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Your data matches 5 different statistics following compositions of up to 3 maps.
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Matching statistic: St000007
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 1
[1,0,1,0]
=> [2,1] => [1,2] => [2,1] => 2
[1,1,0,0]
=> [1,2] => [2,1] => [1,2] => 1
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => [2,3,1] => 2
[1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => [3,1,2] => 2
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => [3,2,1] => 3
[1,1,0,1,0,0]
=> [2,1,3] => [3,2,1] => [2,1,3] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [2,3,1] => [1,2,3] => 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => [2,3,4,1] => 2
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,1,2,3] => [3,4,1,2] => 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,4,2,3] => [3,4,2,1] => 3
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,3,1,2] => [4,2,1,3] => 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,4,1,2] => [4,1,2,3] => 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,4,3] => [2,4,3,1] => 3
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,1,3,2] => [4,3,1,2] => 3
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,4,3,2] => [4,3,2,1] => 4
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [4,3,2,1] => [3,2,1,4] => 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,4,2,1] => [3,1,2,4] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,3,4,2] => [4,2,3,1] => 3
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [4,2,3,1] => [2,3,1,4] => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [3,2,4,1] => [2,1,3,4] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => [2,3,4,5,1] => 2
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [5,1,2,3,4] => [3,4,5,1,2] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,5,2,3,4] => [3,4,5,2,1] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,4,1,2,3] => [4,5,2,1,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [4,5,1,2,3] => [4,5,1,2,3] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,5,3,4] => [2,4,5,3,1] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,1,4,2,3] => [4,5,3,1,2] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,5,4,2,3] => [4,5,3,2,1] => 4
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,4,3,1,2] => [5,3,2,1,4] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [4,5,3,1,2] => [5,3,1,2,4] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,4,5,2,3] => [4,5,2,3,1] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,3,4,1,2] => [5,2,3,1,4] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [4,3,5,1,2] => [5,2,1,3,4] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [3,4,5,1,2] => [5,1,2,3,4] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,5,4] => [2,3,5,4,1] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [5,1,2,4,3] => [3,5,4,1,2] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,5,2,4,3] => [3,5,4,2,1] => 4
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [5,4,1,3,2] => [5,4,2,1,3] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [4,5,1,3,2] => [5,4,1,2,3] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,2,5,4,3] => [2,5,4,3,1] => 4
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [5,1,4,3,2] => [5,4,3,1,2] => 4
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,5,4,3,2] => [5,4,3,2,1] => 5
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [5,4,3,2,1] => [4,3,2,1,5] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,5,3,2,1] => [4,3,1,2,5] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,5,3,2] => [5,4,2,3,1] => 4
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [5,3,4,2,1] => [4,2,3,1,5] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [4,3,5,2,1] => [4,2,1,3,5] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [3,4,5,2,1] => [4,1,2,3,5] => 1
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000456
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 22%●distinct values known / distinct values provided: 9%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 22%●distinct values known / distinct values provided: 9%
Values
[1,0]
=> [1] => ([],1)
=> ([],1)
=> ? = 1
[1,0,1,0]
=> [2,1] => ([],2)
=> ([],2)
=> ? = 2
[1,1,0,0]
=> [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,0,1,0,1,0]
=> [3,2,1] => ([],3)
=> ([],3)
=> ? = 2
[1,0,1,1,0,0]
=> [2,3,1] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 2
[1,1,0,0,1,0]
=> [3,1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 3
[1,1,0,1,0,0]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => ([],4)
=> ([],4)
=> ? = 2
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? = 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? = 3
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? = 3
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 3
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 4
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 3
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => ([],5)
=> ([],5)
=> ? = 2
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> ? = 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> ? = 3
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> ? = 3
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 4
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 3
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => ([(3,4)],5)
=> ([(3,4)],5)
=> ? = 3
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 3
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 4
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ? = 3
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ? = 3
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 4
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ? = 4
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 5
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 4
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 3
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ? = 3
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 4
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 3
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 3
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => ([],6)
=> ([],6)
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => ([(4,5)],6)
=> ([(4,5)],6)
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => ([(4,5)],6)
=> ([(4,5)],6)
=> ? = 3
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => ([(3,4),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => ([(4,5)],6)
=> ([(4,5)],6)
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => ([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ? = 4
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,6] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [4,5,3,2,1,6] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [5,3,4,2,1,6] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,1,6] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,1,6] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [5,4,2,3,1,6] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [4,5,2,3,1,6] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [5,3,2,4,1,6] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,1,6] => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,1,6] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,1,6] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,1,6] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,1,6] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,1,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [5,4,3,1,2,6] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [4,5,3,1,2,6] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [5,3,4,1,2,6] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [4,3,5,1,2,6] => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,4,5,1,2,6] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [5,4,2,1,3,6] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,5,2,1,3,6] => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,3,2,1,4,6] => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1,5,6] => ([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,2,1,5,6] => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,3,1,4,6] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [4,2,3,1,5,6] => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,2,4,1,5,6] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,3,4,1,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1
