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Your data matches 84 different statistics following compositions of up to 3 maps.
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Matching statistic: St000422
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ 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] => [2] => ([],2)
=> 0
[2,1] => [1,2] => [2] => ([],2)
=> 0
[1,2,3] => [1,2,3] => [3] => ([],3)
=> 0
[1,3,2] => [1,2,3] => [3] => ([],3)
=> 0
[2,1,3] => [1,2,3] => [3] => ([],3)
=> 0
[2,3,1] => [1,2,3] => [3] => ([],3)
=> 0
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
=> 0
[1,2,4,3] => [1,2,3,4] => [4] => ([],4)
=> 0
[1,3,2,4] => [1,2,3,4] => [4] => ([],4)
=> 0
[1,3,4,2] => [1,2,3,4] => [4] => ([],4)
=> 0
[2,1,3,4] => [1,2,3,4] => [4] => ([],4)
=> 0
[2,1,4,3] => [1,2,3,4] => [4] => ([],4)
=> 0
[2,3,1,4] => [1,2,3,4] => [4] => ([],4)
=> 0
[2,3,4,1] => [1,2,3,4] => [4] => ([],4)
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,2,5,4,3] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,3,2,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,3,5,4,2] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,4,2,5,3] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,4,3,5,2] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,5,2,3,4] => [1,2,5,4,3] => [3,1,1] => ([(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] => [3,1,1] => ([(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] => [5] => ([],5)
=> 0
[2,1,3,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[2,1,4,3,5] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[2,1,4,5,3] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[2,1,5,3,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,1,5,4,3] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,3,1,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[2,3,1,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[2,3,4,1,5] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[2,3,4,5,1] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[2,3,5,1,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,3,5,4,1] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,4,1,5,3] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,4,3,5,1] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,5,1,3,4] => [1,2,5,4,3] => [3,1,1] => ([(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] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[3,1,4,5,2] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[3,1,5,2,4] => [1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[3,2,4,5,1] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
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 Kn equals 2n−2. For this reason, we do not define the energy of the empty graph.
Matching statistic: St000264
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 40%●distinct values known / distinct values provided: 33%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 40%●distinct values known / distinct values provided: 33%
Values
[1] => [1] => [1] => ([],1)
=> ? = 0 - 3
[1,2] => [1,2] => [2] => ([],2)
=> ? = 0 - 3
[2,1] => [1,2] => [2] => ([],2)
=> ? = 0 - 3
[1,2,3] => [1,2,3] => [3] => ([],3)
=> ? = 0 - 3
[1,3,2] => [1,2,3] => [3] => ([],3)
=> ? = 0 - 3
[2,1,3] => [1,2,3] => [3] => ([],3)
=> ? = 0 - 3
[2,3,1] => [1,2,3] => [3] => ([],3)
=> ? = 0 - 3
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
=> ? = 0 - 3
[1,2,4,3] => [1,2,3,4] => [4] => ([],4)
=> ? = 0 - 3
[1,3,2,4] => [1,2,3,4] => [4] => ([],4)
=> ? = 0 - 3
[1,3,4,2] => [1,2,3,4] => [4] => ([],4)
=> ? = 0 - 3
[2,1,3,4] => [1,2,3,4] => [4] => ([],4)
=> ? = 0 - 3
[2,1,4,3] => [1,2,3,4] => [4] => ([],4)
=> ? = 0 - 3
[2,3,1,4] => [1,2,3,4] => [4] => ([],4)
=> ? = 0 - 3
[2,3,4,1] => [1,2,3,4] => [4] => ([],4)
=> ? = 0 - 3
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 3
[1,2,3,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 3
[1,2,4,3,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 3
[1,2,4,5,3] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 3
[1,2,5,3,4] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 3
[1,2,5,4,3] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 3
[1,3,2,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 3
[1,3,2,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 3
[1,3,4,2,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 3
[1,3,4,5,2] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 3
[1,3,5,2,4] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 3
[1,3,5,4,2] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 3
[1,4,2,5,3] => [1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 4 - 3
[1,4,3,5,2] => [1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 4 - 3
[1,5,2,3,4] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 6 - 3
[1,5,3,2,4] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 6 - 3
[2,1,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 3
[2,1,3,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 3
[2,1,4,3,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 3
[2,1,4,5,3] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 3
[2,1,5,3,4] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 3
[2,1,5,4,3] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 3
[2,3,1,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 3
[2,3,1,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 3
