3,4,5,6,10} - 1
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? ∊ {3,4,5,6,10} - 1
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? ∊ {3,4,5,6,10} - 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ([(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)
=> ? ∊ {3,4,5,6,10} - 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? ∊ {3,4,5,6,10} - 1
[[1],[2],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? ∊ {6,6,6,6,6,6} - 1
[[1],[3],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? ∊ {6,6,6,6,6,6} - 1
[[1],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? ∊ {6,6,6,6,6,6} - 1
[[2],[3],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? ∊ {6,6,6,6,6,6} - 1
[[2],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? ∊ {6,6,6,6,6,6} - 1
[[3],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? ∊ {6,6,6,6,6,6} - 1
[[1,1,1],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,12,12,12,12,24} - 1
[[1,1,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,12,12,12,12,24} - 1
[[1,1,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,12,12,12,12,24} - 1
[[1,2,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,12,12,12,12,24} - 1
[[1,2,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,12,12,12,12,24} - 1
[[1,3,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,12,12,12,12,24} - 1
[[2,2,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,12,12,12,12,24} - 1
[[2,2,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,12,12,12,12,24} - 1
[[2,3,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,12,12,12,12,24} - 1
[[3,3,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,12,12,12,12,24} - 1
[[1,1],[2,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(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)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,12,12,12,12,24} - 1
[[1,1],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(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)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,12,12,12,12,24} - 1
[[1,1],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(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)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,12,12,12,12,24} - 1
[[1,2],[2,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,12,12,12,12,24} - 1
[[1,2],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(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)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,12,12,12,12,24} - 1
[[1,3],[2,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,12,12,12,12,24} - 1
[[1,2],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(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)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,12,12,12,12,24} - 1
[[1,3],[3,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,12,12,12,12,24} - 1
[[1,3],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(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)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,12,12,12,12,24} - 1
[[2,2],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(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)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,12,12,12,12,24} - 1
[[2,2],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(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)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,8,8,8,8,12,12,12,12,24} - 1
Description
Number of indecomposable injective modules with projective dimension 2.
Matching statistic: St000454
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 36%●distinct values known / distinct values provided: 8%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 36%●distinct values known / distinct values provided: 8%
Values
[[1,2]]
=> [1,2] => [1,2] => ([],2)
=> 0 = 1 - 1
[[2,2]]
=> [1,2] => [1,2] => ([],2)
=> 0 = 1 - 1
[[1],[2]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[[1,3]]
=> [1,2] => [1,2] => ([],2)
=> 0 = 1 - 1
[[2,3]]
=> [1,2] => [1,2] => ([],2)
=> 0 = 1 - 1
[[3,3]]
=> [1,2] => [1,2] => ([],2)
=> 0 = 1 - 1
[[1],[3]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[[2],[3]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[[1,1,2]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[[1,2,2]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[[2,2,2]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[[1,1],[2]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? = 3 - 1
[[1,2],[2]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 1 = 2 - 1
[[1,4]]
=> [1,2] => [1,2] => ([],2)
=> 0 = 1 - 1
[[2,4]]
=> [1,2] => [1,2] => ([],2)
=> 0 = 1 - 1
[[3,4]]
=> [1,2] => [1,2] => ([],2)
=> 0 = 1 - 1
[[4,4]]
=> [1,2] => [1,2] => ([],2)
=> 0 = 1 - 1
[[1],[4]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[[2],[4]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[[3],[4]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[[1,1,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[[1,3,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[[2,2,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[[2,3,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[[3,3,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[[1,1],[3]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {3,3,3,6} - 1
[[1,2],[3]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {3,3,3,6} - 1
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 1 = 2 - 1
[[1,3],[3]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 1 = 2 - 1
[[2,2],[3]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {3,3,3,6} - 1
[[2,3],[3]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 1 = 2 - 1
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ? ∊ {3,3,3,6} - 1
[[1,1,1,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[[1,1,1],[2]]
=> [4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {3,4,6} - 1
[[1,1,2],[2]]
=> [3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {3,4,6} - 1
[[1,2,2],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 1 = 2 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,4,6} - 1
[[1,5]]
=> [1,2] => [1,2] => ([],2)
=> 0 = 1 - 1
[[2,5]]
=> [1,2] => [1,2] => ([],2)
=> 0 = 1 - 1
[[3,5]]
=> [1,2] => [1,2] => ([],2)
=> 0 = 1 - 1
[[4,5]]
=> [1,2] => [1,2] => ([],2)
=> 0 = 1 - 1
[[5,5]]
=> [1,2] => [1,2] => ([],2)
=> 0 = 1 - 1
[[1],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[[2],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[[3],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[[4],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[[1,1,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[[1,2,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[[1,3,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[[1,4,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[[2,2,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[[2,3,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[[2,4,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[[3,3,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[[1,1],[4]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {3,3,3,3,3,3,6,6,6} - 1
