Processing math: 100%

Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St000835
Mapping
St000835: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1
[2]
 => 2
[1,1]
 => 0
[3]
 => 3
[2,1]
 => 1
[1,1,1]
 => 1
[4]
 => 4
[3,1]
 => 2
[2,2]
 => 0
[2,1,1]
 => 0
[1,1,1,1]
 => 0
[5]
 => 5
[4,1]
 => 3
[3,2]
 => 1
[3,1,1]
 => 1
[2,2,1]
 => 1
[2,1,1,1]
 => 1
[1,1,1,1,1]
 => 1
[6]
 => 6
[5,1]
 => 4
[4,2]
 => 2
[4,1,1]
 => 2
[3,3]
 => 0
[3,2,1]
 => 0
[3,1,1,1]
 => 0
[2,2,2]
 => 2
[2,2,1,1]
 => 0
[2,1,1,1,1]
 => 0
[1,1,1,1,1,1]
 => 0
[7]
 => 7
[6,1]
 => 5
[5,2]
 => 3
[5,1,1]
 => 3
[4,3]
 => 1
[4,2,1]
 => 1
[4,1,1,1]
 => 1
[3,3,1]
 => 1
[3,2,2]
 => 1
[3,2,1,1]
 => 1
[3,1,1,1,1]
 => 1
[2,2,2,1]
 => 1
[2,2,1,1,1]
 => 1
[2,1,1,1,1,1]
 => 1
[1,1,1,1,1,1,1]
 => 1
[8]
 => 8
[7,1]
 => 6
[6,2]
 => 4
[6,1,1]
 => 4
[5,3]
 => 2
[5,2,1]
 => 2
Description
The minimal difference in size when partitioning the integer partition into two subpartitions. This is the optimal value of the optimisation version of the partition problem [1].
Matching statistic: St001330
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 62%
Values
[1]
 => 1 => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[2]
 => 0 => [2] => ([],2)
 => 1 = 0 + 1
[1,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[3]
 => 1 => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[2,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4]
 => 0 => [2] => ([],2)
 => 1 = 0 + 1
[3,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[2,2]
 => 00 => [3] => ([],3)
 => 1 = 0 + 1
[2,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 1
[1,1,1,1]
 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[5]
 => 1 => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[4,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[3,2]
 => 10 => [1,2] => ([(1,2)],3)
 => 2 = 1 + 1
[3,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[2,2,1]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,1,1,1]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,1,1,1]
 => 11111 => [1,1,1,1,1,1] => ([(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 = 5 + 1
[6]
 => 0 => [2] => ([],2)
 => 1 = 0 + 1
[5,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[4,2]
 => 00 => [3] => ([],3)
 => 1 = 0 + 1
[4,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,2} + 1
[3,3]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[3,2,1]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,2} + 1
[3,1,1,1]
 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[2,2,2]
 => 000 => [4] => ([],4)
 => 1 = 0 + 1
[2,2,1,1]
 => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,2} + 1
[2,1,1,1,1]
 => 01111 => [2,1,1,1,1] => ([(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)
 => ? ∊ {0,0,0,2} + 1
[1,1,1,1,1,1]
 => 111111 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(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),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[7]
 => 1 => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[6,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[5,2]
 => 10 => [1,2] => ([(1,2)],3)
 => 2 = 1 + 1
[5,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,3]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[4,2,1]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[4,1,1,1]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1} + 1
[3,3,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[3,2,2]
 => 100 => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[3,2,1,1]
 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1} + 1
[3,1,1,1,1]
 => 11111 => [1,1,1,1,1,1] => ([(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 = 5 + 1
[2,2,2,1]
 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[2,2,1,1,1]
 => 00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1} + 1
[2,1,1,1,1,1]
 => 011111 => [2,1,1,1,1,1] => ([(0,2),(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),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,1} + 1
[1,1,1,1,1,1,1]
 => 1111111 => [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => 8 = 7 + 1
[8]
 => 0 => [2] => ([],2)
 => 1 = 0 + 1
[7,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[6,2]
 => 00 => [3] => ([],3)
 => 1 = 0 + 1
[6,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,2,8} + 1
[5,3]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[5,2,1]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,2,8} + 1
[5,1,1,1]
 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[4,4]
 => 00 => [3] => ([],3)
 => 1 = 0 + 1
[4,3,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,2,8} + 1
[4,2,2]
 => 000 => [4] => ([],4)
 => 1 = 0 + 1
[4,2,1,1]
 => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,2,8} + 1
[4,1,1,1,1]
 => 01111 => [2,1,1,1,1] => ([(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,2,8} + 1
[3,3,2]
 => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,3,1,1]
 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[3,2,2,1]
 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,2,8} + 1
[3,2,1,1,1]
 => 10111 => [1,2,1,1,1] => ([(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,2,8} + 1
[3,1,1,1,1,1]
 => 111111 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(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),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[2,2,2,2]
 => 0000 => [5] => ([],5)
 => 1 = 0 + 1
[2,2,2,1,1]
 => 00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,2,8} + 1
[2,2,1,1,1,1]
 => 001111 => [3,1,1,1,1] => ([(0,3),(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),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,2,8} + 1
[2,1,1,1,1,1,1]
 => 0111111 => [2,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,2,8} + 1
[1,1,1,1,1,1,1,1]
 => 11111111 => [1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {0,0,0,0,0,0,0,0,0,2,8} + 1
[9]
 => 1 => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[8,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[7,2]
 => 10 => [1,2] => ([(1,2)],3)
 => 2 = 1 + 1
[7,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[6,3]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[6,1,1,1]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,9} + 1
[5,2,1,1]
 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,9} + 1
[4,3,1,1]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,9} + 1
[4,2,1,1,1]
 => 00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,9} + 1
[4,1,1,1,1,1]
 => 011111 => [2,1,1,1,1,1] => ([(0,2),(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),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,9} + 1
[3,3,2,1]
 => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,9} + 1
[3,2,2,1,1]
 => 10011 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,9} + 1
[3,2,1,1,1,1]
 => 101111 => [1,2,1,1,1,1] => ([(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),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,9} + 1
[2,2,2,1,1,1]
 => 000111 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,9} + 1
[2,2,1,1,1,1,1]
 => 0011111 => [3,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,9} + 1
[2,1,1,1,1,1,1,1]
 => 01111111 => [2,1,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,9} + 1
[1,1,1,1,1,1,1,1,1]
 => 111111111 => [1,1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,9} + 1
[8,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,8,10} + 1
[7,2,1]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,8,10} + 1
[6,3,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,8,10} + 1
[6,2,1,1]
 => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,8,10} + 1
[6,1,1,1,1]
 => 01111 => [2,1,1,1,1] => ([(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,8,10} + 1
[5,4,1]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,8,10} + 1
[5,2,2,1]
 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,8,10} + 1
[5,2,1,1,1]
 => 10111 => [1,2,1,1,1] => ([(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,8,10} + 1
[4,4,1,1]
 => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,8,10} + 1
[4,3,3]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,8,10} + 1
[4,3,2,1]
 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,8,10} + 1
[4,3,1,1,1]
 => 01111 => [2,1,1,1,1] => ([(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,8,10} + 1
[4,2,2,1,1]
 => 00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,8,10} + 1
[4,2,1,1,1,1]
 => 001111 => [3,1,1,1,1] => ([(0,3),(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),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,8,10} + 1
[4,1,1,1,1,1,1]
 => 0111111 => [2,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,8,10} + 1
[3,3,2,1,1]
 => 11011 => [1,1,2,1,1] => ([(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)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,8,10} + 1
[3,2,2,2,1]
 => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,8,10} + 1
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.