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Your data matches 58 different statistics following compositions of up to 3 maps.
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Matching statistic: St000422
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,2] => [1,2] => [1,2] => ([],2)
=> 0
[2,1] => [1,2] => [1,2] => ([],2)
=> 0
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
=> 0
[1,3,2] => [1,2,3] => [1,2,3] => ([],3)
=> 0
[2,1,3] => [1,2,3] => [1,2,3] => ([],3)
=> 0
[2,3,1] => [1,2,3] => [1,2,3] => ([],3)
=> 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 2
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 2
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 2
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,2,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,3,5,4,2] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,5,2,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[1,5,3,2,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[2,1,5,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,1,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[2,3,5,1,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,3,5,4,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,5,1,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[2,5,3,1,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[3,1,4,2,5] => [1,3,4,2,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 2
[3,2,4,1,5] => [1,3,4,2,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 2
[3,5,4,1,2] => [1,3,4,2,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 2
Description
The energy of a graph, if it is integral.
The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3].
The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.
Matching statistic: St000632
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000632: Posets ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 33%
Mp00209: Permutations —pattern poset⟶ Posets
St000632: Posets ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 33%
Values
[1] => [1] => ([],1)
=> 0
[1,2] => [1,2] => ([(0,1)],2)
=> 0
[2,1] => [1,2] => ([(0,1)],2)
=> 0
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[1,3,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[2,1,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,3,2,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[2,1,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[3,1,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[3,2,1,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[3,4,1,2] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[3,4,2,1] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[1,2,5,4,3] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[1,3,2,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[1,3,5,4,2] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[1,5,2,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 6
[1,5,3,2,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 6
[2,1,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,1,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,1,5,3,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,1,5,4,3] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,3,1,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,3,1,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,3,4,1,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,3,5,1,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,3,5,4,1] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,5,1,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 6
[2,5,3,1,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 6
[3,1,4,2,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[3,2,4,1,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[3,5,4,1,2] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[3,5,4,2,1] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[4,1,2,3,5] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 4
[4,1,3,2,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[4,1,5,2,3] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[4,2,1,3,5] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 4
[4,2,3,1,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[4,2,5,1,3] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[4,3,1,2,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[4,3,2,1,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[4,3,5,1,2] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[4,3,5,2,1] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[4,5,1,3,2] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 4
[4,5,2,3,1] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 4
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,5,3,6,4] => [1,2,3,5,6,4] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,2,5,4,6,3] => [1,2,3,5,6,4] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,2,6,3,5,4] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,2,6,4,5,3] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,2,6,5,3,4] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,2,6,5,4,3] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,3,5,2,6,4] => [1,2,3,5,6,4] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,3,5,4,6,2] => [1,2,3,5,6,4] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,3,6,2,5,4] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,3,6,4,5,2] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,3,6,5,2,4] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,3,6,5,4,2] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,4,2,3,5,6] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 2
[1,4,2,3,6,5] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 2
[1,4,3,2,5,6] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 2
[1,4,3,2,6,5] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 2
[1,4,5,2,3,6] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 2
[1,4,5,2,6,3] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 2
[1,4,5,3,2,6] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 2
[1,4,5,3,6,2] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 2
[1,5,2,3,6,4] => [1,2,5,6,4,3] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? = 6
[1,5,3,2,6,4] => [1,2,5,6,4,3] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? = 6
Description
The jump number of the poset.
A jump in a linear extension $e_1, \dots, e_n$ of a poset $P$ is a pair $(e_i, e_{i+1})$ so that $e_{i+1}$ does not cover $e_i$ in $P$. The jump number of a poset is the minimal number of jumps in linear extensions of a poset.
