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Your data matches 8 different statistics following compositions of up to 3 maps.
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Matching statistic: St001880
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Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[3,2,1] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[4,2,3,1] => [4,2,3,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[5,2,3,4,1] => [5,2,4,3,1] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[5,2,4,3,1] => [5,2,4,3,1] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[5,3,2,4,1] => [5,3,2,4,1] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[5,3,4,2,1] => [5,3,4,2,1] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[5,4,2,3,1] => [5,4,2,3,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[6,2,3,4,5,1] => [6,2,5,4,3,1] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 4
[6,2,3,5,4,1] => [6,2,5,4,3,1] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 4
[6,2,4,3,5,1] => [6,2,5,4,3,1] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 4
[6,2,4,5,3,1] => [6,2,5,4,3,1] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 4
[6,2,5,3,4,1] => [6,2,5,4,3,1] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 4
[6,2,5,4,3,1] => [6,2,5,4,3,1] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 4
[6,3,2,4,5,1] => [6,3,2,5,4,1] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 4
[6,3,2,5,4,1] => [6,3,2,5,4,1] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 4
[6,3,4,2,5,1] => [6,3,5,2,4,1] => [1,4,2,5,3,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[6,3,4,5,2,1] => [6,3,5,4,2,1] => [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 5
[6,3,5,2,4,1] => [6,3,5,2,4,1] => [1,4,2,5,3,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[6,3,5,4,2,1] => [6,3,5,4,2,1] => [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 5
[6,4,2,3,5,1] => [6,4,2,5,3,1] => [1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[6,4,2,5,3,1] => [6,4,2,5,3,1] => [1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[6,4,3,2,5,1] => [6,4,3,2,5,1] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 4
[6,4,3,5,2,1] => [6,4,3,5,2,1] => [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 5
[6,4,5,3,2,1] => [6,4,5,3,2,1] => [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
[6,5,2,3,4,1] => [6,5,2,4,3,1] => [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 5
[6,5,2,4,3,1] => [6,5,2,4,3,1] => [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 5
[6,5,3,2,4,1] => [6,5,3,2,4,1] => [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 5
[6,5,3,4,2,1] => [6,5,3,4,2,1] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6
[6,5,4,2,3,1] => [6,5,4,2,3,1] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
[6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[7,2,3,4,5,6,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 4
[7,2,3,4,6,5,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 4
[7,2,3,5,4,6,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 4
[7,2,3,5,6,4,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 4
[7,2,3,6,4,5,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 4
[7,2,3,6,5,4,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 4
[7,2,4,3,5,6,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 4
[7,2,4,3,6,5,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 4
[7,2,4,5,3,6,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 4
[7,2,4,5,6,3,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 4
[7,2,4,6,3,5,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 4
[7,2,4,6,5,3,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 4
[7,2,5,3,4,6,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 4
[7,2,5,3,6,4,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 4
[7,2,5,4,3,6,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 4
[7,2,5,4,6,3,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 4
[7,2,5,6,3,4,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 4
[7,2,5,6,4,3,1] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 4
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001004
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Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St001004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St001004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[3,2,1] => [3,2,1] => [3,1,2] => [1,2] => 2 = 3 - 1