[[1,2],[4]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {3,3,3,3,3,3,6,6,6} - 1
[[1,3],[4]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {3,3,3,3,3,3,6,6,6} - 1
[[2,2],[4]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {3,3,3,3,3,3,6,6,6} - 1
[[2,3],[4]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {3,3,3,3,3,3,6,6,6} - 1
[[3,3],[4]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {3,3,3,3,3,3,6,6,6} - 1
[[1],[2],[4]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ? ∊ {3,3,3,3,3,3,6,6,6} - 1
[[1],[3],[4]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ? ∊ {3,3,3,3,3,3,6,6,6} - 1
[[2],[3],[4]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ? ∊ {3,3,3,3,3,3,6,6,6} - 1
[[1,1,1],[3]]
=> [4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} - 1
[[1,1,2],[3]]
=> [4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} - 1
[[1,1,3],[2]]
=> [3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} - 1
[[1,1,3],[3]]
=> [3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} - 1
[[1,2,2],[3]]
=> [4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} - 1
[[1,2,3],[3]]
=> [3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} - 1
[[2,2,2],[3]]
=> [4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} - 1
[[2,2,3],[3]]
=> [3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} - 1
[[1,1],[2,3]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} - 1
[[1,1],[3,3]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} - 1
[[1,2],[2,3]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} - 1
[[1,2],[3,3]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} - 1
[[2,2],[3,3]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} - 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} - 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} - 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} - 1
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {4,5,6,10} - 1
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {4,5,6,10} - 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,5,6,10} - 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,5,6,10} - 1
[[1,1],[5]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6} - 1
[[1,2],[5]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6} - 1
[[1,3],[5]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6} - 1
[[1,4],[5]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6} - 1
[[2,2],[5]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6} - 1
[[2,3],[5]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6} - 1
[[2,4],[5]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6} - 1
[[3,3],[5]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6} - 1
[[3,4],[5]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6} - 1
[[4,4],[5]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6} - 1
[[1],[2],[5]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6} - 1
[[1],[3],[5]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6} - 1
[[1],[4],[5]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6} - 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: St001232
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 16% ●values known / values provided: 32%●distinct values known / distinct values provided: 16%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 16% ●values known / values provided: 32%●distinct values known / distinct values provided: 16%
Values
[[1,2]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[2,2]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[1],[2]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[1,3]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[2,3]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[3,3]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[1],[3]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[2],[3]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[1,1,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[2,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,1],[2]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2 - 1
[[1,2],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,4]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[2,4]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[3,4]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[4,4]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[1],[4]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[2],[4]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[3],[4]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[1,1,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[2,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[2,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[3,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,1],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,6} - 1
[[1,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,6} - 1
[[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[[2,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,6} - 1
[[2,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[[1],[2],[3]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,6} - 1
[[1,1,1,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[1,1,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[1,2,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[2,2,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[1,1,1],[2]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,3,6} - 1
[[1,1,2],[2]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {2,3,6} - 1
[[1,2,2],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {2,3,6} - 1
[[1,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[2,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[3,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[4,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[5,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[1],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[2],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[3],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[4],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[1,1,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,2,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,3,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,4,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[2,2,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[2,3,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[2,4,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[3,3,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,1],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,6,6,6} - 1
[[1,2],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,6,6,6} - 1
[[1,3],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,6,6,6} - 1
[[2,2],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,6,6,6} - 1
[[2,3],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,6,6,6} - 1