Matching statistic: St001633
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001633: Posets ⟶ ℤResult quality: 17% ●values known / values provided: 19%●distinct values known / distinct values provided: 17%
Mp00209: Permutations —pattern poset⟶ Posets
St001633: Posets ⟶ ℤResult quality: 17% ●values known / values provided: 19%●distinct values known / distinct values provided: 17%
Values
[1] => [1] => ([],1)
=> 0
[1,2] => [1,2] => ([(0,1)],2)
=> 0
[2,1] => [1,2] => ([(0,1)],2)
=> 0
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[1,3,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[2,1,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,3,2,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[2,1,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[3,1,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[3,2,1,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[3,4,1,2] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[3,4,2,1] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[1,2,5,4,3] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[1,3,2,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[1,3,5,4,2] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[1,5,2,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 6
[1,5,3,2,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 6
[2,1,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,1,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,1,5,3,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,1,5,4,3] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,3,1,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,3,1,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,3,4,1,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,3,5,1,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,3,5,4,1] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,5,1,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 6
[2,5,3,1,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 6
[3,1,4,2,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[3,2,4,1,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[3,5,4,1,2] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[3,5,4,2,1] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[4,1,2,3,5] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 4
[4,1,3,2,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[4,1,5,2,3] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[4,2,1,3,5] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 4
[4,2,3,1,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[4,2,5,1,3] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[4,3,1,2,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[4,3,2,1,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[4,3,5,1,2] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[4,3,5,2,1] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[4,5,1,3,2] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 4
[4,5,2,3,1] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 4
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,5,3,6,4] => [1,2,3,5,6,4] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,2,5,4,6,3] => [1,2,3,5,6,4] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,2,6,3,5,4] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,2,6,4,5,3] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,2,6,5,3,4] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,2,6,5,4,3] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,3,5,2,6,4] => [1,2,3,5,6,4] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,3,5,4,6,2] => [1,2,3,5,6,4] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,3,6,2,5,4] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,3,6,4,5,2] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,3,6,5,2,4] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,3,6,5,4,2] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,4,2,3,5,6] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 2
[1,4,2,3,6,5] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 2
[1,4,3,2,5,6] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 2
[1,4,3,2,6,5] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 2
[1,4,5,2,3,6] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 2
[1,4,5,2,6,3] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 2
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
Description
The number of simple modules with projective dimension two in the incidence algebra of the poset.
Matching statistic: St000307
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 17% ●values known / values provided: 19%●distinct values known / distinct values provided: 17%
Mp00209: Permutations —pattern poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 17% ●values known / values provided: 19%●distinct values known / distinct values provided: 17%
Values
[1] => [1] => ([],1)
=> 1 = 0 + 1
[1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,3,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,1,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,1,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,1,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2 + 1
[3,2,1,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2 + 1
[3,4,1,2] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2 + 1
[3,4,2,1] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2 + 1
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 + 1
[1,2,5,4,3] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 + 1
[1,3,2,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,5,2,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 + 1
[1,3,5,4,2] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 + 1
[1,5,2,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 6 + 1
[1,5,3,2,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 6 + 1
[2,1,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,5,3,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 + 1
[2,1,5,4,3] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 + 1
[2,3,1,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,1,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,4,1,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,5,1,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 + 1
[2,3,5,4,1] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 + 1
[2,5,1,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 6 + 1
[2,5,3,1,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 6 + 1
[3,1,4,2,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2 + 1
[3,2,4,1,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2 + 1
[3,5,4,1,2] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2 + 1
[3,5,4,2,1] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2 + 1
[4,1,2,3,5] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 4 + 1
[4,1,3,2,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2 + 1
[4,1,5,2,3] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2 + 1
[4,2,1,3,5] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 4 + 1
[4,2,3,1,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2 + 1
[4,2,5,1,3] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2 + 1
[4,3,1,2,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2 + 1
[4,3,2,1,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2 + 1
[4,3,5,1,2] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2 + 1
[4,3,5,2,1] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2 + 1
[4,5,1,3,2] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 4 + 1
[4,5,2,3,1] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 4 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,5,3,6,4] => [1,2,3,5,6,4] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4 + 1
[1,2,5,4,6,3] => [1,2,3,5,6,4] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4 + 1
[1,2,6,3,5,4] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4 + 1
[1,2,6,4,5,3] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4 + 1
[1,2,6,5,3,4] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4 + 1
[1,2,6,5,4,3] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4 + 1
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,5,2,6,4] => [1,2,3,5,6,4] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4 + 1
[1,3,5,4,6,2] => [1,2,3,5,6,4] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4 + 1
[1,3,6,2,5,4] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4 + 1
[1,3,6,4,5,2] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4 + 1
[1,3,6,5,2,4] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4 + 1
[1,3,6,5,4,2] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4 + 1
[1,4,2,3,5,6] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 2 + 1
[1,4,2,3,6,5] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 2 + 1
[1,4,3,2,5,6] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 2 + 1
[1,4,3,2,6,5] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 2 + 1
[1,4,5,2,3,6] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 2 + 1
[1,4,5,2,6,3] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 2 + 1
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
Description
The number of rowmotion orbits of a poset.
Rowmotion is an operation on order ideals in a poset $P$. It sends an order ideal $I$ to the order ideal generated by the minimal antichain of $P \setminus I$.