[2,3,4,1,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 3
[2,3,4,5,1] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 3
[2,3,5,1,4] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 3
[2,3,5,4,1] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 3
[2,4,1,5,3] => [1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 4 - 3
[2,4,3,5,1] => [1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 4 - 3
[2,5,1,3,4] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 6 - 3
[2,5,3,1,4] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 6 - 3
[3,1,4,5,2] => [1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> ? = 4 - 3
[3,1,5,2,4] => [1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 6 - 3
[3,2,4,5,1] => [1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> ? = 4 - 3
[3,2,5,1,4] => [1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 6 - 3
[4,1,2,5,3] => [1,4,5,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 6 - 3
[4,2,1,5,3] => [1,4,5,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 6 - 3
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [6] => ([],6)
=> ? = 0 - 3
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [6] => ([],6)
=> ? = 0 - 3
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [6] => ([],6)
=> ? = 0 - 3
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [6] => ([],6)
=> ? = 0 - 3
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [6] => ([],6)
=> ? = 0 - 3
[1,5,2,3,4,6] => [1,2,5,4,3,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[1,5,3,2,4,6] => [1,2,5,4,3,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[1,5,6,2,4,3] => [1,2,5,4,3,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[1,5,6,3,4,2] => [1,2,5,4,3,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[1,6,2,3,5,4] => [1,2,6,4,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[1,6,2,4,3,5] => [1,2,6,5,3,4] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[1,6,3,2,5,4] => [1,2,6,4,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[1,6,3,4,2,5] => [1,2,6,5,3,4] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[1,6,4,2,3,5] => [1,2,6,5,3,4] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[1,6,4,3,2,5] => [1,2,6,5,3,4] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[1,6,5,2,3,4] => [1,2,6,4,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[1,6,5,3,2,4] => [1,2,6,4,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[2,5,1,3,4,6] => [1,2,5,4,3,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[2,5,3,1,4,6] => [1,2,5,4,3,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[2,5,6,1,4,3] => [1,2,5,4,3,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[2,5,6,3,4,1] => [1,2,5,4,3,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[2,6,1,3,5,4] => [1,2,6,4,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[2,6,1,4,3,5] => [1,2,6,5,3,4] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[2,6,3,1,5,4] => [1,2,6,4,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[2,6,3,4,1,5] => [1,2,6,5,3,4] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[2,6,4,1,3,5] => [1,2,6,5,3,4] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[2,6,4,3,1,5] => [1,2,6,5,3,4] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[2,6,5,1,3,4] => [1,2,6,4,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[2,6,5,3,1,4] => [1,2,6,4,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[3,1,5,2,4,6] => [1,3,5,4,2,6] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[3,1,6,2,5,4] => [1,3,6,4,2,5] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[3,1,6,4,2,5] => [1,3,6,5,2,4] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[3,2,5,1,4,6] => [1,3,5,4,2,6] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[3,2,6,1,5,4] => [1,3,6,4,2,5] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[3,2,6,4,1,5] => [1,3,6,5,2,4] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[3,4,6,1,2,5] => [1,3,6,5,2,4] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[3,4,6,2,1,5] => [1,3,6,5,2,4] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[3,5,6,1,2,4] => [1,3,6,4,2,5] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[3,5,6,2,1,4] => [1,3,6,4,2,5] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[3,6,5,1,4,2] => [1,3,5,4,2,6] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[3,6,5,2,4,1] => [1,3,5,4,2,6] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[4,1,2,5,3,6] => [1,4,5,3,2,6] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[4,1,2,6,5,3] => [1,4,6,3,2,5] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[4,1,3,6,2,5] => [1,4,6,5,2,3] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[4,2,1,5,3,6] => [1,4,5,3,2,6] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[4,2,1,6,5,3] => [1,4,6,3,2,5] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[4,2,3,6,1,5] => [1,4,6,5,2,3] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St001869
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001869: Graphs ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001869: Graphs ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,2] => [1,2] => [2] => ([],2)
=> 0
[2,1] => [1,2] => [2] => ([],2)
=> 0
[1,2,3] => [1,2,3] => [3] => ([],3)
=> 0
[1,3,2] => [1,2,3] => [3] => ([],3)
=> 0
[2,1,3] => [1,2,3] => [3] => ([],3)
=> 0
[2,3,1] => [1,2,3] => [3] => ([],3)
=> 0
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
=> 0
