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Your data matches 23 different statistics following compositions of up to 3 maps.
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Matching statistic: St001729
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Mp00080: Set partitions —to permutation⟶ Permutations
St001729: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001729: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => 0
{{1,2}}
=> [2,1] => 1
{{1},{2}}
=> [1,2] => 0
{{1,2,3}}
=> [2,3,1] => 1
{{1,2},{3}}
=> [2,1,3] => 1
{{1,3},{2}}
=> [3,2,1] => 1
{{1},{2,3}}
=> [1,3,2] => 1
{{1},{2},{3}}
=> [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => 1
{{1,2,3},{4}}
=> [2,3,1,4] => 1
{{1,2,4},{3}}
=> [2,4,3,1] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => 2
{{1,2},{3},{4}}
=> [2,1,3,4] => 1
{{1,3,4},{2}}
=> [3,2,4,1] => 1
{{1,3},{2,4}}
=> [3,4,1,2] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => 2
{{1},{2,3,4}}
=> [1,3,4,2] => 1
{{1},{2,3},{4}}
=> [1,3,2,4] => 1
{{1,4},{2},{3}}
=> [4,2,3,1] => 1
{{1},{2,4},{3}}
=> [1,4,3,2] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => 2
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => 2
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => 2
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => 1
Description
The number of visible descents of a permutation.
A visible descent of a permutation $\pi$ is a position $i$ such that $\pi(i+1) \leq \min(i, \pi(i))$.
Matching statistic: St001792
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001792: Graphs ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001792: Graphs ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
{{1,2}}
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
{{1},{2}}
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
{{1,2,3}}
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
{{1,2},{3}}
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
{{1,3},{2}}
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
{{1},{2,3}}
=> [1,3,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
{{1},{2},{3}}
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1,2,3,4}}
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,2},{3,4}}
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,3},{2,4}}
=> [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 = 1 + 1
{{1,3},{2},{4}}
=> [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 = 1 + 1
{{1,4},{2,3}}
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
{{1},{2,3,4}}
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => ([(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 = 1 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 3 = 2 + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
{{1,3},{2,4},{5}}
=> [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 = 1 + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 3 = 2 + 1
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2} + 1
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2} + 1
{{1,2,4},{3},{5,6}}
=> [2,4,3,1,6,5] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2} + 1
{{1,2,5,6},{3,4}}
=> [2,5,4,3,6,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2} + 1
{{1,2},{3,4,6},{5}}
=> [2,1,4,6,5,3] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2} + 1
{{1,2},{3,5,6},{4}}
=> [2,1,5,4,6,3] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2} + 1
{{1,2},{3,5},{4,6}}
=> [2,1,5,6,3,4] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2} + 1
{{1,2},{3,5},{4},{6}}
=> [2,1,5,4,3,6] => ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2} + 1
{{1,2},{3,6},{4,5}}
=> [2,1,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2} + 1
{{1,2},{3,6},{4},{5}}
=> [2,1,6,4,5,3] => ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2} + 1
{{1,2},{3},{4,6},{5}}
=> [2,1,3,6,5,4] => ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2} + 1
{{1,3,4},{2},{5,6}}
=> [3,2,4,1,6,5] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2} + 1
{{1,3},{2,4},{5,6}}
=> [3,4,1,2,6,5] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2} + 1
{{1,3},{2},{4,5,6}}
=> [3,2,1,5,6,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2} + 1
{{1,3},{2},{4,5},{6}}
=> [3,2,1,5,4,6] => ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2} + 1
{{1,3},{2},{4,6},{5}}
=> [3,2,1,6,5,4] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2} + 1
{{1,3},{2},{4},{5,6}}
=> [3,2,1,4,6,5] => ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2} + 1
{{1,4,5,6},{2,3}}
=> [4,3,2,5,6,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2} + 1
{{1,4},{2,3},{5,6}}
=> [4,3,2,1,6,5] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2} + 1
{{1},{2,3},{4,6},{5}}
=> [1,3,2,6,5,4] => ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2} + 1
{{1,4},{2},{3},{5,6}}
=> [4,2,3,1,6,5] => ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2} + 1
{{1},{2,4},{3},{5,6}}
=> [1,4,3,2,6,5] => ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2} + 1
Description
The arboricity of a graph.