[4,2,3,1] => [4,2,3,1] => [4,3,1,2] => [3,1,2] => 3 = 4 - 1
[4,3,2,1] => [4,3,2,1] => [4,1,2,3] => [1,2,3] => 3 = 4 - 1
[5,2,3,4,1] => [5,2,4,3,1] => [5,4,1,3,2] => [4,1,3,2] => 3 = 4 - 1
[5,2,4,3,1] => [5,2,4,3,1] => [5,4,1,3,2] => [4,1,3,2] => 3 = 4 - 1
[5,3,2,4,1] => [5,3,2,4,1] => [5,4,2,1,3] => [4,2,1,3] => 3 = 4 - 1
[5,3,4,2,1] => [5,3,4,2,1] => [5,1,4,2,3] => [1,4,2,3] => 4 = 5 - 1
[5,4,2,3,1] => [5,4,2,3,1] => [5,3,1,2,4] => [3,1,2,4] => 4 = 5 - 1
[5,4,3,2,1] => [5,4,3,2,1] => [5,1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[6,2,3,4,5,1] => [6,2,5,4,3,1] => [6,5,1,3,4,2] => [5,1,3,4,2] => 3 = 4 - 1
[6,2,3,5,4,1] => [6,2,5,4,3,1] => [6,5,1,3,4,2] => [5,1,3,4,2] => 3 = 4 - 1
[6,2,4,3,5,1] => [6,2,5,4,3,1] => [6,5,1,3,4,2] => [5,1,3,4,2] => 3 = 4 - 1
[6,2,4,5,3,1] => [6,2,5,4,3,1] => [6,5,1,3,4,2] => [5,1,3,4,2] => 3 = 4 - 1
[6,2,5,3,4,1] => [6,2,5,4,3,1] => [6,5,1,3,4,2] => [5,1,3,4,2] => 3 = 4 - 1
[6,2,5,4,3,1] => [6,2,5,4,3,1] => [6,5,1,3,4,2] => [5,1,3,4,2] => 3 = 4 - 1
[6,3,2,4,5,1] => [6,3,2,5,4,1] => [6,5,2,1,4,3] => [5,2,1,4,3] => 3 = 4 - 1
[6,3,2,5,4,1] => [6,3,2,5,4,1] => [6,5,2,1,4,3] => [5,2,1,4,3] => 3 = 4 - 1
[6,3,4,2,5,1] => [6,3,5,2,4,1] => [6,4,5,1,2,3] => [4,5,1,2,3] => 5 = 6 - 1
[6,3,4,5,2,1] => [6,3,5,4,2,1] => [6,1,5,2,4,3] => [1,5,2,4,3] => 4 = 5 - 1
[6,3,5,2,4,1] => [6,3,5,2,4,1] => [6,4,5,1,2,3] => [4,5,1,2,3] => 5 = 6 - 1
[6,3,5,4,2,1] => [6,3,5,4,2,1] => [6,1,5,2,4,3] => [1,5,2,4,3] => 4 = 5 - 1
[6,4,2,3,5,1] => [6,4,2,5,3,1] => [6,5,1,2,3,4] => [5,1,2,3,4] => 5 = 6 - 1
[6,4,2,5,3,1] => [6,4,2,5,3,1] => [6,5,1,2,3,4] => [5,1,2,3,4] => 5 = 6 - 1
[6,4,3,2,5,1] => [6,4,3,2,5,1] => [6,5,2,3,1,4] => [5,2,3,1,4] => 3 = 4 - 1
[6,4,3,5,2,1] => [6,4,3,5,2,1] => [6,1,5,3,2,4] => [1,5,3,2,4] => 4 = 5 - 1
[6,4,5,3,2,1] => [6,4,5,3,2,1] => [6,1,2,5,3,4] => [1,2,5,3,4] => 5 = 6 - 1
[6,5,2,3,4,1] => [6,5,2,4,3,1] => [6,4,1,3,2,5] => [4,1,3,2,5] => 4 = 5 - 1
[6,5,2,4,3,1] => [6,5,2,4,3,1] => [6,4,1,3,2,5] => [4,1,3,2,5] => 4 = 5 - 1
[6,5,3,2,4,1] => [6,5,3,2,4,1] => [6,4,2,1,3,5] => [4,2,1,3,5] => 4 = 5 - 1
[6,5,3,4,2,1] => [6,5,3,4,2,1] => [6,1,4,2,3,5] => [1,4,2,3,5] => 5 = 6 - 1
[6,5,4,2,3,1] => [6,5,4,2,3,1] => [6,3,1,2,4,5] => [3,1,2,4,5] => 5 = 6 - 1
[6,5,4,3,2,1] => [6,5,4,3,2,1] => [6,1,2,3,4,5] => [1,2,3,4,5] => 5 = 6 - 1
[7,2,3,4,5,6,1] => [7,2,6,5,4,3,1] => [7,6,1,3,4,5,2] => [6,1,3,4,5,2] => 3 = 4 - 1
[7,2,3,4,6,5,1] => [7,2,6,5,4,3,1] => [7,6,1,3,4,5,2] => [6,1,3,4,5,2] => 3 = 4 - 1
[7,2,3,5,4,6,1] => [7,2,6,5,4,3,1] => [7,6,1,3,4,5,2] => [6,1,3,4,5,2] => 3 = 4 - 1
[7,2,3,5,6,4,1] => [7,2,6,5,4,3,1] => [7,6,1,3,4,5,2] => [6,1,3,4,5,2] => 3 = 4 - 1
[7,2,3,6,4,5,1] => [7,2,6,5,4,3,1] => [7,6,1,3,4,5,2] => [6,1,3,4,5,2] => 3 = 4 - 1
[7,2,3,6,5,4,1] => [7,2,6,5,4,3,1] => [7,6,1,3,4,5,2] => [6,1,3,4,5,2] => 3 = 4 - 1
[7,2,4,3,5,6,1] => [7,2,6,5,4,3,1] => [7,6,1,3,4,5,2] => [6,1,3,4,5,2] => 3 = 4 - 1
[7,2,4,3,6,5,1] => [7,2,6,5,4,3,1] => [7,6,1,3,4,5,2] => [6,1,3,4,5,2] => 3 = 4 - 1
[7,2,4,5,3,6,1] => [7,2,6,5,4,3,1] => [7,6,1,3,4,5,2] => [6,1,3,4,5,2] => 3 = 4 - 1
[7,2,4,5,6,3,1] => [7,2,6,5,4,3,1] => [7,6,1,3,4,5,2] => [6,1,3,4,5,2] => 3 = 4 - 1
[7,2,4,6,3,5,1] => [7,2,6,5,4,3,1] => [7,6,1,3,4,5,2] => [6,1,3,4,5,2] => 3 = 4 - 1
[7,2,4,6,5,3,1] => [7,2,6,5,4,3,1] => [7,6,1,3,4,5,2] => [6,1,3,4,5,2] => 3 = 4 - 1
[7,2,5,3,4,6,1] => [7,2,6,5,4,3,1] => [7,6,1,3,4,5,2] => [6,1,3,4,5,2] => 3 = 4 - 1
[7,2,5,3,6,4,1] => [7,2,6,5,4,3,1] => [7,6,1,3,4,5,2] => [6,1,3,4,5,2] => 3 = 4 - 1
[7,2,5,4,3,6,1] => [7,2,6,5,4,3,1] => [7,6,1,3,4,5,2] => [6,1,3,4,5,2] => 3 = 4 - 1
[7,2,5,4,6,3,1] => [7,2,6,5,4,3,1] => [7,6,1,3,4,5,2] => [6,1,3,4,5,2] => 3 = 4 - 1
[7,2,5,6,3,4,1] => [7,2,6,5,4,3,1] => [7,6,1,3,4,5,2] => [6,1,3,4,5,2] => 3 = 4 - 1
[7,2,5,6,4,3,1] => [7,2,6,5,4,3,1] => [7,6,1,3,4,5,2] => [6,1,3,4,5,2] => 3 = 4 - 1
Description
The number of indices that are either left-to-right maxima or right-to-left minima.
The (bivariate) generating function for this statistic is (essentially) given in [1], the mid points of a $321$ pattern in the permutation are those elements which are neither left-to-right maxima nor a right-to-left minima, see [[St000371]] and [[St000372]].