[1,2,4,3] => [1,2,3,4] => [4] => ([],4)
=> 0
[1,3,2,4] => [1,2,3,4] => [4] => ([],4)
=> 0
[1,3,4,2] => [1,2,3,4] => [4] => ([],4)
=> 0
[2,1,3,4] => [1,2,3,4] => [4] => ([],4)
=> 0
[2,1,4,3] => [1,2,3,4] => [4] => ([],4)
=> 0
[2,3,1,4] => [1,2,3,4] => [4] => ([],4)
=> 0
[2,3,4,1] => [1,2,3,4] => [4] => ([],4)
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,2,5,4,3] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,3,2,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,3,5,4,2] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,4,2,5,3] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,4,3,5,2] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,5,2,3,4] => [1,2,5,4,3] => [3,1,1] => ([(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] => [3,1,1] => ([(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] => [5] => ([],5)
=> 0
[2,1,3,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[2,1,4,3,5] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[2,1,4,5,3] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[2,1,5,3,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,1,5,4,3] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,3,1,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[2,3,1,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[2,3,4,1,5] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[2,3,4,5,1] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[2,3,5,1,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,3,5,4,1] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,4,1,5,3] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,4,3,5,1] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,5,1,3,4] => [1,2,5,4,3] => [3,1,1] => ([(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] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[3,1,4,5,2] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[3,1,5,2,4] => [1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[3,2,4,5,1] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [7] => ([],7)
=> ? = 0
[1,2,3,4,5,7,6] => [1,2,3,4,5,6,7] => [7] => ([],7)
=> ? = 0
[1,2,3,4,6,5,7] => [1,2,3,4,5,6,7] => [7] => ([],7)
=> ? = 0
[1,2,3,4,6,7,5] => [1,2,3,4,5,6,7] => [7] => ([],7)
=> ? = 0
[1,2,3,5,4,6,7] => [1,2,3,4,5,6,7] => [7] => ([],7)
=> ? = 0
[1,2,3,5,4,7,6] => [1,2,3,4,5,6,7] => [7] => ([],7)
=> ? = 0
[1,2,3,5,6,4,7] => [1,2,3,4,5,6,7] => [7] => ([],7)
=> ? = 0
[1,2,3,5,6,7,4] => [1,2,3,4,5,6,7] => [7] => ([],7)
=> ? = 0
[1,2,4,3,5,6,7] => [1,2,3,4,5,6,7] => [7] => ([],7)
=> ? = 0
[1,2,4,3,5,7,6] => [1,2,3,4,5,6,7] => [7] => ([],7)
=> ? = 0
[1,2,4,3,6,5,7] => [1,2,3,4,5,6,7] => [7] => ([],7)
=> ? = 0
[1,2,4,3,6,7,5] => [1,2,3,4,5,6,7] => [7] => ([],7)
=> ? = 0
[1,2,4,5,3,6,7] => [1,2,3,4,5,6,7] => [7] => ([],7)
=> ? = 0
[1,2,4,5,3,7,6] => [1,2,3,4,5,6,7] => [7] => ([],7)
=> ? = 0
[1,2,4,5,6,3,7] => [1,2,3,4,5,6,7] => [7] => ([],7)
=> ? = 0
[1,2,4,5,6,7,3] => [1,2,3,4,5,6,7] => [7] => ([],7)
=> ? = 0
[1,2,5,3,4,6,7] => [1,2,3,5,4,6,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 4
[1,2,5,3,4,7,6] => [1,2,3,5,4,6,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 4
[1,2,5,4,3,6,7] => [1,2,3,5,4,6,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 4
[1,2,5,4,3,7,6] => [1,2,3,5,4,6,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 4
[1,2,5,6,3,4,7] => [1,2,3,5,4,6,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 4
[1,2,5,6,3,7,4] => [1,2,3,5,4,6,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 4
[1,2,5,6,4,3,7] => [1,2,3,5,4,6,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 4
[1,2,5,6,4,7,3] => [1,2,3,5,4,6,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 4
[1,2,6,3,5,4,7] => [1,2,3,6,4,5,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 4
[1,2,6,3,7,4,5] => [1,2,3,6,4,5,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 4
[1,2,6,4,5,3,7] => [1,2,3,6,4,5,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 4
[1,2,6,4,7,3,5] => [1,2,3,6,4,5,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 4
[1,2,6,5,3,4,7] => [1,2,3,6,4,5,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 4
[1,2,6,5,4,3,7] => [1,2,3,6,4,5,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 4
[1,2,6,5,7,3,4] => [1,2,3,6,4,5,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 4
[1,2,6,5,7,4,3] => [1,2,3,6,4,5,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 4
[1,2,7,3,5,6,4] => [1,2,3,7,4,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 4
[1,2,7,3,6,5,4] => [1,2,3,7,4,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 4
[1,2,7,4,5,6,3] => [1,2,3,7,4,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 4
[1,2,7,4,6,5,3] => [1,2,3,7,4,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 4
[1,2,7,5,3,6,4] => [1,2,3,7,4,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 4
[1,2,7,5,4,6,3] => [1,2,3,7,4,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 4
[1,2,7,5,6,3,4] => [1,2,3,7,4,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 4
[1,2,7,5,6,4,3] => [1,2,3,7,4,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 4
[1,3,2,4,5,6,7] => [1,2,3,4,5,6,7] => [7] => ([],7)
=> ? = 0
[1,3,2,4,5,7,6] => [1,2,3,4,5,6,7] => [7] => ([],7)
=> ? = 0
[1,3,2,4,6,5,7] => [1,2,3,4,5,6,7] => [7] => ([],7)
=> ? = 0
[1,3,2,4,6,7,5] => [1,2,3,4,5,6,7] => [7] => ([],7)
=> ? = 0
[1,3,2,5,4,6,7] => [1,2,3,4,5,6,7] => [7] => ([],7)
=> ? = 0