This is the minimum number of forests that covers all edges of the graph.
Matching statistic: St000668
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 85%●distinct values known / distinct values provided: 75%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 85%●distinct values known / distinct values provided: 75%
Values
{{1}}
=> [1] => [1]
=> []
=> ? = 0
{{1,2}}
=> [2,1] => [1,1]
=> [1]
=> ? ∊ {0,1}
{{1},{2}}
=> [1,2] => [2]
=> []
=> ? ∊ {0,1}
{{1,2,3}}
=> [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,1,1,1}
{{1,2},{3}}
=> [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,1,1,1}
{{1,3},{2}}
=> [3,2,1] => [1,1,1]
=> [1,1]
=> 1
{{1},{2,3}}
=> [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,1,1,1}
{{1},{2},{3}}
=> [1,2,3] => [3]
=> []
=> ? ∊ {0,1,1,1}
{{1,2,3,4}}
=> [2,3,4,1] => [3,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1}
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1}
{{1,2,4},{3}}
=> [2,4,3,1] => [2,1,1]
=> [1,1]
=> 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,2]
=> [2]
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [3,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1}
{{1,3,4},{2}}
=> [3,2,4,1] => [2,1,1]
=> [1,1]
=> 1
{{1,3},{2,4}}
=> [3,4,1,2] => [2,2]
=> [2]
=> 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> 1
{{1},{2,3,4}}
=> [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1}
{{1},{2,3},{4}}
=> [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1}
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1}
{{1},{2},{3},{4}}
=> [1,2,3,4] => [4]
=> []
=> ? ∊ {0,1,1,1,1,1,1}
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [4,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2}
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2}
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [3,1,1]
=> [1,1]
=> 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,2]
=> [2]
=> 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [4,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2}
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [3,1,1]
=> [1,1]
=> 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [3,2]
=> [2]
=> 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [3,1,1]
=> [1,1]
=> 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [3,2]
=> [2]
=> 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [3,2]
=> [2]
=> 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [3,1,1]
=> [1,1]
=> 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [3,2]
=> [2]
=> 2
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [4,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2}
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [3,1,1]
=> [1,1]
=> 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [2,2,1]
=> [2,1]
=> 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [3,1,1]
=> [1,1]
=> 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [3,1,1]
=> [1,1]
=> 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,2]
=> [2]
=> 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,2]
=> [2]
=> 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,2,1]
=> [2,1]
=> 2
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [2,2,1]
=> [2,1]
=> 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,1,1]
=> [1,1]
=> 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [2,1,1,1]
=> [1,1,1]
=> 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [2,2,1]
=> [2,1]
=> 2
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [2,1,1,1]
=> [1,1,1]
=> 1
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> 1
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [4,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2}
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [4,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2}
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [2,1,1,1]
=> [1,1,1]
=> 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [3,2]
=> [2]
=> 2
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [4,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2}
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [3,1,1]
=> [1,1]
=> 1
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [2,2,1]
=> [2,1]
=> 2
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [2,2,1]
=> [2,1]
=> 2
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [3,1,1]
=> [1,1]
=> 1
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 1
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [3,2]
=> [2]
=> 2
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> 1
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [4,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2}
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2}
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [3,1,1]
=> [1,1]
=> 1
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 1
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2}
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2}
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [5,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3}
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [5,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3}
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => [5,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3}
{{1,2,3},{4},{5},{6}}
=> [2,3,1,4,5,6] => [5,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3}
{{1,2},{3},{4},{5},{6}}
=> [2,1,3,4,5,6] => [5,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3}
{{1},{2,3,4,5,6}}
=> [1,3,4,5,6,2] => [5,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3}
{{1},{2,3,4,5},{6}}
=> [1,3,4,5,2,6] => [5,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3}
{{1},{2,3,4},{5},{6}}
=> [1,3,4,2,5,6] => [5,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3}
{{1},{2,3},{4},{5},{6}}
=> [1,3,2,4,5,6] => [5,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3}
{{1},{2},{3,4,5,6}}
=> [1,2,4,5,6,3] => [5,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3}
{{1},{2},{3,4,5},{6}}
=> [1,2,4,5,3,6] => [5,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3}
{{1},{2},{3,4},{5},{6}}
=> [1,2,4,3,5,6] => [5,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3}
{{1},{2},{3},{4,5,6}}
=> [1,2,3,5,6,4] => [5,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3}
{{1},{2},{3},{4,5},{6}}
=> [1,2,3,5,4,6] => [5,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3}
{{1},{2},{3},{4},{5,6}}
=> [1,2,3,4,6,5] => [5,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3}
{{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => [6]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3}
Description
The least common multiple of the parts of the partition.