Matching statistic: St000259
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 19%●distinct values known / distinct values provided: 17%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 19%●distinct values known / distinct values provided: 17%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 0
[1,2] => [1,2] => ([],2)
=> ([],1)
=> 0
[2,1] => [1,2] => ([],2)
=> ([],1)
=> 0
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[1,3,2] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,1,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,3,1] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,2,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,2,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,4,2] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,1,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,1,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,1,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,4,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 6
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 6
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 6
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 6
[3,1,4,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[3,2,4,1,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[3,5,4,1,2] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[3,5,4,2,1] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,1,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 4
[4,1,3,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,1,5,2,3] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,2,1,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 4
[4,2,3,1,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,2,5,1,3] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,3,1,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,3,2,1,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,3,5,1,2] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,3,5,2,1] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,5,1,3,2] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 4
[4,5,2,3,1] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 4
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,5,3,6,4] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,2,5,4,6,3] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,2,6,3,5,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,2,6,4,5,3] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,2,6,5,3,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,2,6,5,4,3] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,5,2,6,4] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,3,5,4,6,2] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,3,6,2,5,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,3,6,4,5,2] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,3,6,5,2,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,3,6,5,4,2] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,4,2,3,5,6] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,4,2,3,6,5] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,4,3,2,5,6] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,4,3,2,6,5] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,4,5,2,3,6] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,4,5,2,6,3] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000260
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 19%●distinct values known / distinct values provided: 17%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 19%●distinct values known / distinct values provided: 17%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 0
[1,2] => [1,2] => ([],2)
=> ([],1)
=> 0
[2,1] => [1,2] => ([],2)
=> ([],1)
=> 0
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[1,3,2] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,1,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,3,1] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,2,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,2,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,4,2] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,1,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,1,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,1,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,4,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 6
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 6
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 6
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 6
[3,1,4,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[3,2,4,1,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[3,5,4,1,2] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[3,5,4,2,1] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,1,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 4
[4,1,3,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,1,5,2,3] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,2,1,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 4
[4,2,3,1,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,2,5,1,3] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,3,1,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,3,2,1,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,3,5,1,2] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,3,5,2,1] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,5,1,3,2] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 4
[4,5,2,3,1] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 4
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,5,3,6,4] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,2,5,4,6,3] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,2,6,3,5,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,2,6,4,5,3] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,2,6,5,3,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,2,6,5,4,3] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,5,2,6,4] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,3,5,4,6,2] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,3,6,2,5,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,3,6,4,5,2] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,3,6,5,2,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,3,6,5,4,2] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,4,2,3,5,6] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,4,2,3,6,5] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,4,3,2,5,6] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,4,3,2,6,5] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,4,5,2,3,6] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,4,5,2,6,3] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000302
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000302: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 19%●distinct values known / distinct values provided: 17%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000302: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 19%●distinct values known / distinct values provided: 17%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 0
[1,2] => [1,2] => ([],2)
=> ([],1)
=> 0
[2,1] => [1,2] => ([],2)
=> ([],1)
=> 0
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[1,3,2] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,1,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,3,1] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,2,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,2,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,4,2] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,1,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,1,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,1,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,4,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 6
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 6
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 6
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 6
[3,1,4,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[3,2,4,1,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[3,5,4,1,2] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[3,5,4,2,1] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,1,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 4
[4,1,3,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,1,5,2,3] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,2,1,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 4
[4,2,3,1,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,2,5,1,3] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,3,1,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,3,2,1,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,3,5,1,2] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,3,5,2,1] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,5,1,3,2] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 4
[4,5,2,3,1] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 4
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,5,3,6,4] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,2,5,4,6,3] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,2,6,3,5,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,2,6,4,5,3] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,2,6,5,3,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,2,6,5,4,3] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,5,2,6,4] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,3,5,4,6,2] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,3,6,2,5,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,3,6,4,5,2] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,3,6,5,2,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,3,6,5,4,2] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,4,2,3,5,6] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,4,2,3,6,5] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,4,3,2,5,6] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,4,3,2,6,5] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,4,5,2,3,6] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,4,5,2,6,3] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
Description
The determinant of the distance matrix of a connected graph.