[1,3,2,5,4,7,6] => [1,2,3,4,5,6,7] => [7] => ([],7)
=> ? = 0
[1,3,2,5,6,4,7] => [1,2,3,4,5,6,7] => [7] => ([],7)
=> ? = 0
[1,3,2,5,6,7,4] => [1,2,3,4,5,6,7] => [7] => ([],7)
=> ? = 0
[1,3,4,2,5,6,7] => [1,2,3,4,5,6,7] => [7] => ([],7)
=> ? = 0
[1,3,4,2,5,7,6] => [1,2,3,4,5,6,7] => [7] => ([],7)
=> ? = 0
Description
The maximum cut size of a graph.
A '''cut''' is a set of edges which connect different sides of a vertex partition V=A⊔B.
Matching statistic: St000456
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 67%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => ([],1)
=> ([],1)
=> ? = 0 - 3
[1,2] => [1,2] => ([],2)
=> ([],1)
=> ? = 0 - 3
[2,1] => [1,2] => ([],2)
=> ([],1)
=> ? = 0 - 3
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> ? = 0 - 3
[1,3,2] => [1,2,3] => ([],3)
=> ([],1)
=> ? = 0 - 3
[2,1,3] => [1,2,3] => ([],3)
=> ([],1)
=> ? = 0 - 3
[2,3,1] => [1,2,3] => ([],3)
=> ([],1)
=> ? = 0 - 3
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? = 0 - 3
[1,2,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? = 0 - 3
[1,3,2,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? = 0 - 3
[1,3,4,2] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? = 0 - 3
[2,1,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? = 0 - 3
[2,1,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? = 0 - 3
[2,3,1,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? = 0 - 3
[2,3,4,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? = 0 - 3
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 - 3
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 - 3
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 - 3
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 - 3
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 - 3
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 - 3
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 - 3
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 - 3
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 6 - 3
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 6 - 3
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 - 3
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 - 3
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 - 3
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 - 3
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1 = 4 - 3
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1 = 4 - 3
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 - 3
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 - 3
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 - 3
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 - 3
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1 = 4 - 3
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1 = 4 - 3
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 4 - 3
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 4 - 3
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 6 - 3
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 6 - 3
[3,1,4,5,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 4 - 3
[3,1,5,2,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 6 - 3
[3,2,4,5,1] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 4 - 3
[3,2,5,1,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 6 - 3
[4,1,2,5,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 6 - 3
[4,2,1,5,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 6 - 3
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[1,2,5,3,4,6] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,2,5,4,3,6] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,2,5,6,3,4] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,2,5,6,4,3] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,2,6,3,5,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,2,6,4,5,3] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,2,6,5,3,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,2,6,5,4,3] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[1,3,5,2,4,6] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,3,5,4,2,6] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,3,5,6,2,4] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,3,5,6,4,2] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,3,6,2,5,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,3,6,4,5,2] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,3,6,5,2,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,3,6,5,4,2] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,4,2,5,3,6] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,4,2,6,5,3] => [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,4,3,5,2,6] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,4,3,6,5,2] => [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,4,5,6,2,3] => [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,4,5,6,3,2] => [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,4,6,5,2,3] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,4,6,5,3,2] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,5,2,3,4,6] => [1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 6 - 3
[1,5,2,4,6,3] => [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,5,3,2,4,6] => [1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 6 - 3
[1,5,3,4,6,2] => [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
Description
The monochromatic index of a connected graph.