Matching statistic: St000260
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 100%
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => ([],1)
=> 0
{{1,2}}
=> [2,1] => [1,2] => ([],2)
=> ? = 0
{{1},{2}}
=> [1,2] => [2,1] => ([(0,1)],2)
=> 1
{{1,2,3}}
=> [2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
{{1,2},{3}}
=> [2,1,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
{{1,3},{2}}
=> [3,2,1] => [1,2,3] => ([],3)
=> ? ∊ {0,1}
{{1},{2,3}}
=> [1,3,2] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,1}
{{1},{2},{3}}
=> [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1
{{1,2,3,4}}
=> [2,3,4,1] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
{{1,2,3},{4}}
=> [2,3,1,4] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
{{1,2,4},{3}}
=> [2,4,3,1] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,1,1,1,1,1}
{{1,2},{3,4}}
=> [2,1,4,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,1,1,1,1,1}
{{1,2},{3},{4}}
=> [2,1,3,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
{{1,3,4},{2}}
=> [3,2,4,1] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,2,3,4] => ([],4)
=> ? ∊ {0,1,1,1,1,1}
{{1},{2,3,4}}
=> [1,3,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
{{1},{2,3},{4}}
=> [1,3,2,4] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,1,1,1,1,1}
{{1},{2,4},{3}}
=> [1,4,3,2] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,1,1,1,1,1}
{{1},{2},{3,4}}
=> [1,2,4,3] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,1,1,1,1,1}
{{1},{2},{3},{4}}
=> [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [2,1,3,4,5] => ([(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [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)
=> 1
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [3,4,5,1,2,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [3,4,5,2,1,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => [3,4,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [3,4,1,2,5,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [3,4,2,1,5,6] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
{{1,2,3},{4},{5,6}}
=> [2,3,1,4,6,5] => [3,4,2,5,1,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
{{1,2,4,6},{3},{5}}
=> [2,4,3,6,5,1] => [3,5,4,1,2,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
{{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => [3,5,1,4,2,6] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
{{1,2,4},{3},{5,6}}
=> [2,4,3,1,6,5] => [3,5,4,2,1,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
{{1,2},{3,4,6},{5}}
=> [2,1,4,6,5,3] => [3,2,5,1,4,6] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
{{1,2},{3,4},{5,6}}
=> [2,1,4,3,6,5] => [3,2,5,4,1,6] => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
{{1,2,6},{3,5},{4}}
=> [2,6,5,4,3,1] => [3,1,2,4,5,6] => ([(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
{{1,2,6},{3},{4,5}}
=> [2,6,3,5,4,1] => [3,1,5,2,4,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
{{1,2},{3,6},{4,5}}
=> [2,1,6,5,4,3] => [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
{{1,2},{3,6},{4},{5}}
=> [2,1,6,4,5,3] => [3,2,1,6,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
{{1,2},{3},{4,6},{5}}
=> [2,1,3,6,5,4] => [3,2,4,1,5,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
{{1,2},{3},{4},{5,6}}
=> [2,1,3,4,6,5] => [3,2,4,5,1,6] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
{{1,3,4,6},{2},{5}}
=> [3,2,4,6,5,1] => [4,3,5,1,2,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
{{1,3,4},{2},{5,6}}
=> [3,2,4,1,6,5] => [4,3,5,2,1,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
{{1,3,6},{2,4,5}}
=> [3,4,6,5,2,1] => [4,5,1,2,3,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
{{1,3},{2,4,6},{5}}
=> [3,4,1,6,5,2] => [4,5,2,1,3,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
{{1,3},{2,4},{5,6}}
=> [3,4,1,2,6,5] => [4,5,2,3,1,6] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
{{1,3,5},{2,6},{4}}
=> [3,6,5,4,1,2] => [4,1,2,3,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000259
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 75%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 75%
Values
{{1}}
=> [1] => ([],1)
=> ([],1)
=> 0
{{1,2}}
=> [2] => ([],2)
=> ([],1)
=> 0
{{1},{2}}
=> [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
{{1,2,3}}
=> [3] => ([],3)
=> ([],1)
=> 0
{{1,2},{3}}
=> [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
{{1,3},{2}}
=> [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
{{1},{2,3}}
=> [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1
{{1},{2},{3}}
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
{{1,2,3,4}}
=> [4] => ([],4)
=> ([],1)
=> 0
{{1,2,3},{4}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
{{1,2,4},{3}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
{{1,2},{3,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1}
{{1,2},{3},{4}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
{{1,3,4},{2}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
{{1,3},{2,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1}
{{1,3},{2},{4}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
{{1,4},{2,3}}
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1}
{{1},{2,3,4}}
=> [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1}
{{1},{2,3},{4}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
{{1,4},{2},{3}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
{{1},{2,4},{3}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
{{1},{2},{3,4}}
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1}
{{1},{2},{3},{4}}
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
{{1,2,3,4,5}}
=> [5] => ([],5)
=> ([],1)
=> 0
{{1,2,3,4},{5}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
{{1,2,3,5},{4}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
{{1,2,3},{4,5}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2,3},{4},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
{{1,2,4,5},{3}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