[[3,3],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,6,6,6} - 1
[[1],[2],[4]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,6,6,6} - 1
[[1],[3],[4]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,6,6,6} - 1
[[2],[3],[4]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,6,6,6} - 1
[[1,1,1],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,5,6,6,6,6,6,8,12} - 1
[[1,1,2],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,5,6,6,6,6,6,8,12} - 1
[[1,1,3],[2]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,2,3,3,3,3,5,6,6,6,6,6,8,12} - 1
[[1,1,3],[3]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,2,3,3,3,3,5,6,6,6,6,6,8,12} - 1
[[1,2,2],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,5,6,6,6,6,6,8,12} - 1
[[1,2,3],[3]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,2,3,3,3,3,5,6,6,6,6,6,8,12} - 1
[[2,2,2],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,5,6,6,6,6,6,8,12} - 1
[[2,2,3],[3]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,2,3,3,3,3,5,6,6,6,6,6,8,12} - 1
[[1,1],[2,3]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,3,3,3,3,5,6,6,6,6,6,8,12} - 1
[[1,1],[3,3]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,3,3,3,3,5,6,6,6,6,6,8,12} - 1
[[1,2],[2,3]]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,5,6,6,6,6,6,8,12} - 1
[[1,2],[3,3]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,3,3,3,3,5,6,6,6,6,6,8,12} - 1
[[2,2],[3,3]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,3,3,3,3,5,6,6,6,6,6,8,12} - 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,5,6,6,6,6,6,8,12} - 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,5,6,6,6,6,6,8,12} - 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,2,3,3,3,3,5,6,6,6,6,6,8,12} - 1
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,3,4,6,10} - 1
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,3,4,6,10} - 1
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {2,3,4,6,10} - 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,3,4,6,10} - 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ? ∊ {2,3,4,6,10} - 1
[[1,1],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,6,6,6,6,6,6} - 1
[[1,2],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,6,6,6,6,6,6} - 1
[[1,3],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,6,6,6,6,6,6} - 1
[[1,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,6,6,6,6,6,6} - 1
[[2,2],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,6,6,6,6,6,6} - 1
[[2,3],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,6,6,6,6,6,6} - 1
[[2,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,6,6,6,6,6,6} - 1
[[3,3],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,6,6,6,6,6,6} - 1
[[3,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,6,6,6,6,6,6} - 1
[[4,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,6,6,6,6,6,6} - 1
[[1],[2],[5]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,6,6,6,6,6,6} - 1
[[1],[3],[5]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,6,6,6,6,6,6} - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001330
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 25%●distinct values known / distinct values provided: 14%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 25%●distinct values known / distinct values provided: 14%
Values
[[1,2]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[2,2]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[2]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[1,3]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[2,3]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[3,3]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[3]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[2],[3]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[1,1,2]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,2,2]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2,2,2]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,1],[2]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3} + 1
[[1,2],[2]]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3} + 1
[[1,4]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[2,4]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[3,4]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[4,4]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[4]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[2],[4]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[3],[4]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[1,1,3]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,2,3]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,3,3]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2,2,3]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2,3,3]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[3,3,3]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,1],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,3,6} + 1
[[1,2],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,3,6} + 1
[[1,3],[2]]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,3,6} + 1
[[1,3],[3]]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,3,6} + 1
[[2,2],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,3,6} + 1
[[2,3],[3]]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,3,3,6} + 1
[[1],[2],[3]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[1,1,1,2]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,1,2,2]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,2,2,2]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[2,2,2,2]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,1,1],[2]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,4,6} + 1
[[1,1,2],[2]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,4,6} + 1
[[1,2,2],[2]]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,4,6} + 1
[[1,1],[2,2]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,3,4,6} + 1
[[1,5]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[2,5]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[3,5]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[4,5]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[5,5]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[5]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[2],[5]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[3],[5]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[4],[5]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[1,1,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,2,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,3,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,4,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2,2,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2,3,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2,4,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[3,3,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[3,4,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[4,4,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,1],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,6,6,6} + 1