Matching statistic: St001005
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St001005: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 80%
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St001005: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 80%
Values
[3,2,1] => [3,2,1] => [3,2,1] => [3,1,2] => 3
[4,2,3,1] => [4,2,3,1] => [3,1,4,2] => [3,4,1,2] => 4
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [4,1,2,3] => 4
[5,2,3,4,1] => [5,2,4,3,1] => [4,3,1,5,2] => [4,5,1,3,2] => 4
[5,2,4,3,1] => [5,2,4,3,1] => [4,3,1,5,2] => [4,5,1,3,2] => 4
[5,3,2,4,1] => [5,3,2,4,1] => [4,1,5,3,2] => [4,5,2,1,3] => 4
[5,3,4,2,1] => [5,3,4,2,1] => [4,2,1,5,3] => [4,1,5,2,3] => 5
[5,4,2,3,1] => [5,4,2,3,1] => [3,1,5,4,2] => [3,5,1,2,4] => 5
[5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => [5,1,2,3,4] => 5
[6,2,3,4,5,1] => [6,2,5,4,3,1] => [5,4,3,1,6,2] => [5,6,1,3,4,2] => 4
[6,2,3,5,4,1] => [6,2,5,4,3,1] => [5,4,3,1,6,2] => [5,6,1,3,4,2] => 4
[6,2,4,3,5,1] => [6,2,5,4,3,1] => [5,4,3,1,6,2] => [5,6,1,3,4,2] => 4
[6,2,4,5,3,1] => [6,2,5,4,3,1] => [5,4,3,1,6,2] => [5,6,1,3,4,2] => 4
[6,2,5,3,4,1] => [6,2,5,4,3,1] => [5,4,3,1,6,2] => [5,6,1,3,4,2] => 4
[6,2,5,4,3,1] => [6,2,5,4,3,1] => [5,4,3,1,6,2] => [5,6,1,3,4,2] => 4
[6,3,2,4,5,1] => [6,3,2,5,4,1] => [5,4,1,6,3,2] => [5,6,2,1,4,3] => 4
[6,3,2,5,4,1] => [6,3,2,5,4,1] => [5,4,1,6,3,2] => [5,6,2,1,4,3] => 4
[6,3,4,2,5,1] => [6,3,5,2,4,1] => [4,1,5,2,6,3] => [4,5,6,1,2,3] => 6
[6,3,4,5,2,1] => [6,3,5,4,2,1] => [5,4,2,1,6,3] => [5,1,6,2,4,3] => 5
[6,3,5,2,4,1] => [6,3,5,2,4,1] => [4,1,5,2,6,3] => [4,5,6,1,2,3] => 6
[6,3,5,4,2,1] => [6,3,5,4,2,1] => [5,4,2,1,6,3] => [5,1,6,2,4,3] => 5
[6,4,2,3,5,1] => [6,4,2,5,3,1] => [5,3,1,6,4,2] => [5,6,1,2,3,4] => 6
[6,4,2,5,3,1] => [6,4,2,5,3,1] => [5,3,1,6,4,2] => [5,6,1,2,3,4] => 6
[6,4,3,2,5,1] => [6,4,3,2,5,1] => [5,1,6,4,3,2] => [5,6,2,3,1,4] => 4
[6,4,3,5,2,1] => [6,4,3,5,2,1] => [5,2,1,6,4,3] => [5,1,6,3,2,4] => 5
[6,4,5,3,2,1] => [6,4,5,3,2,1] => [5,3,2,1,6,4] => [5,1,2,6,3,4] => 6
[6,5,2,3,4,1] => [6,5,2,4,3,1] => [4,3,1,6,5,2] => [4,6,1,3,2,5] => 5
[6,5,2,4,3,1] => [6,5,2,4,3,1] => [4,3,1,6,5,2] => [4,6,1,3,2,5] => 5
[6,5,3,2,4,1] => [6,5,3,2,4,1] => [4,1,6,5,3,2] => [4,6,2,1,3,5] => 5
[6,5,3,4,2,1] => [6,5,3,4,2,1] => [4,2,1,6,5,3] => [4,1,6,2,3,5] => 6
[6,5,4,2,3,1] => [6,5,4,2,3,1] => [3,1,6,5,4,2] => [3,6,1,2,4,5] => 6
[6,5,4,3,2,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [6,1,2,3,4,5] => 6
[7,2,3,4,5,6,1] => [7,2,6,5,4,3,1] => [6,5,4,3,1,7,2] => [6,7,1,3,4,5,2] => ? = 4
[7,2,3,4,6,5,1] => [7,2,6,5,4,3,1] => [6,5,4,3,1,7,2] => [6,7,1,3,4,5,2] => ? = 4
[7,2,3,5,4,6,1] => [7,2,6,5,4,3,1] => [6,5,4,3,1,7,2] => [6,7,1,3,4,5,2] => ? = 4
[7,2,3,5,6,4,1] => [7,2,6,5,4,3,1] => [6,5,4,3,1,7,2] => [6,7,1,3,4,5,2] => ? = 4
[7,2,3,6,4,5,1] => [7,2,6,5,4,3,1] => [6,5,4,3,1,7,2] => [6,7,1,3,4,5,2] => ? = 4
[7,2,3,6,5,4,1] => [7,2,6,5,4,3,1] => [6,5,4,3,1,7,2] => [6,7,1,3,4,5,2] => ? = 4
[7,2,4,3,5,6,1] => [7,2,6,5,4,3,1] => [6,5,4,3,1,7,2] => [6,7,1,3,4,5,2] => ? = 4
[7,2,4,3,6,5,1] => [7,2,6,5,4,3,1] => [6,5,4,3,1,7,2] => [6,7,1,3,4,5,2] => ? = 4
[7,2,4,5,3,6,1] => [7,2,6,5,4,3,1] => [6,5,4,3,1,7,2] => [6,7,1,3,4,5,2] => ? = 4
[7,2,4,5,6,3,1] => [7,2,6,5,4,3,1] => [6,5,4,3,1,7,2] => [6,7,1,3,4,5,2] => ? = 4