{{1,2,4},{3,5}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2,4},{3},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
{{1,2,5},{3,4}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2},{3,4,5}}
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2},{3,4},{5}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
{{1,2,5},{3},{4}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
{{1,2},{3,5},{4}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
{{1,2},{3},{4,5}}
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
{{1,3,4,5},{2}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
{{1,3,4},{2,5}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3,4},{2},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
{{1,3,5},{2,4}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3},{2,4,5}}
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3},{2,4},{5}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
{{1,3,5},{2},{4}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
{{1,3},{2,5},{4}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
{{1,3},{2},{4,5}}
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
{{1,4,5},{2,3}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,4},{2,3,5}}
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,4},{2,3},{5}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
{{1,5},{2,3,4}}
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2,3,4,5}}
=> [1,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2,3,4},{5}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
{{1,5},{2,3},{4}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
{{1},{2,3,5},{4}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
{{1},{2,3},{4,5}}
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2,3},{4},{5}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
{{1,4,5},{2},{3}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
{{1,4},{2,5},{3}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
{{1,4},{2},{3,5}}
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,4},{2},{3},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
{{1,5},{2,4},{3}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
{{1},{2,4,5},{3}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
{{1},{2,4},{3,5}}
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2,4},{3},{5}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
{{1,5},{2},{3,4}}
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2,5},{3,4}}
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2},{3,4,5}}
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2},{3,4},{5}}
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
{{1,5},{2},{3},{4}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
{{1},{2,5},{3},{4}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
{{1},{2},{3,5},{4}}
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
{{1},{2},{3},{4,5}}
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,2,3,4,5,6}}
=> [6] => ([],6)
=> ([],1)
=> 0
{{1,2,3,4},{5,6}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2,3,5},{4,6}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2,3,6},{4,5}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2,3},{4,5,6}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2,3},{4},{5,6}}
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2,4,5},{3,6}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2,4,6},{3,5}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2,4},{3,5,6}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2,4},{3},{5,6}}
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2,5,6},{3,4}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2,5},{3,4,6}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2,6},{3,4,5}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2},{3,4,5,6}}
=> [2,4] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2},{3,4},{5,6}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2,5},{3},{4,6}}
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2},{3,5},{4,6}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2,6},{3},{4,5}}
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2},{3,6},{4,5}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2},{3},{4,5,6}}
=> [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2},{3},{4},{5,6}}
=> [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,3,4,5},{2,6}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,3,4,6},{2,5}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,3,4},{2,5,6}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,3,4},{2},{5,6}}
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St001330
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 75%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 75%
Values
{{1}}
=> [1] => [1] => ([],1)
=> 1 = 0 + 1
{{1,2}}
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
{{1},{2}}
=> [1,2] => [1,2] => ([],2)
=> 1 = 0 + 1
{{1,2,3}}
=> [2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 2 = 1 + 1
{{1,3},{2}}
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => ([(1,2)],3)
=> 2 = 1 + 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => ([],3)
=> 1 = 0 + 1
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 3 = 2 + 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 1 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 2 = 1 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