Matching statistic: St000466
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000466: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 19%●distinct values known / distinct values provided: 17%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000466: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 19%●distinct values known / distinct values provided: 17%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 0
[1,2] => [1,2] => ([],2)
=> ([],1)
=> 0
[2,1] => [1,2] => ([],2)
=> ([],1)
=> 0
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[1,3,2] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,1,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,3,1] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,2,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,2,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,4,2] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,1,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,1,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,1,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,4,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 6
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 6
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 6
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 6
[3,1,4,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[3,2,4,1,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[3,5,4,1,2] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[3,5,4,2,1] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,1,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 4
[4,1,3,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,1,5,2,3] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,2,1,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 4
[4,2,3,1,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,2,5,1,3] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,3,1,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,3,2,1,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,3,5,1,2] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,3,5,2,1] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,5,1,3,2] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 4
[4,5,2,3,1] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 4
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,5,3,6,4] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,2,5,4,6,3] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,2,6,3,5,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,2,6,4,5,3] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,2,6,5,3,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,2,6,5,4,3] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,5,2,6,4] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,3,5,4,6,2] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,3,6,2,5,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,3,6,4,5,2] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,3,6,5,2,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,3,6,5,4,2] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,4,2,3,5,6] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,4,2,3,6,5] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,4,3,2,5,6] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,4,3,2,6,5] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,4,5,2,3,6] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,4,5,2,6,3] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
Description
The Gutman (or modified Schultz) index of a connected graph.
This is
$$\sum_{\{u,v\}\subseteq V} d(u)d(v)d(u,v)$$
where $d(u)$ is the degree of vertex $u$ and $d(u,v)$ is the distance between vertices $u$ and $v$.
For trees on $n$ vertices, the modified Schultz index is related to the Wiener index via $S^\ast(T)=4W(T)-(n-1)(2n-1)$ [1].
Matching statistic: St000467
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000467: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 19%●distinct values known / distinct values provided: 17%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000467: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 19%●distinct values known / distinct values provided: 17%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 0
[1,2] => [1,2] => ([],2)
=> ([],1)
=> 0
[2,1] => [1,2] => ([],2)
=> ([],1)
=> 0
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[1,3,2] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,1,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,3,1] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,2,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,2,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,4,2] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,1,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,1,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,1,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,4,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 6
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 6
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 6
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 6
[3,1,4,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[3,2,4,1,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[3,5,4,1,2] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[3,5,4,2,1] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,1,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 4
[4,1,3,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,1,5,2,3] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,2,1,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 4
