Your data matches 1 statistic following compositions of up to 3 maps.
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Matching statistic: St000456
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1
[1,1,0,1,0,0]
 => [2,1,3] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 4
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [3,1,4,5,2] => ([(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] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 6
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 3
[1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 6
[1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,3,1] => [5,4,3,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [5,6,4,2,3,1] => [4,5,3,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [6,4,5,2,3,1] => [5,3,4,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [5,4,6,2,3,1] => [4,3,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => 4
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [4,5,6,2,3,1] => [3,4,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 3
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,4,1] => [5,4,2,1,6,3] => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [5,6,3,2,4,1] => [4,5,2,1,6,3] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 4
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [6,4,3,2,5,1] => [5,3,2,1,6,4] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [6,3,4,2,5,1] => [5,2,3,1,6,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => 3
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [6,5,2,3,4,1] => [5,4,6,2,1,3] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 7
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [5,6,2,3,4,1] => [4,5,6,2,1,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [6,4,2,3,5,1] => [5,3,6,2,1,4] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 6
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [5,4,2,3,6,1] => [4,3,6,2,1,5] => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [4,5,2,3,6,1] => [3,4,6,2,1,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [6,3,2,4,5,1] => [5,2,6,3,1,4] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [5,3,2,4,6,1] => [4,2,6,3,1,5] => ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 4
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [4,3,2,5,6,1] => [3,2,6,4,1,5] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,6,1] => [2,3,6,4,1,5] => ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => 2
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,5,1] => [5,6,2,3,1,4] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => 6
[1,0,1,1,1,1,0,0,0,1,0,0]
 => [5,2,3,4,6,1] => [4,6,2,3,1,5] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
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.