[[1,2],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,6,6,6} + 1
[[1,4],[2]]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,6,6,6} + 1
[[1,3],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,6,6,6} + 1
[[1,4],[3]]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,6,6,6} + 1
[[1,4],[4]]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,6,6,6} + 1
[[2,2],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,6,6,6} + 1
[[2,3],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,6,6,6} + 1
[[2,4],[3]]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,6,6,6} + 1
[[2,4],[4]]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,6,6,6} + 1
[[3,3],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,6,6,6} + 1
[[3,4],[4]]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,3,3,3,6,6,6} + 1
[[1],[2],[4]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[1],[3],[4]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[1,1,1],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} + 1
[[1,1,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} + 1
[[1,1,3],[2]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} + 1
[[1,1,3],[3]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} + 1
[[1,2,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} + 1
[[1,2,3],[2]]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} + 1
[[1,2,3],[3]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} + 1
[[1,3,3],[2]]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} + 1
[[1,3,3],[3]]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} + 1
[[2,2,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} + 1
[[2,2,3],[3]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} + 1
[[2,3,3],[3]]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} + 1
[[1,1],[2,3]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} + 1
[[1,1],[3,3]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} + 1
[[1,2],[2,3]]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} + 1
[[1,2],[3,3]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} + 1
[[2,2],[3,3]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} + 1
[[1,1],[2],[3]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} + 1
[[1,2],[2],[3]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} + 1
[[1,3],[2],[3]]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,5,6,6,6,6,6,8,12} + 1
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,3,4,5,6,10} + 1
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,3,4,5,6,10} + 1
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,3,4,5,6,10} + 1
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,3,4,5,6,10} + 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? ∊ {2,3,4,5,6,10} + 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? ∊ {2,3,4,5,6,10} + 1
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: St001568
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 9%●distinct values known / distinct values provided: 3%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 9%●distinct values known / distinct values provided: 3%
Values
[[1,2]]
=> ([(0,1)],2)
=> [1]
=> ? ∊ {1,1,2}
[[2,2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {1,1,2}
[[1],[2]]
=> ([],1)
=> [1]
=> ? ∊ {1,1,2}
[[1,3]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {1,2,2}
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
[[1],[3]]
=> ([(0,1)],2)
=> [1]
=> ? ∊ {1,2,2}
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {1,2,2}
[[1,1,2]]
=> ([(0,1)],2)
=> [1]
=> ? ∊ {1,1,1,2,3}
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {1,1,1,2,3}
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? ∊ {1,1,1,2,3}
[[1,1],[2]]
=> ([],1)
=> [1]
=> ? ∊ {1,1,1,2,3}
[[1,2],[2]]
=> ([(0,1)],2)
=> [1]
=> ? ∊ {1,1,1,2,3}
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? ∊ {1,2,2,2}
[[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 1
[[3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> ? ∊ {1,2,2,2}
[[4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> ? ∊ {1,2,2,2}
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {1,2,2,2}
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {1,1,1,2,2,2,3,3,3,6}
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 1
[[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> ? ∊ {1,1,1,2,2,2,3,3,3,6}
[[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> ? ∊ {1,1,1,2,2,2,3,3,3,6}
[[1,1],[3]]
=> ([(0,1)],2)
=> [1]
=> ? ∊ {1,1,1,2,2,2,3,3,3,6}
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {1,1,1,2,2,2,3,3,3,6}
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {1,1,1,2,2,2,3,3,3,6}
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? ∊ {1,1,1,2,2,2,3,3,3,6}
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? ∊ {1,1,1,2,2,2,3,3,3,6}
[[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [6,6,6,2]
=> ? ∊ {1,1,1,2,2,2,3,3,3,6}
[[1],[2],[3]]
=> ([],1)
=> [1]
=> ? ∊ {1,1,1,2,2,2,3,3,3,6}
[[1,1,1,2]]
=> ([(0,1)],2)
=> [1]
=> ? ∊ {1,1,1,1,2,3,4,6}
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {1,1,1,1,2,3,4,6}
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? ∊ {1,1,1,1,2,3,4,6}
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? ∊ {1,1,1,1,2,3,4,6}
[[1,1,1],[2]]
=> ([],1)
=> [1]
=> ? ∊ {1,1,1,1,2,3,4,6}
[[1,1,2],[2]]
=> ([(0,1)],2)
=> [1]
=> ? ∊ {1,1,1,1,2,3,4,6}
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {1,1,1,1,2,3,4,6}
[[1,1],[2,2]]
=> ([],1)
=> [1]
=> ? ∊ {1,1,1,1,2,3,4,6}
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? ∊ {1,1,1,1,2,2,2,2}
[[2,5]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> [8,4,2]
=> ? ∊ {1,1,1,1,2,2,2,2}
[[3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [24,24,24,24,14]
=> ? ∊ {1,1,1,1,2,2,2,2}
[[4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {1,1,1,1,2,2,2,2}
[[5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {1,1,1,1,2,2,2,2}
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? ∊ {1,1,1,1,2,2,2,2}
[[2],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 1
[[3],[5]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> ? ∊ {1,1,1,1,2,2,2,2}
[[4],[5]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> ? ∊ {1,1,1,1,2,2,2,2}
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6}
[[1,2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 1
[[1,3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6}
[[1,4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6}
[[2,2,4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> [9,9,9,9,3,3]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6}
[[2,3,4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> ?
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6}
[[2,4,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ?
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6}
[[3,3,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ?
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6}
[[3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> ?
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6}
[[4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> ?