This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path.
For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St001118
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001118: Graphs ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 67%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001118: Graphs ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => ([],1)
=> ([],1)
=> ? = 0 - 3
[1,2] => [1,2] => ([],2)
=> ([],1)
=> ? = 0 - 3
[2,1] => [1,2] => ([],2)
=> ([],1)
=> ? = 0 - 3
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> ? = 0 - 3
[1,3,2] => [1,2,3] => ([],3)
=> ([],1)
=> ? = 0 - 3
[2,1,3] => [1,2,3] => ([],3)
=> ([],1)
=> ? = 0 - 3
[2,3,1] => [1,2,3] => ([],3)
=> ([],1)
=> ? = 0 - 3
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? = 0 - 3
[1,2,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? = 0 - 3
[1,3,2,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? = 0 - 3
[1,3,4,2] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? = 0 - 3
[2,1,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? = 0 - 3
[2,1,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? = 0 - 3
[2,3,1,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? = 0 - 3
[2,3,4,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? = 0 - 3
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 - 3
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 - 3
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 - 3
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 - 3
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 - 3
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 - 3
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 - 3
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 - 3
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 6 - 3
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 6 - 3
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 - 3
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 - 3
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 - 3
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 - 3
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1 = 4 - 3
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1 = 4 - 3
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 - 3
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 - 3
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 - 3
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 - 3
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1 = 4 - 3
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1 = 4 - 3
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 4 - 3
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 4 - 3
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 6 - 3
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 6 - 3
[3,1,4,5,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 4 - 3
[3,1,5,2,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 6 - 3
[3,2,4,5,1] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 4 - 3
[3,2,5,1,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 6 - 3
[4,1,2,5,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 6 - 3
[4,2,1,5,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 6 - 3
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[1,2,5,3,4,6] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,2,5,4,3,6] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,2,5,6,3,4] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,2,5,6,4,3] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,2,6,3,5,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,2,6,4,5,3] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,2,6,5,3,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,2,6,5,4,3] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[1,3,5,2,4,6] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,3,5,4,2,6] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,3,5,6,2,4] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,3,5,6,4,2] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,3,6,2,5,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,3,6,4,5,2] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,3,6,5,2,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,3,6,5,4,2] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,4,2,5,3,6] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,4,2,6,5,3] => [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,4,3,5,2,6] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,4,3,6,5,2] => [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,4,5,6,2,3] => [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,4,5,6,3,2] => [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,4,6,5,2,3] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,4,6,5,3,2] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,5,2,3,4,6] => [1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 6 - 3
[1,5,2,4,6,3] => [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[1,5,3,2,4,6] => [1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 6 - 3
[1,5,3,4,6,2] => [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1 = 4 - 3
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 0 - 3
Description
The acyclic chromatic index of a graph.
An acyclic edge coloring of a graph is a proper colouring of the edges of a graph such that the union of the edges colored with any two given colours is a forest.
The smallest number of colours such that such a colouring exists is the acyclic chromatic index.
