Your data matches 2 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000698
Mapping
Mp00252: Permutations —restriction⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3,4] => [1,2,3] => [1,0,1,0,1,0]
 => [2,1]
 => 0
[1,2,4,3] => [1,2,3] => [1,0,1,0,1,0]
 => [2,1]
 => 0
[1,3,2,4] => [1,3,2] => [1,0,1,1,0,0]
 => [1,1]
 => 1
[1,3,4,2] => [1,3,2] => [1,0,1,1,0,0]
 => [1,1]
 => 1
[1,4,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => [2,1]
 => 0
[1,4,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => [1,1]
 => 1
[2,1,3,4] => [2,1,3] => [1,1,0,0,1,0]
 => [2]
 => 1
[2,1,4,3] => [2,1,3] => [1,1,0,0,1,0]
 => [2]
 => 1
[2,4,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => [2]
 => 1
[4,1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => [2,1]
 => 0
[4,1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => [1,1]
 => 1
[4,2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => [2]
 => 1
[1,2,3,4,5] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 0
[1,2,3,5,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 0
[1,2,4,3,5] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 2
[1,2,4,5,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 2
[1,2,5,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 0
[1,2,5,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 2
[1,3,2,4,5] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 2
[1,3,2,5,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 2
[1,3,4,2,5] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 2
[1,3,4,5,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 2
[1,3,5,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 2
[1,3,5,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 2
[1,4,2,3,5] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1
[1,4,2,5,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1
[1,4,3,2,5] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1
[1,4,3,5,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1
[1,4,5,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1
[1,4,5,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1
[1,5,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 0
[1,5,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 2
[1,5,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 2
[1,5,3,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 2
[1,5,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1
[1,5,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1
[2,1,3,4,5] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 2
[2,1,3,5,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 2
[2,1,4,3,5] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 2
[2,1,4,5,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 2
[2,1,5,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 2
[2,1,5,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 2
[2,3,1,4,5] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 2
[2,3,1,5,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 2
[2,3,4,1,5] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 0
[2,3,4,5,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 0
[2,3,5,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 2
[2,3,5,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 0
[2,4,1,3,5] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 1
[2,4,1,5,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 1
Description
The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. For any positive integer $k$, one associates a $k$-core to a partition by repeatedly removing all rim hooks of size $k$. This statistic counts the $2$-rim hooks that are removed in this process to obtain a $2$-core.
Matching statistic: St001330
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 14%
Values
[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)
 => ? = 0 + 2
[1,3,2,4] => [1,3,2,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[1,3,4,2] => [1,3,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[1,4,2,3] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[1,4,3,2] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[2,1,3,4] => [1,3,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[2,1,4,3] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[2,4,1,3] => [1,3,2,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 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
[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)
 => ? = 0 + 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)
 => ? = 2 + 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)
 => ? = 2 + 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)
 => ? = 0 + 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)
 => ? = 2 + 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)
 => ? = 2 + 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)
 => ? = 2 + 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)
 => ? = 2 + 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)
 => ? = 2 + 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)
 => ? = 2 + 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)
 => ? = 2 + 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)
 => ? = 1 + 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)
 => ? = 1 + 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)
 => ? = 1 + 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)
 => ? = 1 + 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)
 => ? = 1 + 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)
 => ? = 1 + 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)
 => ? = 0 + 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)
 => ? = 2 + 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)
 => ? = 2 + 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)
 => ? = 2 + 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)
 => ? = 1 + 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)
 => ? = 1 + 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)
 => ? = 2 + 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)
 => ? = 2 + 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)
 => ? = 2 + 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)
 => ? = 2 + 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)
 => ? = 2 + 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)
 => ? = 2 + 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)
 => ? = 2 + 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)
 => ? = 2 + 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)
 => ? = 0 + 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)
 => ? = 2 + 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)
 => ? = 0 + 2
[2,4,1,3,5] => [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)
 => ? = 1 + 2
[2,4,1,5,3] => [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)
 => ? = 1 + 2
[2,4,3,1,5] => [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)
 => ? = 1 + 2
[2,4,3,5,1] => [1,2,4,3,5] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
[2,4,5,1,3] => [1,3,2,4,5] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
[2,4,5,3,1] => [1,2,4,5,3] => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 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
[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
[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
[6,3,4,5,1,2] => [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
[6,3,4,5,2,1] => [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
[2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[3,4,5,6,7,1,2] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[3,4,5,6,7,2,1] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[4,5,6,7,1,2,3] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[4,5,6,7,2,3,1] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[4,5,6,7,3,1,2] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[4,5,6,7,3,2,1] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[7,2,3,4,5,6,1] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[7,3,4,5,6,1,2] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[7,3,4,5,6,2,1] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[7,4,5,6,1,2,3] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[7,4,5,6,2,3,1] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[7,4,5,6,3,1,2] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[7,4,5,6,3,2,1] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[5,6,7,8,4,3,2,1] => [1,2,3,4,5,6,7,8] => [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
[4,5,6,7,8,3,2,1] => [1,2,3,4,5,6,7,8] => [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
[5,6,7,8,3,4,2,1] => [1,2,3,4,5,6,7,8] => [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
[5,6,7,8,4,2,3,1] => [1,2,3,4,5,6,7,8] => [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
[4,5,6,7,8,2,3,1] => [1,2,3,4,5,6,7,8] => [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
[8,5,6,7,2,3,4,1] => [1,2,3,4,5,6,7,8] => [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
[5,6,7,8,2,3,4,1] => [1,2,3,4,5,6,7,8] => [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
[5,6,7,8,4,3,1,2] => [1,2,3,4,5,6,7,8] => [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
[4,5,6,7,8,3,1,2] => [1,2,3,4,5,6,7,8] => [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
[5,6,7,8,3,4,1,2] => [1,2,3,4,5,6,7,8] => [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
[8,3,4,5,6,7,1,2] => [1,2,3,4,5,6,7,8] => [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
[3,4,5,6,7,8,1,2] => [1,2,3,4,5,6,7,8] => [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
[8,5,6,7,4,1,2,3] => [1,2,3,4,5,6,7,8] => [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
[5,6,7,8,4,1,2,3] => [1,2,3,4,5,6,7,8] => [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
[4,5,6,7,8,1,2,3] => [1,2,3,4,5,6,7,8] => [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
[8,5,6,7,1,2,3,4] => [1,2,3,4,5,6,7,8] => [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
[5,6,7,8,1,2,3,4] => [1,2,3,4,5,6,7,8] => [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
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.