[7,2,4,6,3,5,1] => [7,2,6,5,4,3,1] => [6,5,4,3,1,7,2] => [6,7,1,3,4,5,2] => ? = 4
[7,2,4,6,5,3,1] => [7,2,6,5,4,3,1] => [6,5,4,3,1,7,2] => [6,7,1,3,4,5,2] => ? = 4
[7,2,5,3,4,6,1] => [7,2,6,5,4,3,1] => [6,5,4,3,1,7,2] => [6,7,1,3,4,5,2] => ? = 4
[7,2,5,3,6,4,1] => [7,2,6,5,4,3,1] => [6,5,4,3,1,7,2] => [6,7,1,3,4,5,2] => ? = 4
[7,2,5,4,3,6,1] => [7,2,6,5,4,3,1] => [6,5,4,3,1,7,2] => [6,7,1,3,4,5,2] => ? = 4
[7,2,5,4,6,3,1] => [7,2,6,5,4,3,1] => [6,5,4,3,1,7,2] => [6,7,1,3,4,5,2] => ? = 4
[7,2,5,6,3,4,1] => [7,2,6,5,4,3,1] => [6,5,4,3,1,7,2] => [6,7,1,3,4,5,2] => ? = 4
[7,2,5,6,4,3,1] => [7,2,6,5,4,3,1] => [6,5,4,3,1,7,2] => [6,7,1,3,4,5,2] => ? = 4
[7,2,6,3,4,5,1] => [7,2,6,5,4,3,1] => [6,5,4,3,1,7,2] => [6,7,1,3,4,5,2] => ? = 4
[7,2,6,3,5,4,1] => [7,2,6,5,4,3,1] => [6,5,4,3,1,7,2] => [6,7,1,3,4,5,2] => ? = 4
[7,2,6,4,3,5,1] => [7,2,6,5,4,3,1] => [6,5,4,3,1,7,2] => [6,7,1,3,4,5,2] => ? = 4
[7,2,6,4,5,3,1] => [7,2,6,5,4,3,1] => [6,5,4,3,1,7,2] => [6,7,1,3,4,5,2] => ? = 4
[7,2,6,5,3,4,1] => [7,2,6,5,4,3,1] => [6,5,4,3,1,7,2] => [6,7,1,3,4,5,2] => ? = 4
[7,2,6,5,4,3,1] => [7,2,6,5,4,3,1] => [6,5,4,3,1,7,2] => [6,7,1,3,4,5,2] => ? = 4
[7,3,2,4,5,6,1] => [7,3,2,6,5,4,1] => [6,5,4,1,7,3,2] => [6,7,2,1,4,5,3] => ? = 4
[7,3,2,4,6,5,1] => [7,3,2,6,5,4,1] => [6,5,4,1,7,3,2] => [6,7,2,1,4,5,3] => ? = 4
[7,3,2,5,4,6,1] => [7,3,2,6,5,4,1] => [6,5,4,1,7,3,2] => [6,7,2,1,4,5,3] => ? = 4
[7,3,2,5,6,4,1] => [7,3,2,6,5,4,1] => [6,5,4,1,7,3,2] => [6,7,2,1,4,5,3] => ? = 4
[7,3,2,6,4,5,1] => [7,3,2,6,5,4,1] => [6,5,4,1,7,3,2] => [6,7,2,1,4,5,3] => ? = 4
[7,3,2,6,5,4,1] => [7,3,2,6,5,4,1] => [6,5,4,1,7,3,2] => [6,7,2,1,4,5,3] => ? = 4
[7,3,4,2,5,6,1] => [7,3,6,2,5,4,1] => [5,4,1,6,2,7,3] => [5,6,7,1,4,2,3] => ? = 6
[7,3,4,2,6,5,1] => [7,3,6,2,5,4,1] => [5,4,1,6,2,7,3] => [5,6,7,1,4,2,3] => ? = 6
[7,3,4,5,2,6,1] => [7,3,6,5,2,4,1] => [4,1,6,5,2,7,3] => [4,6,7,1,2,5,3] => ? = 6
[7,3,4,5,6,2,1] => [7,3,6,5,4,2,1] => [6,5,4,2,1,7,3] => [6,1,7,2,4,5,3] => ? = 5
[7,3,4,6,2,5,1] => [7,3,6,5,2,4,1] => [4,1,6,5,2,7,3] => [4,6,7,1,2,5,3] => ? = 6
[7,3,4,6,5,2,1] => [7,3,6,5,4,2,1] => [6,5,4,2,1,7,3] => [6,1,7,2,4,5,3] => ? = 5
[7,3,5,2,4,6,1] => [7,3,6,2,5,4,1] => [5,4,1,6,2,7,3] => [5,6,7,1,4,2,3] => ? = 6
[7,3,5,2,6,4,1] => [7,3,6,2,5,4,1] => [5,4,1,6,2,7,3] => [5,6,7,1,4,2,3] => ? = 6
[7,3,5,4,2,6,1] => [7,3,6,5,2,4,1] => [4,1,6,5,2,7,3] => [4,6,7,1,2,5,3] => ? = 6
[7,3,5,4,6,2,1] => [7,3,6,5,4,2,1] => [6,5,4,2,1,7,3] => [6,1,7,2,4,5,3] => ? = 5
[7,3,5,6,2,4,1] => [7,3,6,5,2,4,1] => [4,1,6,5,2,7,3] => [4,6,7,1,2,5,3] => ? = 6
[7,3,5,6,4,2,1] => [7,3,6,5,4,2,1] => [6,5,4,2,1,7,3] => [6,1,7,2,4,5,3] => ? = 5
[7,3,6,2,4,5,1] => [7,3,6,2,5,4,1] => [5,4,1,6,2,7,3] => [5,6,7,1,4,2,3] => ? = 6
[7,3,6,2,5,4,1] => [7,3,6,2,5,4,1] => [5,4,1,6,2,7,3] => [5,6,7,1,4,2,3] => ? = 6
[7,3,6,4,2,5,1] => [7,3,6,5,2,4,1] => [4,1,6,5,2,7,3] => [4,6,7,1,2,5,3] => ? = 6
[7,3,6,4,5,2,1] => [7,3,6,5,4,2,1] => [6,5,4,2,1,7,3] => [6,1,7,2,4,5,3] => ? = 5
[7,3,6,5,2,4,1] => [7,3,6,5,2,4,1] => [4,1,6,5,2,7,3] => [4,6,7,1,2,5,3] => ? = 6
[7,3,6,5,4,2,1] => [7,3,6,5,4,2,1] => [6,5,4,2,1,7,3] => [6,1,7,2,4,5,3] => ? = 5
[7,4,2,3,5,6,1] => [7,4,2,6,5,3,1] => [6,5,3,1,7,4,2] => [6,7,1,2,3,5,4] => ? = 6
[7,4,2,3,6,5,1] => [7,4,2,6,5,3,1] => [6,5,3,1,7,4,2] => [6,7,1,2,3,5,4] => ? = 6
Description
The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both.