Matching statistic: St000632
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000632: Posets ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 33%
Mp00209: Permutations —pattern poset⟶ Posets
St000632: Posets ⟶ ℤResult quality: 16% ●values known / values provided: 16%●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
[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
[1,4,2,5,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 4
[1,4,3,5,2] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 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
[2,4,1,5,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 4
[2,4,3,5,1] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 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
[3,1,4,5,2] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 4
[3,1,5,2,4] => [1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 6
[3,2,4,5,1] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 4
[3,2,5,1,4] => [1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 6
[4,1,2,5,3] => [1,4,5,3,2] => ([(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)
=> ? = 6
[4,2,1,5,3] => [1,4,5,3,2] => ([(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)
=> ? = 6
[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
[1,2,5,3,4,6] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,2,5,4,3,6] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,2,5,6,3,4] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,2,5,6,4,3] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 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
[1,3,5,2,4,6] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,3,5,4,2,6] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,3,5,6,2,4] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,3,5,6,4,2] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 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
[1,4,2,5,3,6] => [1,2,4,5,3,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 4
[1,4,2,6,5,3] => [1,2,4,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ? = 4
[1,4,3,5,2,6] => [1,2,4,5,3,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 4
[1,4,3,6,5,2] => [1,2,4,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ? = 4
[1,4,5,6,2,3] => [1,2,4,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ? = 4
[1,4,5,6,3,2] => [1,2,4,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ? = 4
[1,4,6,5,2,3] => [1,2,4,5,3,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 4
[1,4,6,5,3,2] => [1,2,4,5,3,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 4
[1,5,2,3,4,6] => [1,2,5,4,3,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? = 6
[1,5,2,4,6,3] => [1,2,5,6,3,4] => ([(0,2),(0,3),(0,4),(1,8),(1,9),(2,1),(2,10),(2,11),(3,6),(3,7),(3,11),(4,6),(4,7),(4,10),(6,14),(7,12),(7,14),(8,13),(8,15),(9,13),(9,15),(10,8),(10,12),(10,14),(11,9),(11,12),(11,14),(12,13),(12,15),(13,5),(14,15),(15,5)],16)
=> ? = 4
[1,5,3,2,4,6] => [1,2,5,4,3,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? = 6
[1,5,3,4,6,2] => [1,2,5,6,3,4] => ([(0,2),(0,3),(0,4),(1,8),(1,9),(2,1),(2,10),(2,11),(3,6),(3,7),(3,11),(4,6),(4,7),(4,10),(6,14),(7,12),(7,14),(8,13),(8,15),(9,13),(9,15),(10,8),(10,12),(10,14),(11,9),(11,12),(11,14),(12,13),(12,15),(13,5),(14,15),(15,5)],16)
=> ? = 4
[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 jump number of the poset.
A jump in a linear extension e1,…,en of a poset P is a pair (ei,ei+1) so that ei+1 does not cover ei 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 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001633: Posets ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 33%
Mp00209: Permutations —pattern poset⟶ Posets
St001633: Posets ⟶ ℤResult quality: 16% ●values known / values provided: 16%●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
[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
[1,4,2,5,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 4
[1,4,3,5,2] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 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
[2,4,1,5,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 4
[2,4,3,5,1] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 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
[3,1,4,5,2] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 4
[3,1,5,2,4] => [1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 6
[3,2,4,5,1] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 4
[3,2,5,1,4] => [1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 6
[4,1,2,5,3] => [1,4,5,3,2] => ([(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)
=> ? = 6
[4,2,1,5,3] => [1,4,5,3,2] => ([(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)
=> ? = 6
[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
[1,2,5,3,4,6] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,2,5,4,3,6] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,2,5,6,3,4] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,2,5,6,4,3] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 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