{{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 2 = 1 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 2 = 1 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1 = 0 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [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} + 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [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} + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = 2 + 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [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} + 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 2 = 1 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [3,4,5,1,2] => ([(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} + 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 2 = 1 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> 2 = 1 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,1,3,5,2] => ([(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} + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [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} + 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,4,5,1,3] => ([(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} + 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [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} + 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,1,5,2,3] => ([(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} + 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [3,2,4,1,5] => ([(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} + 1
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [4,2,3,5,1] => ([(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} + 1
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [3,2,4,5,1] => ([(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} + 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = 2 + 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 2 = 1 + 1
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 2 = 1 + 1
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [3,4,1,5,2] => ([(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} + 1
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [3,4,2,5,1] => ([(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} + 1
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [2,4,3,5,1] => ([(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} + 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,4,3,5,2] => ([(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} + 1
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3} + 1
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [4,6,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3} + 1
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [4,5,1,2,3,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3} + 1
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3} + 1
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3} + 1
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => [3,6,1,2,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3} + 1
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [5,1,2,4,6,3] => ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3} + 1
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => [3,5,1,2,4,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3} + 1
{{1,2,4,6},{3,5}}
=> [2,4,5,6,3,1] => [5,3,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3} + 1
{{1,2,4,6},{3},{5}}
=> [2,4,3,6,5,1] => [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3} + 1
{{1,2,5,6},{3,4}}
=> [2,5,4,3,6,1] => [4,3,6,1,2,5] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3} + 1
{{1,2,5},{3,4,6}}
=> [2,5,4,6,1,3] => [5,1,2,6,3,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3} + 1
{{1,2,5},{3,4},{6}}
=> [2,5,4,3,1,6] => [4,3,5,1,2,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3} + 1
{{1,2,6},{3,4,5}}
=> [2,6,4,5,3,1] => [5,3,4,6,1,2] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3} + 1
{{1,2,6},{3,4},{5}}
=> [2,6,4,3,5,1] => [4,3,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3} + 1
{{1,2,5,6},{3},{4}}
=> [2,5,3,4,6,1] => [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3} + 1
{{1,2,5},{3,6},{4}}
=> [2,5,6,4,1,3] => [4,5,1,2,6,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3} + 1
{{1,2,5},{3},{4,6}}
=> [2,5,3,6,1,4] => [3,5,1,2,6,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3} + 1
{{1,2,5},{3},{4},{6}}
=> [2,5,3,4,1,6] => [3,4,5,1,2,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3} + 1
{{1,2,6},{3,5},{4}}
=> [2,6,5,4,3,1] => [4,5,3,6,1,2] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3} + 1
{{1,2,6},{3},{4,5}}
=> [2,6,3,5,4,1] => [3,5,4,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3} + 1
{{1,2},{3,6},{4,5}}
=> [2,1,6,5,4,3] => [2,1,5,4,6,3] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3} + 1
{{1,2,6},{3},{4},{5}}
=> [2,6,3,4,5,1] => [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3} + 1
{{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => [5,1,3,4,6,2] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3} + 1
{{1,3,4,6},{2,5}}
=> [3,5,4,6,2,1] => [5,2,6,1,3,4] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3} + 1
{{1,3,4},{2,5,6}}
=> [3,5,4,1,6,2] => [4,1,3,6,2,5] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3} + 1
{{1,3,4},{2,5},{6}}
=> [3,5,4,1,2,6] => [4,1,3,5,2,6] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3} + 1
{{1,3,4,6},{2},{5}}
=> [3,2,4,6,5,1] => [2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3} + 1
{{1,3,4},{2,6},{5}}
=> [3,6,4,1,5,2] => [4,1,3,5,6,2] => ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3} + 1
{{1,3,5,6},{2,4}}
=> [3,4,5,2,6,1] => [4,2,6,1,3,5] => ([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3} + 1
{{1,3,5},{2,4,6}}
=> [3,4,5,6,1,2] => [5,1,3,6,2,4] => ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3} + 1
{{1,3,5},{2,4},{6}}
=> [3,4,5,2,1,6] => [4,2,5,1,3,6] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3} + 1
{{1,3,6},{2,4,5}}
=> [3,4,6,5,2,1] => [5,2,4,6,1,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3} + 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: St001630
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 50%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 50%
Values
{{1}}
=> [1] => [[1],[]]
=> ([],1)
=> ? = 0
{{1,2}}
=> [2] => [[2],[]]
=> ([],1)
=> ? ∊ {0,1}
{{1},{2}}
=> [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,1}
{{1,2,3}}
=> [3] => [[3],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1}
{{1,2},{3}}
=> [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,1,1,1,1}
{{1,3},{2}}
=> [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,1,1,1,1}