Matching statistic: St001029
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00156: Graphs —line graph⟶ Graphs
St001029: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00156: Graphs —line graph⟶ Graphs
St001029: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 100%
Values
[3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[4,2,3,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[4,3,2,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[5,2,3,4,1] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[5,2,4,3,1] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> 3 = 4 - 1
[5,3,2,4,1] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[5,3,4,2,1] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[5,4,2,3,1] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[5,4,3,2,1] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
[6,2,3,4,5,1] => [5,6,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 4 - 1
[6,2,3,5,4,1] => [4,5,1,6,3,2] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,6),(0,7),(1,4),(1,5),(2,3),(2,5),(2,7),(2,8),(3,4),(3,6),(3,8),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 4 - 1
[6,2,4,3,5,1] => [5,6,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 4 - 1
[6,2,4,5,3,1] => [3,5,6,1,4,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,7),(1,8),(2,3),(2,6),(2,8),(3,4),(3,6),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 4 - 1
[6,2,5,3,4,1] => [4,6,5,3,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 4 - 1
[6,2,5,4,3,1] => [3,4,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> ? = 4 - 1
[6,3,2,4,5,1] => [5,3,6,2,4,1] => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 4 - 1
[6,3,2,5,4,1] => [4,3,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> ? = 4 - 1
[6,3,4,2,5,1] => [5,4,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 6 - 1
[6,3,4,5,2,1] => [2,5,6,3,4,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,6),(2,8),(3,5),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 - 1
[6,3,5,2,4,1] => [4,5,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 6 - 1
[6,3,5,4,2,1] => [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,6),(1,2),(1,5),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[6,4,2,3,5,1] => [5,4,2,6,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 6 - 1
[6,4,2,5,3,1] => [3,5,4,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> ? = 6 - 1
[6,4,3,2,5,1] => [5,3,4,6,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 4 - 1
[6,4,3,5,2,1] => [2,5,4,6,3,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 - 1
[6,4,5,3,2,1] => [2,3,5,6,4,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 6 - 1
[6,5,2,3,4,1] => [4,5,2,3,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,6),(2,8),(3,5),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 - 1
[6,5,2,4,3,1] => [3,4,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,6),(1,2),(1,5),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[6,5,3,2,4,1] => [4,3,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 - 1
[6,5,3,4,2,1] => [2,4,5,3,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 6 - 1
[6,5,4,2,3,1] => [3,4,2,5,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 6 - 1
[6,5,4,3,2,1] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[7,2,3,4,5,6,1] => [6,7,2,3,4,5,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,3,4,6,5,1] => [5,6,1,3,7,4,2] => ([(0,4),(0,5),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,3,5,4,6,1] => [6,7,2,5,3,4,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,3,5,6,4,1] => [4,6,1,7,3,5,2] => ([(0,3),(0,5),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,3,6,4,5,1] => [5,7,2,6,4,3,1] => ([(0,2),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,3,6,5,4,1] => [4,5,2,6,7,3,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,4,3,5,6,1] => [6,7,4,2,3,5,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,4,3,6,5,1] => [5,6,4,1,7,3,2] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,4,5,3,6,1] => [6,7,5,2,4,3,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,4,5,6,3,1] => [3,6,7,1,4,5,2] => ([(0,3),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,4,6,3,5,1] => [5,7,6,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,4,6,5,3,1] => [3,5,6,2,7,4,1] => ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,5,3,4,6,1] => [6,7,5,3,2,4,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,5,3,6,4,1] => [4,6,7,5,3,1,2] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,5,4,3,6,1] => [6,7,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,5,4,6,3,1] => [3,6,7,5,1,4,2] => ([(0,4),(0,6),(1,2),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,5,6,3,4,1] => [4,7,6,3,2,5,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,5,6,4,3,1] => [3,4,6,7,2,5,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,6,3,4,5,1] => [5,7,6,3,4,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,6,3,5,4,1] => [4,5,7,6,1,3,2] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,6,4,3,5,1] => [5,7,4,6,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,6,4,5,3,1] => [3,5,7,6,4,1,2] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 4 - 1
[7,2,6,5,3,4,1] => [4,7,5,3,6,2,1] => ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,6,5,4,3,1] => [3,4,5,6,7,1,2] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,8),(1,9),(2,3),(2,4),(2,7),(2,9),(3,4),(3,6),(3,9),(4,5),(4,9),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],10)
=> ? = 4 - 1
[7,3,2,4,5,6,1] => [6,3,7,2,4,5,1] => ([(0,3),(0,5),(0,6),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,3,2,4,6,5,1] => [5,3,6,1,7,4,2] => ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,3,2,5,4,6,1] => [6,3,7,5,2,4,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,3,2,5,6,4,1] => [4,3,6,7,1,5,2] => ([(0,3),(0,4),(0,6),(1,2),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 4 - 1
[7,3,2,6,4,5,1] => [5,3,7,6,4,2,1] => ([(0,3),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,3,2,6,5,4,1] => [4,3,5,6,7,2,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,3,4,2,5,6,1] => [6,4,7,3,2,5,1] => ([(0,2),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 6 - 1
[7,6,5,4,3,2,1] => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
Description
The size of the core of a graph.
The core of the graph $G$ is the smallest graph $C$ such that there is a graph homomorphism from $G$ to $C$ and a graph homomorphism from $C$ to $G$.
Matching statistic: St001108
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00156: Graphs —line graph⟶ Graphs
St001108: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00156: Graphs —line graph⟶ Graphs
St001108: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 100%
Values
[3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[4,2,3,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[4,3,2,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[5,2,3,4,1] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[5,2,4,3,1] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> 3 = 4 - 1
[5,3,2,4,1] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[5,3,4,2,1] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[5,4,2,3,1] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[5,4,3,2,1] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
[6,2,3,4,5,1] => [5,6,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 4 - 1
[6,2,3,5,4,1] => [4,5,1,6,3,2] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,6),(0,7),(1,4),(1,5),(2,3),(2,5),(2,7),(2,8),(3,4),(3,6),(3,8),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 4 - 1
[6,2,4,3,5,1] => [5,6,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 4 - 1
[6,2,4,5,3,1] => [3,5,6,1,4,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,7),(1,8),(2,3),(2,6),(2,8),(3,4),(3,6),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 4 - 1
[6,2,5,3,4,1] => [4,6,5,3,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 4 - 1
[6,2,5,4,3,1] => [3,4,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> ? = 4 - 1
[6,3,2,4,5,1] => [5,3,6,2,4,1] => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 4 - 1
[6,3,2,5,4,1] => [4,3,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> ? = 4 - 1
[6,3,4,2,5,1] => [5,4,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 6 - 1
[6,3,4,5,2,1] => [2,5,6,3,4,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,6),(2,8),(3,5),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 - 1
[6,3,5,2,4,1] => [4,5,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 6 - 1
[6,3,5,4,2,1] => [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,6),(1,2),(1,5),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[6,4,2,3,5,1] => [5,4,2,6,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 6 - 1
[6,4,2,5,3,1] => [3,5,4,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> ? = 6 - 1
[6,4,3,2,5,1] => [5,3,4,6,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 4 - 1
[6,4,3,5,2,1] => [2,5,4,6,3,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 - 1
[6,4,5,3,2,1] => [2,3,5,6,4,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 6 - 1
[6,5,2,3,4,1] => [4,5,2,3,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,6),(2,8),(3,5),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 - 1
[6,5,2,4,3,1] => [3,4,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,6),(1,2),(1,5),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[6,5,3,2,4,1] => [4,3,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 - 1
[6,5,3,4,2,1] => [2,4,5,3,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 6 - 1
[6,5,4,2,3,1] => [3,4,2,5,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 6 - 1