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6}
[[1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
[[1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [2]
=> 1
[[1,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> [2]
=> 1
[[2,2],[4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 1
[[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
[[1,2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 1
[[1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
[[1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> [2]
=> 1
[[1,2],[3,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
[[1,2],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 1
[[1,5],[3]]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> [3,2]
=> 1
[[1],[3],[5]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[1],[4],[5]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
[[2],[3],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 1
[[1,1,2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 1
[[1,1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[1,1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
[[1,1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [2]
=> 1
[[1,1,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> [2]
=> 1
[[1,2,2],[4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 1
[[1,2,4],[2]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 1
[[1,1],[3,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[1,1],[4,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
[[1,2],[2,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[1,3],[2,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
[[1,3],[3,4]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> [2]
=> 1
[[1,2],[2],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [2]
=> 1
[[1,3],[2],[4]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [2]
=> 1
[[1,3],[3],[4]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [2]
=> 1
[[1,1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[1,1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
[[1,1,2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 1
[[1,1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[1,1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
[[1,1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> [2]
=> 1
[[1,2,2,3],[2]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 1
Description
The smallest positive integer that does not appear twice in the partition.
Matching statistic: St001118
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Values
[[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {1,1,2}
[[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? ∊ {1,1,2}
[[1],[2]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,2}
[[1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? ∊ {1,2,2}
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[[1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {1,2,2}
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? ∊ {1,2,2}
[[1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {1,1,1,2,3}
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? ∊ {1,1,1,2,3}
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? ∊ {1,1,1,2,3}
[[1,1],[2]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2,3}
[[1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {1,1,1,2,3}
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? ∊ {1,1,2,2}
[[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[[3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? ∊ {1,1,2,2}
[[4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? ∊ {1,1,2,2}
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? ∊ {1,1,2,2}
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? ∊ {1,1,1,1,2,2,3,3,3,6}
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,1,2,2,3,3,3,6}
[[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,2,2,3,3,3,6}
[[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {1,1,1,1,2,2,3,3,3,6}
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? ∊ {1,1,1,1,2,2,3,3,3,6}
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? ∊ {1,1,1,1,2,2,3,3,3,6}
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? ∊ {1,1,1,1,2,2,3,3,3,6}
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? ∊ {1,1,1,1,2,2,3,3,3,6}
[[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? ∊ {1,1,1,1,2,2,3,3,3,6}
[[1],[2],[3]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,3,3,3,6}
[[1,1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {1,1,1,1,2,3,4,6}
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? ∊ {1,1,1,1,2,3,4,6}
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? ∊ {1,1,1,1,2,3,4,6}
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? ∊ {1,1,1,1,2,3,4,6}
[[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,3,4,6}
[[1,1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ? ∊ {1,1,1,1,2,3,4,6}
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? ∊ {1,1,1,1,2,3,4,6}
[[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,3,4,6}
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? ∊ {1,1,1,1,1,2,2,2}
[[2,5]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,1,1,2,2,2}
[[3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {1,1,1,1,1,2,2,2}
[[4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> ? ∊ {1,1,1,1,1,2,2,2}
[[5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? ∊ {1,1,1,1,1,2,2,2}
[[2],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[[3],[5]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,1,1,2,2,2}
[[4],[5]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,6,6,6}
[[1,2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[[1,3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,6,6,6}
[[1,4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,6,6,6}
[[2,2,4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,6,6,6}
[[2,3,4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> ([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,6,6,6}
[[2,4,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,6,6,6}
[[3,3,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,6,6,6}
[[3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> ([(3,4),(3,16),(4,15),(5,6),(5,17),(6,18),(7,17),(7,18),(8,15),(8,16),(9,12),(9,13),(9,14),(9,15),(9,16),(10,11),(10,13),(10,14),(10,15),(10,18),(11,12),(11,14),(11,16),(11,17),(12,13),(12,15),(12,18),(13,16),(13,17),(14,17),(14,18),(15,16),(15,17),(16,18),(17,18)],19)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,6,6,6}
[[4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> ([(4,5),(4,17),(5,16),(6,7),(6,18),(7,19),(8,18),(8,19),(9,16),(9,17),(10,13),(10,14),(10,15),(10,16),(10,17),(11,12),(11,14),(11,15),(11,16),(11,19),(12,13),(12,15),(12,17),(12,18),(13,14),(13,16),(13,19),(14,17),(14,18),(15,18),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,6,6,6}
[[1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
[[1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[[1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
[[1,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1
[[2,2],[4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
[[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[[1,2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[[1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
[[1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[[1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 1
[[1,2],[3,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
[[2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[[1,2],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[[1],[3],[5]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
[[1],[4],[5]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[[2],[3],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[[1,1,2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[[1,1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
[[1,1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[[1,1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
[[1,1,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1
[[1,2,2],[4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[[1,2,4],[2]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[[1,1],[3,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
[[1,1],[4,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[[1,2],[2,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
[[1,3],[2,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[[1,3],[3,4]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 1
[[1,2],[2],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[[1,3],[2],[4]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 1
[[1,3],[3],[4]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
[[1,1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
[[1,1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[[1,1,2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[[1,1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
[[1,1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[[1,1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 1
[[1,2,2,3],[2]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[[1,1,2],[3,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
Description
The acyclic chromatic index of a graph.