{{1},{2,3}}
=> [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1}
{{1},{2},{3}}
=> [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1}
{{1,2,3,4}}
=> [4] => [[4],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2,3},{4}}
=> [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2,4},{3}}
=> [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2},{3,4}}
=> [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2},{3},{4}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3,4},{2}}
=> [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3},{2,4}}
=> [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3},{2},{4}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,4},{2,3}}
=> [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2,3,4}}
=> [1,3] => [[3,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2,3},{4}}
=> [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,4},{2},{3}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2,4},{3}}
=> [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2},{3,4}}
=> [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2},{3},{4}}
=> [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2,3,4,5}}
=> [5] => [[5],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,3,4},{5}}
=> [4,1] => [[4,4],[3]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,3,5},{4}}
=> [4,1] => [[4,4],[3]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,3},{4,5}}
=> [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,3},{4},{5}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,4,5},{3}}
=> [4,1] => [[4,4],[3]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,4},{3,5}}
=> [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,4},{3},{5}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,5},{3,4}}
=> [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2},{3,4,5}}
=> [2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2},{3,4},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2},{3,5},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3,4,5},{2}}
=> [4,1] => [[4,4],[3]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3,4},{2,5}}
=> [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3,4},{2},{5}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3,5},{2,4}}
=> [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3},{2,4,5}}
=> [2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3},{2,4},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3},{2,5},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,4,5},{2,3}}
=> [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,4},{2,3,5}}
=> [2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,4},{2,3},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,5},{2,3,4}}
=> [2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1},{2,3,4,5}}
=> [1,4] => [[4,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1},{2,3,4},{5}}
=> [1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,5},{2,3},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1},{2,3,5},{4}}
=> [1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1},{2,3},{4,5}}
=> [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,4},{2,5},{3}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,5},{2,4},{3}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1},{2,4},{3,5}}
=> [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1},{2,5},{3,4}}
=> [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,3},{4,5,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,3},{4,5},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,3},{4,6},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,4},{3,5,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,4},{3,5},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,4},{3,6},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,5},{3,4,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,5},{3,4},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,6},{3,4,5}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2},{3,4,5},{6}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,2,6},{3,4},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2},{3,4,6},{5}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,2},{3,4},{5,6}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,5},{3,6},{4}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,6},{3,5},{4}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2},{3,5,6},{4}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,2},{3,5},{4,6}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2},{3,6},{4,5}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
{{1,2},{3},{4,5},{6}}
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2},{3},{4,6},{5}}
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,3,4},{2,5,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,4},{2,5},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,4},{2,6},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,5},{2,4,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,5},{2,4},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,6},{2,4,5}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3},{2,4,5},{6}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,3,6},{2,4},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3},{2,4,6},{5}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,3},{2,4},{5,6}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,5},{2,6},{4}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,6},{2,5},{4}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3},{2,5,6},{4}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,3},{2,5},{4,6}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
Matching statistic: St001878
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 50%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 50%
Values
{{1}}
=> [1] => [[1],[]]
=> ([],1)
=> ? = 0
{{1,2}}
=> [2] => [[2],[]]
=> ([],1)
=> ? ∊ {0,1}
{{1},{2}}
=> [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,1}
{{1,2,3}}