[6,5,4,3,2,1] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[7,2,3,4,5,6,1] => [6,7,2,3,4,5,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,3,4,6,5,1] => [5,6,1,3,7,4,2] => ([(0,4),(0,5),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,3,5,4,6,1] => [6,7,2,5,3,4,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,3,5,6,4,1] => [4,6,1,7,3,5,2] => ([(0,3),(0,5),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,3,6,4,5,1] => [5,7,2,6,4,3,1] => ([(0,2),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,3,6,5,4,1] => [4,5,2,6,7,3,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,4,3,5,6,1] => [6,7,4,2,3,5,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,4,3,6,5,1] => [5,6,4,1,7,3,2] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,4,5,3,6,1] => [6,7,5,2,4,3,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,4,5,6,3,1] => [3,6,7,1,4,5,2] => ([(0,3),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,4,6,3,5,1] => [5,7,6,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,4,6,5,3,1] => [3,5,6,2,7,4,1] => ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,5,3,4,6,1] => [6,7,5,3,2,4,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,5,3,6,4,1] => [4,6,7,5,3,1,2] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,5,4,3,6,1] => [6,7,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,5,4,6,3,1] => [3,6,7,5,1,4,2] => ([(0,4),(0,6),(1,2),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,5,6,3,4,1] => [4,7,6,3,2,5,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,5,6,4,3,1] => [3,4,6,7,2,5,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,6,3,4,5,1] => [5,7,6,3,4,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,6,3,5,4,1] => [4,5,7,6,1,3,2] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,6,4,3,5,1] => [5,7,4,6,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,6,4,5,3,1] => [3,5,7,6,4,1,2] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 4 - 1
[7,2,6,5,3,4,1] => [4,7,5,3,6,2,1] => ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,6,5,4,3,1] => [3,4,5,6,7,1,2] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,8),(1,9),(2,3),(2,4),(2,7),(2,9),(3,4),(3,6),(3,9),(4,5),(4,9),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],10)
=> ? = 4 - 1
[7,3,2,4,5,6,1] => [6,3,7,2,4,5,1] => ([(0,3),(0,5),(0,6),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,3,2,4,6,5,1] => [5,3,6,1,7,4,2] => ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,3,2,5,4,6,1] => [6,3,7,5,2,4,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,3,2,5,6,4,1] => [4,3,6,7,1,5,2] => ([(0,3),(0,4),(0,6),(1,2),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 4 - 1
[7,3,2,6,4,5,1] => [5,3,7,6,4,2,1] => ([(0,3),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,3,2,6,5,4,1] => [4,3,5,6,7,2,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,3,4,2,5,6,1] => [6,4,7,3,2,5,1] => ([(0,2),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 6 - 1
[7,6,5,4,3,2,1] => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
Description
The 2-dynamic chromatic number of a graph.
A $k$-dynamic coloring of a graph $G$ is a proper coloring of $G$ in such a way that each vertex $v$ sees at least $\min\{d(v), k\}$ colors in its neighborhood. The $k$-dynamic chromatic number of a graph is the smallest number of colors needed to find an $k$-dynamic coloring.
This statistic records the $2$-dynamic chromatic number of a graph.
Matching statistic: St001494
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00156: Graphs —line graph⟶ Graphs
St001494: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00156: Graphs —line graph⟶ Graphs
St001494: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 100%
Values
[3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[4,2,3,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[4,3,2,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[5,2,3,4,1] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[5,2,4,3,1] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> 3 = 4 - 1
[5,3,2,4,1] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[5,3,4,2,1] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[5,4,2,3,1] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[5,4,3,2,1] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
[6,2,3,4,5,1] => [5,6,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 4 - 1
[6,2,3,5,4,1] => [4,5,1,6,3,2] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,6),(0,7),(1,4),(1,5),(2,3),(2,5),(2,7),(2,8),(3,4),(3,6),(3,8),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 4 - 1
[6,2,4,3,5,1] => [5,6,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 4 - 1
[6,2,4,5,3,1] => [3,5,6,1,4,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,7),(1,8),(2,3),(2,6),(2,8),(3,4),(3,6),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 4 - 1
[6,2,5,3,4,1] => [4,6,5,3,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 4 - 1
[6,2,5,4,3,1] => [3,4,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> ? = 4 - 1
[6,3,2,4,5,1] => [5,3,6,2,4,1] => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 4 - 1
[6,3,2,5,4,1] => [4,3,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> ? = 4 - 1
[6,3,4,2,5,1] => [5,4,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 6 - 1
[6,3,4,5,2,1] => [2,5,6,3,4,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,6),(2,8),(3,5),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 - 1
[6,3,5,2,4,1] => [4,5,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 6 - 1
[6,3,5,4,2,1] => [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,6),(1,2),(1,5),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[6,4,2,3,5,1] => [5,4,2,6,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 6 - 1
[6,4,2,5,3,1] => [3,5,4,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> ? = 6 - 1
[6,4,3,2,5,1] => [5,3,4,6,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 4 - 1
[6,4,3,5,2,1] => [2,5,4,6,3,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 - 1
[6,4,5,3,2,1] => [2,3,5,6,4,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 6 - 1
[6,5,2,3,4,1] => [4,5,2,3,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,6),(2,8),(3,5),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 - 1
[6,5,2,4,3,1] => [3,4,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,6),(1,2),(1,5),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[6,5,3,2,4,1] => [4,3,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 - 1
[6,5,3,4,2,1] => [2,4,5,3,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 6 - 1
[6,5,4,2,3,1] => [3,4,2,5,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 6 - 1
[6,5,4,3,2,1] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[7,2,3,4,5,6,1] => [6,7,2,3,4,5,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,3,4,6,5,1] => [5,6,1,3,7,4,2] => ([(0,4),(0,5),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,3,5,4,6,1] => [6,7,2,5,3,4,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,3,5,6,4,1] => [4,6,1,7,3,5,2] => ([(0,3),(0,5),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,3,6,4,5,1] => [5,7,2,6,4,3,1] => ([(0,2),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,3,6,5,4,1] => [4,5,2,6,7,3,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,4,3,5,6,1] => [6,7,4,2,3,5,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,4,3,6,5,1] => [5,6,4,1,7,3,2] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,4,5,3,6,1] => [6,7,5,2,4,3,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,4,5,6,3,1] => [3,6,7,1,4,5,2] => ([(0,3),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,4,6,3,5,1] => [5,7,6,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,4,6,5,3,1] => [3,5,6,2,7,4,1] => ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,5,3,4,6,1] => [6,7,5,3,2,4,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,5,3,6,4,1] => [4,6,7,5,3,1,2] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,5,4,3,6,1] => [6,7,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,5,4,6,3,1] => [3,6,7,5,1,4,2] => ([(0,4),(0,6),(1,2),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,5,6,3,4,1] => [4,7,6,3,2,5,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,5,6,4,3,1] => [3,4,6,7,2,5,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,6,3,4,5,1] => [5,7,6,3,4,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,6,3,5,4,1] => [4,5,7,6,1,3,2] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,6,4,3,5,1] => [5,7,4,6,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,6,4,5,3,1] => [3,5,7,6,4,1,2] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 4 - 1
[7,2,6,5,3,4,1] => [4,7,5,3,6,2,1] => ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,6,5,4,3,1] => [3,4,5,6,7,1,2] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,8),(1,9),(2,3),(2,4),(2,7),(2,9),(3,4),(3,6),(3,9),(4,5),(4,9),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],10)
=> ? = 4 - 1
[7,3,2,4,5,6,1] => [6,3,7,2,4,5,1] => ([(0,3),(0,5),(0,6),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,3,2,4,6,5,1] => [5,3,6,1,7,4,2] => ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,3,2,5,4,6,1] => [6,3,7,5,2,4,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,3,2,5,6,4,1] => [4,3,6,7,1,5,2] => ([(0,3),(0,4),(0,6),(1,2),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 4 - 1
[7,3,2,6,4,5,1] => [5,3,7,6,4,2,1] => ([(0,3),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,3,2,6,5,4,1] => [4,3,5,6,7,2,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,3,4,2,5,6,1] => [6,4,7,3,2,5,1] => ([(0,2),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 6 - 1
[7,6,5,4,3,2,1] => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
Description
The Alon-Tarsi number of a graph.
