Your data matches 3 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001652
(load all 4 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St001652: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => 1
[1,0,1,0]
 => [2,1] => [2,1] => 1
[1,1,0,0]
 => [1,2] => [1,2] => 2
[1,0,1,0,1,0]
 => [3,2,1] => [3,2,1] => 1
[1,0,1,1,0,0]
 => [2,3,1] => [1,3,2] => 1
[1,1,0,0,1,0]
 => [3,1,2] => [2,3,1] => 2
[1,1,0,1,0,0]
 => [2,1,3] => [2,1,3] => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => 3
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4,3,2,1] => 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,4,3,2] => 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [2,4,3,1] => 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [2,1,4,3] => 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,4,3] => 2
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [3,4,2,1] => 2
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [3,1,4,2] => 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [3,2,4,1] => 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,2,1,4] => 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [1,3,2,4] => 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,3,4,1] => 3
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [2,3,1,4] => 2
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,5,4,3,2] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [2,5,4,3,1] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [2,1,5,4,3] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,2,5,4,3] => 2
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [3,5,4,2,1] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [3,1,5,4,2] => 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [3,2,5,4,1] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [3,2,1,5,4] => 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,3,2,5,4] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [2,3,5,4,1] => 2
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [2,3,1,5,4] => 2
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [2,1,3,5,4] => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,5,4] => 3
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [4,5,3,2,1] => 2
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [4,1,5,3,2] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [4,2,5,3,1] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [4,2,1,5,3] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,4,2,5,3] => 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [4,3,5,2,1] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [4,3,1,5,2] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [4,3,2,5,1] => 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [4,3,2,1,5] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [1,4,3,2,5] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [2,4,3,5,1] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [2,4,3,1,5] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [2,1,4,3,5] => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [1,2,4,3,5] => 2
Description
The length of a longest interval of consecutive numbers. For a permutation $\pi=\pi_1,\dots,\pi_n$, this statistic returns the length of a longest subsequence $\pi_k,\dots,\pi_\ell$ such that $\pi_{i+1} = \pi_i + 1$ for $i\in\{k,\dots,\ell-1\}$.
Matching statistic: St001662
(load all 2 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => [1] => 1
[1,0,1,0]
 => [2,1] => [2,1] => [2,1] => 1
[1,1,0,0]
 => [1,2] => [1,2] => [1,2] => 2
[1,0,1,0,1,0]
 => [3,2,1] => [3,2,1] => [3,2,1] => 1
[1,0,1,1,0,0]
 => [2,3,1] => [1,3,2] => [1,3,2] => 1
[1,1,0,0,1,0]
 => [3,1,2] => [2,3,1] => [3,1,2] => 2
[1,1,0,1,0,0]
 => [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => [1,2,3] => 3
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,4,3,2] => [1,4,3,2] => 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [2,4,3,1] => [4,1,3,2] => 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 2
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [3,4,2,1] => [4,3,1,2] => 2
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [3,1,4,2] => [2,4,1,3] => 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [3,2,4,1] => [4,2,1,3] => 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,3,4,1] => [4,1,2,3] => 3
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [2,3,1,4] => [3,1,2,4] => 2
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [2,5,4,3,1] => [5,1,4,3,2] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [2,1,5,4,3] => [2,1,5,4,3] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [3,5,4,2,1] => [5,4,1,3,2] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [3,1,5,4,2] => [2,5,1,4,3] => 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [3,2,5,4,1] => [5,2,1,4,3] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [3,2,1,5,4] => [3,2,1,5,4] => 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,3,2,5,4] => [1,3,2,5,4] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [2,3,5,4,1] => [5,1,2,4,3] => 2
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [2,3,1,5,4] => [3,1,2,5,4] => 2
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [2,1,3,5,4] => [2,1,3,5,4] => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => 3
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [4,5,3,2,1] => [5,4,3,1,2] => 2
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [4,1,5,3,2] => [2,5,4,1,3] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [4,2,5,3,1] => [5,2,4,1,3] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [4,2,1,5,3] => [3,2,5,1,4] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,4,2,5,3] => [1,3,5,2,4] => 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [4,3,5,2,1] => [5,4,2,1,3] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [4,3,1,5,2] => [3,5,2,1,4] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [4,3,2,5,1] => [5,3,2,1,4] => 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [2,4,3,5,1] => [5,1,3,2,4] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [2,4,3,1,5] => [4,1,3,2,5] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [2,1,4,3,5] => [2,1,4,3,5] => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
Description
The length of the longest factor of consecutive numbers in a permutation.
Matching statistic: St001330
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 2 = 1 + 1
[1,0,1,0]
 => [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,1,0,0]
 => [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,0,1,0,1,0]
 => [4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,0,1,1,0,0]
 => [3,1,4,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 1
[1,1,0,0,1,0]
 => [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[1,1,0,1,0,0]
 => [4,3,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [2,3,4,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [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)
 => 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [6,5,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [6,5,4,1,2,3] => ([(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)
 => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [6,5,3,1,2,4] => ([(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
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [5,1,2,6,3,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [4,1,2,6,5,3] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [6,4,3,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [5,3,1,2,6,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [5,4,1,2,6,3] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [6,5,4,3,1,2] => ([(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)
 => ? = 3 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [6,2,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [6,5,2,1,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [6,4,2,1,3,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [5,2,1,3,6,4] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [6,5,4,2,1,3] => ([(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)
 => ? = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [3,1,6,5,2,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [4,1,6,2,3,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [4,1,6,5,2,3] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [3,1,6,4,2,5] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [5,4,1,6,2,3] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [3,1,6,5,4,2] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => [6,3,2,1,4,5] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => [6,5,3,2,1,4] => ([(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)
 => ? = 2 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => [4,2,1,6,3,5] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => ? = 2 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => [5,2,1,6,3,4] => ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [2,5,4,1,6,3] => [4,2,1,6,5,3] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
 => ? = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => [4,3,1,6,2,5] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [6,3,5,1,2,4] => [5,3,1,6,2,4] => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3] => [4,1,5,2,6,3] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [5,3,4,1,6,2] => [4,3,1,6,5,2] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => [6,4,3,2,1,5] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [2,3,6,5,1,4] => [5,3,2,1,6,4] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [2,6,4,5,1,3] => [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)
 => ? = 2 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [6,3,4,5,1,2] => [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)
 => ? = 3 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [6,5,4,3,2,1] => ([(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
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,1,2,3,4,7,5] => [7,6,1,2,3,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,1,2,3,7,4,6] => [7,5,1,2,3,4,6] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [7,1,2,3,6,4,5] => [6,1,2,3,4,7,5] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [7,1,2,5,3,4,6] => [5,1,2,3,7,4,6] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [7,1,4,2,3,5,6] => [4,1,2,7,3,5,6] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [7,1,4,2,6,3,5] => [4,1,2,6,3,7,5] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [7,3,1,2,4,5,6] => [3,1,7,2,4,5,6] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,1,0,0]
 => [7,3,1,2,6,4,5] => [3,1,6,2,4,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [7,3,1,5,2,4,6] => [3,1,5,2,7,4,6] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => 2 = 1 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => [7,6,5,4,3,2,1] => ([(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
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => [8,7,6,5,4,3,2,1] => ([(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
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.