[1,3,5,2,4,6] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,3,5,4,2,6] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,3,5,6,2,4] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,3,5,6,4,2] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 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
[1,4,2,5,3,6] => [1,2,4,5,3,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 4
[1,4,2,6,5,3] => [1,2,4,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ? = 4
[1,4,3,5,2,6] => [1,2,4,5,3,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 4
[1,4,3,6,5,2] => [1,2,4,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ? = 4
[1,4,5,6,2,3] => [1,2,4,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ? = 4
[1,4,5,6,3,2] => [1,2,4,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ? = 4
[1,4,6,5,2,3] => [1,2,4,5,3,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 4
[1,4,6,5,3,2] => [1,2,4,5,3,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 4
[1,5,2,3,4,6] => [1,2,5,4,3,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? = 6
[1,5,2,4,6,3] => [1,2,5,6,3,4] => ([(0,2),(0,3),(0,4),(1,8),(1,9),(2,1),(2,10),(2,11),(3,6),(3,7),(3,11),(4,6),(4,7),(4,10),(6,14),(7,12),(7,14),(8,13),(8,15),(9,13),(9,15),(10,8),(10,12),(10,14),(11,9),(11,12),(11,14),(12,13),(12,15),(13,5),(14,15),(15,5)],16)
=> ? = 4
[1,5,3,2,4,6] => [1,2,5,4,3,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? = 6
[1,5,3,4,6,2] => [1,2,5,6,3,4] => ([(0,2),(0,3),(0,4),(1,8),(1,9),(2,1),(2,10),(2,11),(3,6),(3,7),(3,11),(4,6),(4,7),(4,10),(6,14),(7,12),(7,14),(8,13),(8,15),(9,13),(9,15),(10,8),(10,12),(10,14),(11,9),(11,12),(11,14),(12,13),(12,15),(13,5),(14,15),(15,5)],16)
=> ? = 4
[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: St000298
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 33%
Mp00209: Permutations —pattern poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 33%
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
[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
[1,4,2,5,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 4 + 1
[1,4,3,5,2] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 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
[2,4,1,5,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 4 + 1
[2,4,3,5,1] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 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
[3,1,4,5,2] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 4 + 1
[3,1,5,2,4] => [1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 6 + 1
[3,2,4,5,1] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 4 + 1
[3,2,5,1,4] => [1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 6 + 1
[4,1,2,5,3] => [1,4,5,3,2] => ([(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)
=> ? = 6 + 1
[4,2,1,5,3] => [1,4,5,3,2] => ([(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)
=> ? = 6 + 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
[1,2,5,3,4,6] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4 + 1
[1,2,5,4,3,6] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4 + 1
[1,2,5,6,3,4] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4 + 1
[1,2,5,6,4,3] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 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
[1,3,5,2,4,6] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4 + 1
[1,3,5,4,2,6] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4 + 1
[1,3,5,6,2,4] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4 + 1
[1,3,5,6,4,2] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 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
[1,4,2,5,3,6] => [1,2,4,5,3,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 4 + 1
[1,4,2,6,5,3] => [1,2,4,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ? = 4 + 1
[1,4,3,5,2,6] => [1,2,4,5,3,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 4 + 1
[1,4,3,6,5,2] => [1,2,4,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ? = 4 + 1
[1,4,5,6,2,3] => [1,2,4,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ? = 4 + 1
[1,4,5,6,3,2] => [1,2,4,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ? = 4 + 1
[1,4,6,5,2,3] => [1,2,4,5,3,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 4 + 1
[1,4,6,5,3,2] => [1,2,4,5,3,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 4 + 1
[1,5,2,3,4,6] => [1,2,5,4,3,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? = 6 + 1
[1,5,2,4,6,3] => [1,2,5,6,3,4] => ([(0,2),(0,3),(0,4),(1,8),(1,9),(2,1),(2,10),(2,11),(3,6),(3,7),(3,11),(4,6),(4,7),(4,10),(6,14),(7,12),(7,14),(8,13),(8,15),(9,13),(9,15),(10,8),(10,12),(10,14),(11,9),(11,12),(11,14),(12,13),(12,15),(13,5),(14,15),(15,5)],16)
=> ? = 4 + 1
[1,5,3,2,4,6] => [1,2,5,4,3,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? = 6 + 1
[1,5,3,4,6,2] => [1,2,5,6,3,4] => ([(0,2),(0,3),(0,4),(1,8),(1,9),(2,1),(2,10),(2,11),(3,6),(3,7),(3,11),(4,6),(4,7),(4,10),(6,14),(7,12),(7,14),(8,13),(8,15),(9,13),(9,15),(10,8),(10,12),(10,14),(11,9),(11,12),(11,14),(12,13),(12,15),(13,5),(14,15),(15,5)],16)
=> ? = 4 + 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 order dimension or Dushnik-Miller dimension of a poset.
This is the minimal number of linear orderings whose intersection is the given poset.
