Your data matches 2 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001085
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St001085: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => [1] => 0
[1,0,1,0]
 => [1,2] => [1,2] => [1,2] => 0
[1,1,0,0]
 => [2,1] => [2,1] => [2,1] => 0
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
 => [1,3,2] => [3,1,2] => [2,3,1] => 0
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
 => [2,3,1] => [1,3,2] => [1,3,2] => 0
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => [3,2,1] => 0
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [4,1,2,3] => [2,3,4,1] => 0
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [3,1,2,4] => [2,3,1,4] => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,4,1,3] => [3,4,1,2] => 0
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [4,3,1,2] => [2,4,3,1] => 0
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [1,4,2,3] => [1,3,4,2] => 0
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 0
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 0
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [4,1,3,2] => [3,2,4,1] => 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 1
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [1,4,3,2] => [1,4,3,2] => 0
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [5,1,2,3,4] => [2,3,4,5,1] => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [4,1,2,3,5] => [2,3,4,1,5] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [3,5,1,2,4] => [2,4,5,1,3] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [5,4,1,2,3] => [2,3,5,4,1] => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [3,1,2,4,5] => [2,3,1,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,5,1,3,4] => [3,4,5,1,2] => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [2,4,1,3,5] => [3,4,1,2,5] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,3,5,1,4] => [4,5,1,2,3] => 0
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [5,2,4,1,3] => [3,4,1,5,2] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [4,3,1,2,5] => [2,4,3,1,5] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [3,2,5,1,4] => [4,5,1,3,2] => 0
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [2,5,4,1,3] => [3,5,4,1,2] => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [5,4,3,1,2] => [2,5,4,3,1] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [1,5,2,3,4] => [1,3,4,5,2] => 0
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [1,3,5,2,4] => [1,4,5,2,3] => 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [5,1,4,2,3] => [3,4,2,5,1] => 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [1,2,5,3,4] => [1,2,4,5,3] => 0
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [5,1,2,4,3] => [2,4,3,5,1] => 0
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [4,1,3,2,5] => [3,2,4,1,5] => 2
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [3,1,2,5,4] => [2,3,1,5,4] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [2,5,1,4,3] => [4,3,5,1,2] => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [5,4,1,3,2] => [3,2,5,4,1] => 1
Description
The number of occurrences of the vincular pattern |21-3 in a permutation. This is the number of occurrences of the pattern $213$, where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive. In other words, this is the number of ascents whose bottom value is strictly smaller and the top value is strictly larger than the first entry of the permutation.
Matching statistic: St001330
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 50%
Values
[1,0]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 2 = 0 + 2
[1,0,1,0]
 => [3,1,2] => [2,3,1] => ([(0,2),(1,2)],3)
 => 2 = 0 + 2
[1,1,0,0]
 => [2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
 => 2 = 0 + 2
[1,0,1,0,1,0]
 => [4,1,2,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[1,0,1,1,0,0]
 => [3,1,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[1,1,0,0,1,0]
 => [2,4,1,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 3 = 1 + 2
[1,1,0,1,0,0]
 => [4,3,1,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[1,1,1,0,0,0]
 => [2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? = 1 + 2
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ? = 1 + 2
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? = 0 + 2
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ? = 1 + 2
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ? = 1 + 2
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 0 + 2
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [2,3,6,4,5,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [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)
 => ? = 1 + 2
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [2,4,5,3,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [2,6,3,4,1,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [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)
 => ? = 1 + 2
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [6,4,5,2,3,1] => ([(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)
 => ? = 0 + 2
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [3,4,2,5,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [3,4,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [6,3,4,2,5,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [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)
 => ? = 1 + 2
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [4,6,5,2,3,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [3,5,2,4,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [6,2,3,1,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 1 + 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [3,6,4,5,1,2] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 0 + 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 0 + 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [4,5,3,6,1,2] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 0 + 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [6,3,4,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [2,4,5,6,3,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [2,6,4,5,3,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [2,3,5,6,4,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [2,6,3,5,4,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [5,6,2,4,3,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 2 + 2
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [4,5,2,6,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [3,5,2,6,4,1] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [6,2,4,3,1,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ? = 1 + 2
[1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => [6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => [3,5,6,4,1,2] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 1 + 2
[1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => [3,4,6,5,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 1 + 2
[1,1,1,0,0,1,1,0,0,0]
 => [2,5,4,1,6,3] => [6,3,5,4,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,1,1,0,1,0,0,1,0,0]
 => [6,3,5,1,2,4] => [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 0 + 2
[1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3] => [2,3,6,5,4,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[1,1,1,0,1,1,0,0,0,0]
 => [5,3,4,1,6,2] => [6,2,4,1,5,3] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 1 + 2
[1,1,1,1,0,0,0,1,0,0]
 => [2,3,6,5,1,4] => [4,6,5,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,1,1,1,0,0,1,0,0,0]
 => [2,6,4,5,1,3] => [3,5,1,2,6,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[1,1,1,0,1,0,0,0,1,0,1,0]
 => [7,3,4,1,2,5,6] => [2,4,1,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => 2 = 0 + 2
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => [7,1,2,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [8,1,2,3,4,5,6,7] => [2,3,4,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 2 = 0 + 2
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [6,7,8,1,2,3,4,5] => [2,3,4,5,8,1,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 2 = 0 + 2
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => [7,8,4,5,1,2,3,6] => [2,3,5,1,6,8,4,7] => ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
 => 2 = 0 + 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => [8,1,2,3,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 2 = 0 + 2
Description
The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.