An acyclic edge coloring of a graph is a proper colouring of the edges of a graph such that the union of the edges colored with any two given colours is a forest.
The smallest number of colours such that such a colouring exists is the acyclic chromatic index.
Matching statistic: St000259
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 8%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 8%
Values
[[1,2]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,2}
[[2,2]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,2}
[[1],[2]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,3]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,2,2}
[[2,3]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,2,2}
[[3,3]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,2,2}
[[1],[3]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[2],[3]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,1,2]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,2,3}
[[1,2,2]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,2,3}
[[2,2,2]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,2,3}
[[1,1],[2]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,2,3}
[[1,2],[2]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {1,1,1,2,3}
[[1,4]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,2,2,2}
[[2,4]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,2,2,2}
[[3,4]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,2,2,2}
[[4,4]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,2,2,2}
[[1],[4]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[2],[4]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[3],[4]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,1,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,2,2,2,3,3,3,6}
[[1,2,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,2,2,2,3,3,3,6}
[[1,3,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,2,2,2,3,3,3,6}
[[2,2,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,2,2,2,3,3,3,6}
[[2,3,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,2,2,2,3,3,3,6}
[[3,3,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,2,2,2,3,3,3,6}
[[1,1],[3]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,2,2,2,3,3,3,6}
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,2,2,2,3,3,3,6}
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,2,2,2,3,3,3,6}
[[1,3],[3]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,2,2,2,3,3,3,6}
[[2,2],[3]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,2,2,2,3,3,3,6}
[[2,3],[3]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,2,2,2,3,3,3,6}
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,1,1,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,2,3,4,6}
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,2,3,4,6}
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,2,3,4,6}
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,2,3,4,6}
[[1,1,1],[2]]
=> [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,2,3,4,6}
[[1,1,2],[2]]
=> [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,2,3,4,6}
[[1,2,2],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,2,3,4,6}
[[1,1],[2,2]]
=> [3,4,1,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,2,3,4,6}
[[1,5]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,2,2,2,2}
[[2,5]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,2,2,2,2}
[[3,5]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,2,2,2,2}
[[4,5]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,2,2,2,2}
[[5,5]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,2,2,2,2}
[[1],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[2],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[3],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[4],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,1,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6}
[[1,2,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6}
[[1,3,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6}
[[1,4,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6}
[[2,2,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6}
[[2,3,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6}
[[2,4,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6}
[[3,3,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6}
[[3,4,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6}
[[4,4,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6}
[[1,1],[4]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6}
[[1],[2],[4]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1],[3],[4]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[2],[3],[4]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 3
[[1],[6]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[2],[6]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[3],[6]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[4],[6]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[5],[6]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1],[2],[5]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1],[3],[5]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1],[4],[5]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[2],[3],[5]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[2],[4],[5]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[3],[4],[5]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,2],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 3
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 3
[[1,3],[3],[4]]
=> [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 3
[[2,3],[3],[4]]
=> [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 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],[2,3],[3]]
=> [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[[1],[7]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[2],[7]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[3],[7]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[4],[7]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[5],[7]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[6],[7]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1],[2],[6]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1],[3],[6]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1],[4],[6]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1],[5],[6]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[2],[3],[6]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[2],[4],[6]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[2],[5],[6]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[3],[4],[6]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[3],[5],[6]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[4],[5],[6]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,2],[2],[5]]
=> [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 3
[[1,3],[2],[5]]
=> [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 3
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
The following 37 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001545The second Elser number of a connected graph. St001645The pebbling number of a connected graph. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000467The hyper-Wiener index of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001128The exponens consonantiae of a partition. St000736The last entry in the first row of a semistandard tableau. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000103The sum of the entries of a semistandard tableau. St000264The girth of a graph, which is not a tree. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition.
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