Processing math: 100%

Your data matches 4 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000432
(load all 4 compositions to match this statistic)
Mapping
Mp00069: Permutations —complement⟶ Permutations
St000432: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => 0
[2,1] => [1,2] => 0
[1,2,3] => [3,2,1] => 0
[1,3,2] => [3,1,2] => 1
[2,1,3] => [2,3,1] => 1
[2,3,1] => [2,1,3] => 0
[3,1,2] => [1,3,2] => 0
[3,2,1] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [4,3,1,2] => 2
[1,3,2,4] => [4,2,3,1] => 2
[1,3,4,2] => [4,2,1,3] => 2
[1,4,2,3] => [4,1,3,2] => 2
[1,4,3,2] => [4,1,2,3] => 3
[2,1,3,4] => [3,4,2,1] => 2
[2,1,4,3] => [3,4,1,2] => 4
[2,3,1,4] => [3,2,4,1] => 2
[2,3,4,1] => [3,2,1,4] => 0
[2,4,1,3] => [3,1,4,2] => 2
[2,4,3,1] => [3,1,2,4] => 1
[3,1,2,4] => [2,4,3,1] => 2
[3,1,4,2] => [2,4,1,3] => 2
[3,2,1,4] => [2,3,4,1] => 3
[3,2,4,1] => [2,3,1,4] => 1
[3,4,1,2] => [2,1,4,3] => 0
[3,4,2,1] => [2,1,3,4] => 0
[4,1,2,3] => [1,4,3,2] => 0
[4,1,3,2] => [1,4,2,3] => 1
[4,2,1,3] => [1,3,4,2] => 1
[4,2,3,1] => [1,3,2,4] => 0
[4,3,1,2] => [1,2,4,3] => 0
[4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [5,4,3,1,2] => 3
[1,2,4,3,5] => [5,4,2,3,1] => 3
[1,2,4,5,3] => [5,4,2,1,3] => 4
[1,2,5,3,4] => [5,4,1,3,2] => 4
[1,2,5,4,3] => [5,4,1,2,3] => 6
[1,3,2,4,5] => [5,3,4,2,1] => 3
[1,3,2,5,4] => [5,3,4,1,2] => 6
[1,3,4,2,5] => [5,3,2,4,1] => 4
[1,3,4,5,2] => [5,3,2,1,4] => 3
[1,3,5,2,4] => [5,3,1,4,2] => 5
[1,3,5,4,2] => [5,3,1,2,4] => 5
[1,4,2,3,5] => [5,2,4,3,1] => 4
[1,4,2,5,3] => [5,2,4,1,3] => 5
[1,4,3,2,5] => [5,2,3,4,1] => 6
[1,4,3,5,2] => [5,2,3,1,4] => 5
[1,4,5,2,3] => [5,2,1,4,3] => 4
[1,4,5,3,2] => [5,2,1,3,4] => 5
Description
The number of occurrences of the pattern 231 or of the pattern 312 in a permutation.
Matching statistic: St001308
(load all 2 compositions to match this statistic)
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001308: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => ([(0,1)],2)
 => 0
[2,1] => [1,2] => ([],2)
 => 0
[1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
[1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
 => 1
[2,1,3] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[2,3,1] => [1,3,2] => ([(1,2)],3)
 => 0
[3,1,2] => [2,1,3] => ([(1,2)],3)
 => 0
[3,2,1] => [1,2,3] => ([],3)
 => 0
[1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,4,3,2] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[2,1,3,4] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,1,4,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 4
[2,3,1,4] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,3,4,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 0
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 2
[2,4,3,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 1
[3,1,2,4] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 2
[3,2,1,4] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 3
[3,2,4,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => 1
[3,4,1,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 0
[3,4,2,1] => [1,2,4,3] => ([(2,3)],4)
 => 0
[4,1,2,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 0
[4,1,3,2] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 1
[4,2,1,3] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 1
[4,2,3,1] => [1,3,2,4] => ([(2,3)],4)
 => 0
[4,3,1,2] => [2,1,3,4] => ([(2,3)],4)
 => 0
[4,3,2,1] => [1,2,3,4] => ([],4)
 => 0
[1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,2,3,5,4] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,2,4,3,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,2,4,5,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,2,5,3,4] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,2,5,4,3] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 6
[1,3,2,4,5] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,3,2,5,4] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 6
[1,3,4,2,5] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,3,4,5,2] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,3,5,4,2] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,2,5,3] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,4,3,2,5] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 6
[1,4,3,5,2] => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,4,5,2,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 4
[1,4,5,3,2] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
Description
The number of induced paths on three vertices in a graph.