[4,2,3,1,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,2,5,1,3] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,3,1,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,3,2,1,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,3,5,1,2] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,3,5,2,1] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[4,5,1,3,2] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 4
[4,5,2,3,1] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 4
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,5,3,6,4] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,2,5,4,6,3] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,2,6,3,5,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,2,6,4,5,3] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,2,6,5,3,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,2,6,5,4,3] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,5,2,6,4] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,3,5,4,6,2] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,3,6,2,5,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,3,6,4,5,2] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,3,6,5,2,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,3,6,5,4,2] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,4,2,3,5,6] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,4,2,3,6,5] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,4,3,2,5,6] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,4,3,2,6,5] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,4,5,2,3,6] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[1,4,5,2,6,3] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
Description
The hyper-Wiener index of a connected graph.
This is
$$
\sum_{\{u,v\}\subseteq V} d(u,v)+d(u,v)^2.
$$
Matching statistic: St000298
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 17% ●values known / values provided: 19%●distinct values known / distinct values provided: 17%
Mp00209: Permutations —pattern poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 17% ●values known / values provided: 19%●distinct values known / distinct values provided: 17%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,2] => [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,3,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,1,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,1,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,1,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[3,2,1,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[3,4,1,2] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[3,4,2,1] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 + 1
[1,2,5,4,3] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 + 1
[1,3,2,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,5,2,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 + 1
[1,3,5,4,2] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 + 1
[1,5,2,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 6 + 1
[1,5,3,2,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 6 + 1
[2,1,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,5,3,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 + 1
[2,1,5,4,3] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 + 1
[2,3,1,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,1,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,4,1,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,5,1,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 + 1
[2,3,5,4,1] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 + 1
[2,5,1,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 6 + 1
[2,5,3,1,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 6 + 1
[3,1,4,2,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
=> ? = 2 + 1
[3,2,4,1,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
=> ? = 2 + 1
[3,5,4,1,2] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
=> ? = 2 + 1
[3,5,4,2,1] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
=> ? = 2 + 1
[4,1,2,3,5] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(0,3),(0,4),(1,10),(2,7),(2,8),(3,1),(3,5),(3,6),(4,2),(4,5),(4,6),(5,8),(5,10),(6,7),(6,10),(7,9),(8,9),(10,9)],11)
=> ? = 4 + 1
[4,1,3,2,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
=> ? = 2 + 1
[4,1,5,2,3] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
=> ? = 2 + 1
[4,2,1,3,5] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(0,3),(0,4),(1,10),(2,7),(2,8),(3,1),(3,5),(3,6),(4,2),(4,5),(4,6),(5,8),(5,10),(6,7),(6,10),(7,9),(8,9),(10,9)],11)
=> ? = 4 + 1
[4,2,3,1,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
=> ? = 2 + 1
[4,2,5,1,3] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
=> ? = 2 + 1
[4,3,1,2,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
=> ? = 2 + 1
[4,3,2,1,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
=> ? = 2 + 1
[4,3,5,1,2] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
=> ? = 2 + 1
[4,3,5,2,1] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
=> ? = 2 + 1
[4,5,1,3,2] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(0,3),(0,4),(1,10),(2,7),(2,8),(3,1),(3,5),(3,6),(4,2),(4,5),(4,6),(5,8),(5,10),(6,7),(6,10),(7,9),(8,9),(10,9)],11)
=> ? = 4 + 1
[4,5,2,3,1] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(0,3),(0,4),(1,10),(2,7),(2,8),(3,1),(3,5),(3,6),(4,2),(4,5),(4,6),(5,8),(5,10),(6,7),(6,10),(7,9),(8,9),(10,9)],11)
=> ? = 4 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,5,3,6,4] => [1,2,3,5,6,4] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ([(0,2),(0,5),(1,11),(2,6),(2,7),(3,4),(3,9),(3,12),(4,1),(4,8),(4,10),(5,3),(5,6),(5,7),(6,12),(7,9),(7,12),(8,11),(9,8),(9,10),(10,11),(12,10)],13)
=> ? = 4 + 1