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: St001545
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001545: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 20%●distinct values known / distinct values provided: 9%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001545: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 20%●distinct values known / distinct values provided: 9%
Values
[1,0]
=> [1] => ([],1)
=> ([],1)
=> ? = 1 + 1
[1,0,1,0]
=> [1,2] => ([],2)
=> ([],1)
=> ? = 2 + 1
[1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> ([],1)
=> ? = 2 + 1
[1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,1,0,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 3 + 1
[1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> ([],1)
=> ? = 2 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 3 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 3 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 3 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 4 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 3 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 4 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 3 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 4 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 4 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 5 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 4 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 3 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 3 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 4 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 3 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 3 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 2 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 3 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 4 + 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,1,5,6,2,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [3,5,1,2,6,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,6,1,2,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [4,1,5,6,2,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [4,1,6,2,3,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,5,1,2,6,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [4,5,1,6,2,3] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
Description
The second Elser number of a connected graph.
For a connected graph $G$ the $k$-th Elser number is
$$
els_k(G) = (-1)^{|V(G)|+1} \sum_N (-1)^{|E(N)|} |V(N)|^k
$$
where the sum is over all nuclei of $G$, that is, the connected subgraphs of $G$ whose vertex set is a vertex cover of $G$.
It is clear that this number is even. It was shown in [1] that it is non-negative.
Matching statistic: St001198
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 9% ●values known / values provided: 15%●distinct values known / distinct values provided: 9%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 9% ●values known / values provided: 15%●distinct values known / distinct values provided: 9%
Values
[1,0]
=> [1] => [1] => [1,0]
=> ? = 1 + 1
[1,0,1,0]
=> [2,1] => [2,1] => [1,1,0,0]
=> ? = 2 + 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 2 + 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 2 + 1
[1,1,0,0,1,0]
=> [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 3 + 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 4 + 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 4 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 4 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 4 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 5 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 4 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 4 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 4 + 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [4,5,3,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [5,3,4,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [5,4,2,3,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [4,5,2,3,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [5,3,2,4,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,1,6] => [5,3,2,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,1,6] => [5,3,2,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,1,6] => [5,3,2,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,1,6] => [5,2,3,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,1,6] => [5,2,3,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [5,4,3,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [4,5,3,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [5,3,4,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [4,3,5,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,4,5,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [5,4,2,1,3,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,5,2,1,3,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,3,2,1,4,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1,5,6] => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,2,1,5,6] => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,3,1,4,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [4,2,3,1,5,6] => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,2,4,1,5,6] => [4,2,3,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,3,4,1,5,6] => [4,2,3,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2 = 1 + 1
Description
The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001206
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 9% ●values known / values provided: 15%●distinct values known / distinct values provided: 9%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 9% ●values known / values provided: 15%●distinct values known / distinct values provided: 9%
Values
[1,0]
=> [1] => [1] => [1,0]
=> ? = 1 + 1
[1,0,1,0]
=> [2,1] => [2,1] => [1,1,0,0]
=> ? = 2 + 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 2 + 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 2 + 1
[1,1,0,0,1,0]
=> [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 3 + 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 4 + 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 4 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 4 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 4 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 5 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 4 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 4 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 4 + 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [4,5,3,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [5,3,4,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [5,4,2,3,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [4,5,2,3,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [5,3,2,4,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,1,6] => [5,3,2,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,1,6] => [5,3,2,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,1,6] => [5,3,2,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,1,6] => [5,2,3,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,1,6] => [5,2,3,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [5,4,3,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [4,5,3,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [5,3,4,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [4,3,5,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,4,5,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [5,4,2,1,3,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,5,2,1,3,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,3,2,1,4,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1,5,6] => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,2,1,5,6] => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,3,1,4,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [4,2,3,1,5,6] => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,2,4,1,5,6] => [4,2,3,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,3,4,1,5,6] => [4,2,3,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2 = 1 + 1
Description
The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.
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