Matching statistic: St000425
(load all 4 compositions to match this statistic)
Mapping
St000425: Permutations ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 2
[1,3,2,4] => 2
[1,3,4,2] => 2
[1,4,2,3] => 2
[1,4,3,2] => 3
[2,1,3,4] => 2
[2,1,4,3] => 4
[2,3,1,4] => 2
[2,3,4,1] => 0
[2,4,1,3] => 2
[2,4,3,1] => 1
[3,1,2,4] => 2
[3,1,4,2] => 2
[3,2,1,4] => 3
[3,2,4,1] => 1
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 0
[1,2,3,5,4] => 3
[1,2,4,3,5] => 3
[1,2,4,5,3] => 4
[1,2,5,3,4] => 4
[1,2,5,4,3] => 6
[1,3,2,4,5] => 3
[1,3,2,5,4] => 6
[1,3,4,2,5] => 4
[1,3,4,5,2] => 3
[1,3,5,2,4] => 5
[1,3,5,4,2] => 5
[1,4,2,3,5] => 4
[1,4,2,5,3] => 5
[1,4,3,2,5] => 6
[1,4,3,5,2] => 5
[1,4,5,2,3] => 4
[1,4,5,3,2] => 5
[7,4,2,3,5,1,6] => ? = 8
[7,4,2,3,5,6,1] => ? = 4
[7,4,2,3,6,1,5] => ? = 10
[7,4,2,3,6,5,1] => ? = 7
[7,4,2,5,1,3,6] => ? = 9
[7,4,2,5,1,6,3] => ? = 9
[7,4,2,5,3,1,6] => ? = 9
[7,4,2,5,3,6,1] => ? = 5
[7,4,2,5,6,1,3] => ? = 5
[7,4,2,5,6,3,1] => ? = 4
[7,4,2,6,1,3,5] => ? = 9
[7,4,2,6,1,5,3] => ? = 10
[7,4,2,6,3,1,5] => ? = 9
[7,4,2,6,3,5,1] => ? = 6
[7,4,2,6,5,1,3] => ? = 7
[7,4,2,6,5,3,1] => ? = 6
[7,4,3,1,2,5,6] => ? = 10
[7,4,3,1,2,6,5] => ? = 14
[7,4,3,1,5,2,6] => ? = 10
[7,4,3,1,5,6,2] => ? = 8
[7,4,3,1,6,2,5] => ? = 12
[7,4,3,1,6,5,2] => ? = 11
[7,4,3,2,1,5,6] => ? = 12
[7,4,3,2,1,6,5] => ? = 16
[7,4,3,2,5,1,6] => ? = 10
[7,4,3,2,5,6,1] => ? = 6
[7,4,3,2,6,1,5] => ? = 12
[7,4,3,2,6,5,1] => ? = 9
[7,4,3,5,1,2,6] => ? = 8
[7,4,3,5,1,6,2] => ? = 6
[7,4,3,5,2,1,6] => ? = 9
[7,4,3,5,2,6,1] => ? = 5
[7,4,3,5,6,1,2] => ? = 2
[7,4,3,5,6,2,1] => ? = 2
[7,4,3,6,1,2,5] => ? = 8
[7,4,3,6,1,5,2] => ? = 7
[7,4,3,6,2,1,5] => ? = 9
[7,4,3,6,2,5,1] => ? = 6
[7,4,3,6,5,1,2] => ? = 4
[7,4,3,6,5,2,1] => ? = 4
[7,4,5,1,2,3,6] => ? = 6
[7,4,5,1,2,6,3] => ? = 6
[7,4,5,1,3,2,6] => ? = 8
[7,4,5,1,3,6,2] => ? = 6
[7,4,5,1,6,2,3] => ? = 4
[7,4,5,1,6,3,2] => ? = 5
[7,4,5,2,1,3,6] => ? = 8
[7,4,5,2,1,6,3] => ? = 8
[7,4,5,2,3,1,6] => ? = 8
[7,4,5,2,3,6,1] => ? = 4
Description
The number of occurrences of the pattern 132 or of the pattern 213 in a permutation.