[1,2,5,4,6,3] => [1,2,3,5,6,4] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ([(0,2),(0,5),(1,11),(2,6),(2,7),(3,4),(3,9),(3,12),(4,1),(4,8),(4,10),(5,3),(5,6),(5,7),(6,12),(7,9),(7,12),(8,11),(9,8),(9,10),(10,11),(12,10)],13)
=> ? = 4 + 1
[1,2,6,3,5,4] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ([(0,2),(0,5),(1,11),(2,6),(2,7),(3,4),(3,9),(3,12),(4,1),(4,8),(4,10),(5,3),(5,6),(5,7),(6,12),(7,9),(7,12),(8,11),(9,8),(9,10),(10,11),(12,10)],13)
=> ? = 4 + 1
[1,2,6,4,5,3] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ([(0,2),(0,5),(1,11),(2,6),(2,7),(3,4),(3,9),(3,12),(4,1),(4,8),(4,10),(5,3),(5,6),(5,7),(6,12),(7,9),(7,12),(8,11),(9,8),(9,10),(10,11),(12,10)],13)
=> ? = 4 + 1
[1,2,6,5,3,4] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ([(0,2),(0,5),(1,11),(2,6),(2,7),(3,4),(3,9),(3,12),(4,1),(4,8),(4,10),(5,3),(5,6),(5,7),(6,12),(7,9),(7,12),(8,11),(9,8),(9,10),(10,11),(12,10)],13)
=> ? = 4 + 1
[1,2,6,5,4,3] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ([(0,2),(0,5),(1,11),(2,6),(2,7),(3,4),(3,9),(3,12),(4,1),(4,8),(4,10),(5,3),(5,6),(5,7),(6,12),(7,9),(7,12),(8,11),(9,8),(9,10),(10,11),(12,10)],13)
=> ? = 4 + 1
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,5,2,6,4] => [1,2,3,5,6,4] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ([(0,2),(0,5),(1,11),(2,6),(2,7),(3,4),(3,9),(3,12),(4,1),(4,8),(4,10),(5,3),(5,6),(5,7),(6,12),(7,9),(7,12),(8,11),(9,8),(9,10),(10,11),(12,10)],13)
=> ? = 4 + 1
[1,3,5,4,6,2] => [1,2,3,5,6,4] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ([(0,2),(0,5),(1,11),(2,6),(2,7),(3,4),(3,9),(3,12),(4,1),(4,8),(4,10),(5,3),(5,6),(5,7),(6,12),(7,9),(7,12),(8,11),(9,8),(9,10),(10,11),(12,10)],13)
=> ? = 4 + 1
[1,3,6,2,5,4] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ([(0,2),(0,5),(1,11),(2,6),(2,7),(3,4),(3,9),(3,12),(4,1),(4,8),(4,10),(5,3),(5,6),(5,7),(6,12),(7,9),(7,12),(8,11),(9,8),(9,10),(10,11),(12,10)],13)
=> ? = 4 + 1
[1,3,6,4,5,2] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ([(0,2),(0,5),(1,11),(2,6),(2,7),(3,4),(3,9),(3,12),(4,1),(4,8),(4,10),(5,3),(5,6),(5,7),(6,12),(7,9),(7,12),(8,11),(9,8),(9,10),(10,11),(12,10)],13)
=> ? = 4 + 1
[1,3,6,5,2,4] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ([(0,2),(0,5),(1,11),(2,6),(2,7),(3,4),(3,9),(3,12),(4,1),(4,8),(4,10),(5,3),(5,6),(5,7),(6,12),(7,9),(7,12),(8,11),(9,8),(9,10),(10,11),(12,10)],13)
=> ? = 4 + 1
[1,3,6,5,4,2] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ([(0,2),(0,5),(1,11),(2,6),(2,7),(3,4),(3,9),(3,12),(4,1),(4,8),(4,10),(5,3),(5,6),(5,7),(6,12),(7,9),(7,12),(8,11),(9,8),(9,10),(10,11),(12,10)],13)
=> ? = 4 + 1
[1,4,2,3,5,6] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ([(0,2),(0,5),(1,12),(2,6),(2,7),(3,4),(3,8),(3,9),(3,13),(4,1),(4,10),(4,11),(5,3),(5,6),(5,7),(6,9),(6,13),(7,8),(7,13),(8,10),(9,11),(10,12),(11,12),(13,10),(13,11)],14)
=> ? = 2 + 1
[1,4,2,3,6,5] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ([(0,2),(0,5),(1,12),(2,6),(2,7),(3,4),(3,8),(3,9),(3,13),(4,1),(4,10),(4,11),(5,3),(5,6),(5,7),(6,9),(6,13),(7,8),(7,13),(8,10),(9,11),(10,12),(11,12),(13,10),(13,11)],14)
=> ? = 2 + 1
[1,4,3,2,5,6] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ([(0,2),(0,5),(1,12),(2,6),(2,7),(3,4),(3,8),(3,9),(3,13),(4,1),(4,10),(4,11),(5,3),(5,6),(5,7),(6,9),(6,13),(7,8),(7,13),(8,10),(9,11),(10,12),(11,12),(13,10),(13,11)],14)
=> ? = 2 + 1
[1,4,3,2,6,5] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ([(0,2),(0,5),(1,12),(2,6),(2,7),(3,4),(3,8),(3,9),(3,13),(4,1),(4,10),(4,11),(5,3),(5,6),(5,7),(6,9),(6,13),(7,8),(7,13),(8,10),(9,11),(10,12),(11,12),(13,10),(13,11)],14)
=> ? = 2 + 1
[1,4,5,2,3,6] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ([(0,2),(0,5),(1,12),(2,6),(2,7),(3,4),(3,8),(3,9),(3,13),(4,1),(4,10),(4,11),(5,3),(5,6),(5,7),(6,9),(6,13),(7,8),(7,13),(8,10),(9,11),(10,12),(11,12),(13,10),(13,11)],14)
=> ? = 2 + 1
[1,4,5,2,6,3] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ([(0,2),(0,5),(1,12),(2,6),(2,7),(3,4),(3,8),(3,9),(3,13),(4,1),(4,10),(4,11),(5,3),(5,6),(5,7),(6,9),(6,13),(7,8),(7,13),(8,10),(9,11),(10,12),(11,12),(13,10),(13,11)],14)
=> ? = 2 + 1
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
Description
The order dimension or Dushnik-Miller dimension of a poset.
This is the minimal number of linear orderings whose intersection is the given poset.
The following 48 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001964The interval resolution global dimension of a poset. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001902The number of potential covers of a poset. St000908The length of the shortest maximal antichain in a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001472The permanent of the Coxeter matrix of the poset. St001510The number of self-evacuating linear extensions of a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001779The order of promotion on the set of linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001893The flag descent of a signed permutation. St001769The reflection length of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001868The number of alignments of type NE of a signed permutation. St001892The flag excedance statistic of a signed permutation. St001894The depth of a signed permutation. St001855The number of signed permutations less than or equal to a signed permutation in left weak order. St001864The number of excedances of a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St000635The number of strictly order preserving maps of a poset into itself. St001890The maximum magnitude of the Möbius function of a poset. St000896The number of zeros on the main diagonal of an alternating sign matrix.
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