Matching statistic: St000307
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 33%
Mp00209: Permutations —pattern poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 33%
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
[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
[1,4,2,5,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 4 + 1
[1,4,3,5,2] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 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
[2,4,1,5,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 4 + 1
[2,4,3,5,1] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 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
[3,1,4,5,2] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 4 + 1
[3,1,5,2,4] => [1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 6 + 1
[3,2,4,5,1] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 4 + 1
[3,2,5,1,4] => [1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 6 + 1
[4,1,2,5,3] => [1,4,5,3,2] => ([(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)
=> ? = 6 + 1
[4,2,1,5,3] => [1,4,5,3,2] => ([(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)
=> ? = 6 + 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
[1,2,5,3,4,6] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4 + 1
[1,2,5,4,3,6] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4 + 1
[1,2,5,6,3,4] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4 + 1
[1,2,5,6,4,3] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 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
[1,3,5,2,4,6] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4 + 1
[1,3,5,4,2,6] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4 + 1
[1,3,5,6,2,4] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4 + 1
[1,3,5,6,4,2] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 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
[1,4,2,5,3,6] => [1,2,4,5,3,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 4 + 1
[1,4,2,6,5,3] => [1,2,4,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ? = 4 + 1
[1,4,3,5,2,6] => [1,2,4,5,3,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 4 + 1
[1,4,3,6,5,2] => [1,2,4,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ? = 4 + 1
[1,4,5,6,2,3] => [1,2,4,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ? = 4 + 1
[1,4,5,6,3,2] => [1,2,4,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ? = 4 + 1
[1,4,6,5,2,3] => [1,2,4,5,3,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 4 + 1
[1,4,6,5,3,2] => [1,2,4,5,3,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 4 + 1
[1,5,2,3,4,6] => [1,2,5,4,3,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? = 6 + 1
[1,5,2,4,6,3] => [1,2,5,6,3,4] => ([(0,2),(0,3),(0,4),(1,8),(1,9),(2,1),(2,10),(2,11),(3,6),(3,7),(3,11),(4,6),(4,7),(4,10),(6,14),(7,12),(7,14),(8,13),(8,15),(9,13),(9,15),(10,8),(10,12),(10,14),(11,9),(11,12),(11,14),(12,13),(12,15),(13,5),(14,15),(15,5)],16)
=> ? = 4 + 1
[1,5,3,2,4,6] => [1,2,5,4,3,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? = 6 + 1
[1,5,3,4,6,2] => [1,2,5,6,3,4] => ([(0,2),(0,3),(0,4),(1,8),(1,9),(2,1),(2,10),(2,11),(3,6),(3,7),(3,11),(4,6),(4,7),(4,10),(6,14),(7,12),(7,14),(8,13),(8,15),(9,13),(9,15),(10,8),(10,12),(10,14),(11,9),(11,12),(11,14),(12,13),(12,15),(13,5),(14,15),(15,5)],16)
=> ? = 4 + 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∖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: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 33%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 33%
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
[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
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(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
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(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
[3,1,4,5,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[3,1,5,2,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 6
[3,2,4,5,1] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 4
[3,2,5,1,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 6
[4,1,2,5,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 6
[4,2,1,5,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 6
[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
[1,2,5,3,4,6] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,2,5,4,3,6] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,2,5,6,3,4] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,2,5,6,4,3] => [1,2,3,5,4,6] => ([(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
[1,3,5,2,4,6] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,3,5,4,2,6] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,3,5,6,2,4] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,3,5,6,4,2] => [1,2,3,5,4,6] => ([(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
[1,4,2,5,3,6] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,4,2,6,5,3] => [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 4
[1,4,3,5,2,6] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,4,3,6,5,2] => [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 4
[1,4,5,6,2,3] => [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 4
[1,4,5,6,3,2] => [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 4
[1,4,6,5,2,3] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,4,6,5,3,2] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,5,2,3,4,6] => [1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 6
[1,5,2,4,6,3] => [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 4
[1,5,3,2,4,6] => [1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 6
[1,5,3,4,6,2] => [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> ? = 4
[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.
The following 74 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St001632The number of indecomposable injective modules I with dimExt1(I,A)=1 for the incidence algebra A of a poset. St001060The distinguishing index of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. 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. St001964The interval resolution global dimension 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. St001428The number of B-inversions of a signed permutation. St001434The number of negative sum pairs of a signed permutation. St000068The number of minimal elements in 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. St001845The number of join irreducibles minus the rank of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000181The number of connected components of the Hasse diagram for the poset. St001490The number of connected components of a skew partition. St001769The reflection length of a signed permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001848The atomic length of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001862The number of crossings of a signed permutation. St001864The number of excedances of a signed permutation. St001866The nesting alignments of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001892The flag excedance statistic of a signed permutation. St001893The flag descent of a signed permutation. St001894The depth of a signed permutation. St001895The oddness of a signed permutation. St001896The number of right descents of a signed permutations. St001770The number of facets of a certain subword complex associated with the signed permutation. St001851The number of Hecke atoms of a signed permutation. St001852The size of the conjugacy class of the signed permutation. St001855The number of signed permutations less than or equal to a signed permutation in left weak order. St000635The number of strictly order preserving maps of a poset into itself. St001890The maximum magnitude of the Möbius function of a poset. St000753The Grundy value for the game of Kayles on a binary word. St001889The size of the connectivity set of a signed permutation. St000896The number of zeros on the main diagonal of an alternating sign matrix.
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