Matching statistic: St001330
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 11%
Values
[1,2] => [1,2] => [2,1] => ([(0,1)],2)
 => 2 = 0 + 2
[2,1] => [1,2] => [2,1] => ([(0,1)],2)
 => 2 = 0 + 2
[1,2,3] => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
 => 2 = 0 + 2
[1,3,2] => [1,3,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 1 + 2
[2,1,3] => [1,3,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 1 + 2
[2,3,1] => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
 => 2 = 0 + 2
[3,1,2] => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
 => 2 = 0 + 2
[3,2,1] => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
 => 2 = 0 + 2
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[1,2,4,3] => [1,2,4,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 2
[1,3,2,4] => [1,3,2,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 2
[1,3,4,2] => [1,3,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 2
[1,4,2,3] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 2
[1,4,3,2] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 + 2
[2,1,3,4] => [1,3,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 2
[2,1,4,3] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 4 + 2
[2,3,1,4] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 2
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[2,4,1,3] => [1,3,2,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 2
[2,4,3,1] => [1,2,4,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[3,1,2,4] => [1,2,4,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 2
[3,1,4,2] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 2
[3,2,1,4] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 + 2
[3,2,4,1] => [1,2,4,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[3,4,1,2] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[3,4,2,1] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[4,1,2,3] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[4,1,3,2] => [1,3,2,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[4,2,1,3] => [1,3,2,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[4,2,3,1] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[4,3,1,2] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[4,3,2,1] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 2
[1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 2
[1,2,4,5,3] => [1,2,4,5,3] => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 2
[1,2,5,3,4] => [1,2,5,3,4] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 2
[1,2,5,4,3] => [1,2,5,3,4] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 6 + 2
[1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 2
[1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ? = 6 + 2
[1,3,4,2,5] => [1,3,4,2,5] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 2
[1,3,4,5,2] => [1,3,4,5,2] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 2
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 + 2
[1,3,5,4,2] => [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 + 2
[1,4,2,3,5] => [1,4,2,3,5] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 2
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 + 2
[1,4,3,2,5] => [1,4,2,5,3] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 6 + 2
[1,4,3,5,2] => [1,4,2,3,5] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 + 2
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? = 4 + 2
[1,4,5,3,2] => [1,4,5,2,3] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? = 5 + 2
[1,5,2,3,4] => [1,5,2,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 2
[1,5,2,4,3] => [1,5,2,4,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 + 2
[1,5,3,2,4] => [1,5,2,4,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 + 2
[1,5,3,4,2] => [1,5,2,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 + 2
[1,5,4,2,3] => [1,5,2,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 + 2
[1,5,4,3,2] => [1,5,2,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 6 + 2
[2,1,3,4,5] => [1,3,4,5,2] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 2
[2,1,3,5,4] => [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 6 + 2
[2,1,4,3,5] => [1,4,2,3,5] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 6 + 2
[2,1,4,5,3] => [1,4,5,2,3] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? = 7 + 2
[2,1,5,3,4] => [1,5,2,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 7 + 2
[2,1,5,4,3] => [1,5,2,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 9 + 2
[2,3,1,4,5] => [1,4,5,2,3] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? = 4 + 2
[2,3,1,5,4] => [1,5,2,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 7 + 2
[2,3,4,1,5] => [1,5,2,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 2
[2,3,4,5,1] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[2,3,5,1,4] => [1,4,2,3,5] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 2
[2,3,5,4,1] => [1,2,3,5,4] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 2
[3,4,5,1,2] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[3,4,5,2,1] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[4,5,1,2,3] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[4,5,2,3,1] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[4,5,3,1,2] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[4,5,3,2,1] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[5,1,2,3,4] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[5,2,3,4,1] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[5,3,4,1,2] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[5,3,4,2,1] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[5,4,1,2,3] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[5,4,2,3,1] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[5,4,3,1,2] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[5,4,3,2,1] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[2,3,4,5,6,1] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[3,4,5,6,1,2] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[3,4,5,6,2,1] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[4,5,6,1,2,3] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[4,5,6,2,3,1] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[4,5,6,3,1,2] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[4,5,6,3,2,1] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[5,6,1,2,3,4] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[5,6,2,3,4,1] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[5,6,3,4,1,2] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[5,6,3,4,2,1] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[5,6,4,1,2,3] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[5,6,4,2,3,1] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[5,6,4,3,1,2] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[5,6,4,3,2,1] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[6,1,2,3,4,5] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[6,2,3,4,5,1] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 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.