Let $G$ be a graph with vertices $\{1,\dots,n\}$ and edge set $E$. Let $P_G=\prod_{i < j, (i,j)\in E} x_i-x_j$ be its graph polynomial. Then the Alon-Tarsi number is the smallest number $k$ such that $P_G$ contains a monomial with exponents strictly less than $k$.
Matching statistic: St001655
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00156: Graphs —line graph⟶ Graphs
St001655: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00156: Graphs —line graph⟶ Graphs
St001655: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 100%
Values
[3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[4,2,3,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[4,3,2,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[5,2,3,4,1] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[5,2,4,3,1] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> 3 = 4 - 1
[5,3,2,4,1] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[5,3,4,2,1] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[5,4,2,3,1] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[5,4,3,2,1] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
[6,2,3,4,5,1] => [5,6,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 4 - 1
[6,2,3,5,4,1] => [4,5,1,6,3,2] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,6),(0,7),(1,4),(1,5),(2,3),(2,5),(2,7),(2,8),(3,4),(3,6),(3,8),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 4 - 1
[6,2,4,3,5,1] => [5,6,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 4 - 1
[6,2,4,5,3,1] => [3,5,6,1,4,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,7),(1,8),(2,3),(2,6),(2,8),(3,4),(3,6),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 4 - 1
[6,2,5,3,4,1] => [4,6,5,3,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 4 - 1
[6,2,5,4,3,1] => [3,4,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> ? = 4 - 1
[6,3,2,4,5,1] => [5,3,6,2,4,1] => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 4 - 1
[6,3,2,5,4,1] => [4,3,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> ? = 4 - 1
[6,3,4,2,5,1] => [5,4,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 6 - 1
[6,3,4,5,2,1] => [2,5,6,3,4,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,6),(2,8),(3,5),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 - 1
[6,3,5,2,4,1] => [4,5,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 6 - 1
[6,3,5,4,2,1] => [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,6),(1,2),(1,5),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[6,4,2,3,5,1] => [5,4,2,6,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 6 - 1
[6,4,2,5,3,1] => [3,5,4,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> ? = 6 - 1
[6,4,3,2,5,1] => [5,3,4,6,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 4 - 1
[6,4,3,5,2,1] => [2,5,4,6,3,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 - 1
[6,4,5,3,2,1] => [2,3,5,6,4,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 6 - 1
[6,5,2,3,4,1] => [4,5,2,3,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,6),(2,8),(3,5),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 - 1
[6,5,2,4,3,1] => [3,4,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,6),(1,2),(1,5),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[6,5,3,2,4,1] => [4,3,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 - 1
[6,5,3,4,2,1] => [2,4,5,3,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 6 - 1
[6,5,4,2,3,1] => [3,4,2,5,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 6 - 1
[6,5,4,3,2,1] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[7,2,3,4,5,6,1] => [6,7,2,3,4,5,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,3,4,6,5,1] => [5,6,1,3,7,4,2] => ([(0,4),(0,5),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,3,5,4,6,1] => [6,7,2,5,3,4,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,3,5,6,4,1] => [4,6,1,7,3,5,2] => ([(0,3),(0,5),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,3,6,4,5,1] => [5,7,2,6,4,3,1] => ([(0,2),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,3,6,5,4,1] => [4,5,2,6,7,3,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,4,3,5,6,1] => [6,7,4,2,3,5,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,4,3,6,5,1] => [5,6,4,1,7,3,2] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,4,5,3,6,1] => [6,7,5,2,4,3,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,4,5,6,3,1] => [3,6,7,1,4,5,2] => ([(0,3),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,4,6,3,5,1] => [5,7,6,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,4,6,5,3,1] => [3,5,6,2,7,4,1] => ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,5,3,4,6,1] => [6,7,5,3,2,4,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,5,3,6,4,1] => [4,6,7,5,3,1,2] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,5,4,3,6,1] => [6,7,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,5,4,6,3,1] => [3,6,7,5,1,4,2] => ([(0,4),(0,6),(1,2),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,5,6,3,4,1] => [4,7,6,3,2,5,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,5,6,4,3,1] => [3,4,6,7,2,5,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,6,3,4,5,1] => [5,7,6,3,4,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,6,3,5,4,1] => [4,5,7,6,1,3,2] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,6,4,3,5,1] => [5,7,4,6,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,6,4,5,3,1] => [3,5,7,6,4,1,2] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 4 - 1
[7,2,6,5,3,4,1] => [4,7,5,3,6,2,1] => ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,6,5,4,3,1] => [3,4,5,6,7,1,2] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,8),(1,9),(2,3),(2,4),(2,7),(2,9),(3,4),(3,6),(3,9),(4,5),(4,9),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],10)
=> ? = 4 - 1
[7,3,2,4,5,6,1] => [6,3,7,2,4,5,1] => ([(0,3),(0,5),(0,6),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,3,2,4,6,5,1] => [5,3,6,1,7,4,2] => ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,3,2,5,4,6,1] => [6,3,7,5,2,4,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,3,2,5,6,4,1] => [4,3,6,7,1,5,2] => ([(0,3),(0,4),(0,6),(1,2),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 4 - 1
[7,3,2,6,4,5,1] => [5,3,7,6,4,2,1] => ([(0,3),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,3,2,6,5,4,1] => [4,3,5,6,7,2,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,3,4,2,5,6,1] => [6,4,7,3,2,5,1] => ([(0,2),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 6 - 1
[7,6,5,4,3,2,1] => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
Description
The general position number of a graph.