=> [3] => [[3],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1}
{{1,2},{3}}
=> [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,1,1,1,1}
{{1,3},{2}}
=> [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,1,1,1,1}
{{1},{2,3}}
=> [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1}
{{1},{2},{3}}
=> [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1}
{{1,2,3,4}}
=> [4] => [[4],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2,3},{4}}
=> [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2,4},{3}}
=> [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2},{3,4}}
=> [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2},{3},{4}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3,4},{2}}
=> [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3},{2,4}}
=> [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3},{2},{4}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,4},{2,3}}
=> [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2,3,4}}
=> [1,3] => [[3,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2,3},{4}}
=> [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,4},{2},{3}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2,4},{3}}
=> [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2},{3,4}}
=> [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2},{3},{4}}
=> [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2,3,4,5}}
=> [5] => [[5],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,3,4},{5}}
=> [4,1] => [[4,4],[3]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,3,5},{4}}
=> [4,1] => [[4,4],[3]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,3},{4,5}}
=> [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,3},{4},{5}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,4,5},{3}}
=> [4,1] => [[4,4],[3]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,4},{3,5}}
=> [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,4},{3},{5}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,5},{3,4}}
=> [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2},{3,4,5}}
=> [2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2},{3,4},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2},{3,5},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3,4,5},{2}}
=> [4,1] => [[4,4],[3]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3,4},{2,5}}
=> [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3,4},{2},{5}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3,5},{2,4}}
=> [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3},{2,4,5}}
=> [2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3},{2,4},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3},{2,5},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,4,5},{2,3}}
=> [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,4},{2,3,5}}
=> [2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,4},{2,3},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,5},{2,3,4}}
=> [2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1},{2,3,4,5}}
=> [1,4] => [[4,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1},{2,3,4},{5}}
=> [1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,5},{2,3},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1},{2,3,5},{4}}
=> [1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1},{2,3},{4,5}}
=> [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,4},{2,5},{3}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,5},{2,4},{3}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1},{2,4},{3,5}}
=> [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1},{2,5},{3,4}}
=> [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,3},{4,5,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,3},{4,5},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,3},{4,6},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,4},{3,5,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,4},{3,5},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,4},{3,6},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,5},{3,4,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,5},{3,4},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,6},{3,4,5}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2},{3,4,5},{6}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,2,6},{3,4},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2},{3,4,6},{5}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,2},{3,4},{5,6}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,5},{3,6},{4}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,6},{3,5},{4}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2},{3,5,6},{4}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,2},{3,5},{4,6}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2},{3,6},{4,5}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
{{1,2},{3},{4,5},{6}}
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2},{3},{4,6},{5}}
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,3,4},{2,5,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,4},{2,5},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,4},{2,6},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,5},{2,4,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,5},{2,4},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,6},{2,4,5}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3},{2,4,5},{6}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,3,6},{2,4},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3},{2,4,6},{5}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,3},{2,4},{5,6}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,5},{2,6},{4}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,6},{2,5},{4}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3},{2,5,6},{4}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,3},{2,5},{4,6}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St000681
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 75%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 75%
Values
{{1}}
=> [1]
=> []
=> ?
=> ? = 0
{{1,2}}
=> [2]
=> []
=> ?
=> ? ∊ {0,1}
{{1},{2}}
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1}
{{1,2,3}}
=> [3]
=> []
=> ?
=> ? ∊ {0,1,1,1,1}
{{1,2},{3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1}
{{1,3},{2}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1}
{{1},{2,3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1}
{{1},{2},{3}}
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1}
{{1,2,3,4}}
=> [4]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2,3},{4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2,4},{3}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2},{3,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3,4},{2}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3},{2,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,4},{2,3}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2,3,4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2,3,4,5}}
=> [5]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,3,4},{5}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,3,5},{4}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,3},{4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,4,5},{3}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,4},{3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,5},{3,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2},{3,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3,4},{2,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3,5},{2,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3},{2,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,4,5},{2,3}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,4},{2,3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,5},{2,3,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1},{2,3,4,5}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,2},{3},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3},{4},{5,6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,3},{2},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3,6},{2},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3},{2,6},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2},{4},{5,6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,4},{2,3},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1},{2,3,4},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,5},{2,3},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,5},{2,3},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1},{2,3,5},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,6},{2,3},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1},{2,3},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,6},{2,3},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1},{2,3,6},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2,3},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
Description
The Grundy value of Chomp on Ferrers diagrams.
Players take turns and choose a cell of the diagram, cutting off all cells below and to the right of this cell in English notation. The player who is left with the single cell partition looses. The traditional version is played on chocolate bars, see [1].
This statistic is the Grundy value of the partition, that is, the smallest non-negative integer which does not occur as value of a partition obtained by a single move.
Matching statistic: St000707
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000707: Integer partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000707: Integer partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 50%
Values
{{1}}
=> [1]
=> []
=> ?
=> ? = 0
{{1,2}}
=> [2]
=> []
=> ?
=> ? ∊ {0,1}
{{1},{2}}
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1}
{{1,2,3}}
=> [3]
=> []
=> ?
=> ? ∊ {0,1,1,1,1}
{{1,2},{3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1}
{{1,3},{2}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1}
{{1},{2,3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1}
{{1},{2},{3}}
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1}
{{1,2,3,4}}
=> [4]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2,3},{4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2,4},{3}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2},{3,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3,4},{2}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3},{2,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,4},{2,3}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2,3,4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2,3,4,5}}
=> [5]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,3,4},{5}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,3,5},{4}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,3},{4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,4,5},{3}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,4},{3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,5},{3,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2},{3,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3,4},{2,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3,5},{2,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3},{2,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,4,5},{2,3}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,4},{2,3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,5},{2,3,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1},{2,3,4,5}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,2},{3},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3},{4},{5,6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,3},{2},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3,6},{2},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3},{2,6},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2},{4},{5,6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,4},{2,3},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1},{2,3,4},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,5},{2,3},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1,5},{2,3},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1},{2,3,5},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,6},{2,3},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
{{1},{2,3},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,6},{2,3},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1},{2,3,6},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2,3},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
Description
The product of the factorials of the parts.
The following 13 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. 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. St001568The smallest positive integer that does not appear twice in the partition. St000455The second largest eigenvalue of a graph if it is integral. St000454The largest eigenvalue of a graph if it is integral. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001935The number of ascents in a parking function. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001624The breadth of a lattice.
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