A set $S$ of vertices in a graph $G$ is a general position set if no three vertices of $S$ lie on a shortest path between any two of them.
Matching statistic: St001656
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00156: Graphs —line graph⟶ Graphs
St001656: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00156: Graphs —line graph⟶ Graphs
St001656: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 100%
Values
[3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[4,2,3,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[4,3,2,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[5,2,3,4,1] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[5,2,4,3,1] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> 3 = 4 - 1
[5,3,2,4,1] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[5,3,4,2,1] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[5,4,2,3,1] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[5,4,3,2,1] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
[6,2,3,4,5,1] => [5,6,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 4 - 1
[6,2,3,5,4,1] => [4,5,1,6,3,2] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,6),(0,7),(1,4),(1,5),(2,3),(2,5),(2,7),(2,8),(3,4),(3,6),(3,8),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 4 - 1
[6,2,4,3,5,1] => [5,6,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 4 - 1
[6,2,4,5,3,1] => [3,5,6,1,4,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,7),(1,8),(2,3),(2,6),(2,8),(3,4),(3,6),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 4 - 1
[6,2,5,3,4,1] => [4,6,5,3,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 4 - 1
[6,2,5,4,3,1] => [3,4,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> ? = 4 - 1
[6,3,2,4,5,1] => [5,3,6,2,4,1] => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 4 - 1
[6,3,2,5,4,1] => [4,3,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> ? = 4 - 1
[6,3,4,2,5,1] => [5,4,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 6 - 1
[6,3,4,5,2,1] => [2,5,6,3,4,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,6),(2,8),(3,5),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 - 1
[6,3,5,2,4,1] => [4,5,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 6 - 1
[6,3,5,4,2,1] => [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,6),(1,2),(1,5),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[6,4,2,3,5,1] => [5,4,2,6,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 6 - 1
[6,4,2,5,3,1] => [3,5,4,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> ? = 6 - 1
[6,4,3,2,5,1] => [5,3,4,6,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 4 - 1
[6,4,3,5,2,1] => [2,5,4,6,3,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 - 1
[6,4,5,3,2,1] => [2,3,5,6,4,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 6 - 1
[6,5,2,3,4,1] => [4,5,2,3,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,6),(2,8),(3,5),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 - 1
[6,5,2,4,3,1] => [3,4,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,6),(1,2),(1,5),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[6,5,3,2,4,1] => [4,3,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 - 1
[6,5,3,4,2,1] => [2,4,5,3,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 6 - 1
[6,5,4,2,3,1] => [3,4,2,5,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 6 - 1
[6,5,4,3,2,1] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[7,2,3,4,5,6,1] => [6,7,2,3,4,5,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,3,4,6,5,1] => [5,6,1,3,7,4,2] => ([(0,4),(0,5),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,3,5,4,6,1] => [6,7,2,5,3,4,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,3,5,6,4,1] => [4,6,1,7,3,5,2] => ([(0,3),(0,5),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,3,6,4,5,1] => [5,7,2,6,4,3,1] => ([(0,2),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,3,6,5,4,1] => [4,5,2,6,7,3,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,4,3,5,6,1] => [6,7,4,2,3,5,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,4,3,6,5,1] => [5,6,4,1,7,3,2] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,4,5,3,6,1] => [6,7,5,2,4,3,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,4,5,6,3,1] => [3,6,7,1,4,5,2] => ([(0,3),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,4,6,3,5,1] => [5,7,6,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,4,6,5,3,1] => [3,5,6,2,7,4,1] => ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,5,3,4,6,1] => [6,7,5,3,2,4,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,5,3,6,4,1] => [4,6,7,5,3,1,2] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,5,4,3,6,1] => [6,7,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,5,4,6,3,1] => [3,6,7,5,1,4,2] => ([(0,4),(0,6),(1,2),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,5,6,3,4,1] => [4,7,6,3,2,5,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,5,6,4,3,1] => [3,4,6,7,2,5,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,6,3,4,5,1] => [5,7,6,3,4,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,6,3,5,4,1] => [4,5,7,6,1,3,2] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,6,4,3,5,1] => [5,7,4,6,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,6,4,5,3,1] => [3,5,7,6,4,1,2] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 4 - 1
[7,2,6,5,3,4,1] => [4,7,5,3,6,2,1] => ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,2,6,5,4,3,1] => [3,4,5,6,7,1,2] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,8),(1,9),(2,3),(2,4),(2,7),(2,9),(3,4),(3,6),(3,9),(4,5),(4,9),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],10)
=> ? = 4 - 1
[7,3,2,4,5,6,1] => [6,3,7,2,4,5,1] => ([(0,3),(0,5),(0,6),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,3,2,4,6,5,1] => [5,3,6,1,7,4,2] => ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,3,2,5,4,6,1] => [6,3,7,5,2,4,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,3,2,5,6,4,1] => [4,3,6,7,1,5,2] => ([(0,3),(0,4),(0,6),(1,2),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 4 - 1
[7,3,2,6,4,5,1] => [5,3,7,6,4,2,1] => ([(0,3),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,3,2,6,5,4,1] => [4,3,5,6,7,2,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 1
[7,3,4,2,5,6,1] => [6,4,7,3,2,5,1] => ([(0,2),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 6 - 1
[7,6,5,4,3,2,1] => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
Description
The monophonic position number of a graph.
A subset $M$ of the vertex set of a graph is a monophonic position set if no three vertices of $M$ lie on a common induced path. The monophonic position number is the size of a largest monophonic position set.
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