Your data matches 3 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000392
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
Mp00278: Binary words —rowmotion⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [1,2] => 0 => 0 => 0
[1,1,0,0]
 => [2,1] => 1 => 1 => 1
[1,0,1,0,1,0]
 => [1,2,3] => 00 => 00 => 0
[1,0,1,1,0,0]
 => [1,3,2] => 01 => 10 => 1
[1,1,0,0,1,0]
 => [2,1,3] => 10 => 01 => 1
[1,1,0,1,0,0]
 => [2,3,1] => 01 => 10 => 1
[1,1,1,0,0,0]
 => [3,2,1] => 11 => 11 => 2
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 000 => 000 => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 001 => 010 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 010 => 100 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 001 => 010 => 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 011 => 101 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 100 => 001 => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 101 => 110 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 010 => 100 => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 001 => 010 => 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 011 => 101 => 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 110 => 011 => 2
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 101 => 110 => 2
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 101 => 110 => 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 111 => 111 => 3
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0000 => 0000 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 0001 => 0010 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 0010 => 0100 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 0001 => 0010 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 0011 => 0101 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 0100 => 1000 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 0101 => 1010 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 0010 => 0100 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 0001 => 0010 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 0011 => 0101 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 0110 => 1001 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 0101 => 1010 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => 0101 => 1010 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 0111 => 1011 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 1000 => 0001 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 1001 => 0110 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 1010 => 1100 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 1001 => 0110 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 1011 => 1101 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 0100 => 1000 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 0101 => 1010 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 0010 => 0100 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 0001 => 0010 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => 0011 => 0101 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => 0110 => 1001 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => 0101 => 1010 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => 0101 => 1010 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => 0111 => 1011 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 1100 => 0011 => 2
Description
The length of the longest run of ones in a binary word.
Matching statistic: St001235
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 55%
Values
[1,0,1,0]
 => [1,2] => [2] => [2] => 1 = 0 + 1
[1,1,0,0]
 => [2,1] => [1,1] => [1,1] => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,2,3] => [3] => [3] => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,3,2] => [2,1] => [1,2] => 2 = 1 + 1
[1,1,0,0,1,0]
 => [2,1,3] => [1,2] => [2,1] => 2 = 1 + 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,1] => [1,2] => 2 = 1 + 1
[1,1,1,0,0,0]
 => [3,2,1] => [1,1,1] => [1,1,1] => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [4] => [4] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [3,1] => [1,3] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,2] => [2,2] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,1] => [1,3] => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [2,1,1] => [1,2,1] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [1,3] => [3,1] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [1,2,1] => [1,1,2] => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,2] => [2,2] => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [3,1] => [1,3] => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [2,1,1] => [1,2,1] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [1,1,2] => [2,1,1] => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [1,2,1] => [1,1,2] => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [1,2,1] => [1,1,2] => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [1,1,1,1] => [1,1,1,1] => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [5] => [5] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [4,1] => [1,4] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [3,2] => [2,3] => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [4,1] => [1,4] => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [3,1,1] => [1,3,1] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [2,3] => [3,2] => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1] => [1,2,2] => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [3,2] => [2,3] => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [4,1] => [1,4] => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [3,1,1] => [1,3,1] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [2,1,2] => [2,2,1] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [2,2,1] => [1,2,2] => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [2,2,1] => [1,2,2] => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [2,1,1,1] => [1,2,1,1] => 3 = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [1,4] => [4,1] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [1,3,1] => [1,1,3] => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [1,2,2] => [2,1,2] => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [1,3,1] => [1,1,3] => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [1,2,1,1] => [1,1,2,1] => 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,3] => [3,2] => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,2,1] => [1,2,2] => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [3,2] => [2,3] => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [4,1] => [1,4] => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [3,1,1] => [1,3,1] => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [2,1,2] => [2,2,1] => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [2,2,1] => [1,2,2] => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [2,2,1] => [1,2,2] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [2,1,1,1] => [1,2,1,1] => 3 = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [1,1,3] => [3,1,1] => 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7] => [7] => [7] => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,7,6] => [6,1] => [1,6] => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,4,6,5,7] => [5,2] => [2,5] => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,6,7,5] => [6,1] => [1,6] => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,7,6,5] => [5,1,1] => [1,5,1] => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,3,5,4,6,7] => [4,3] => [3,4] => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,3,5,4,7,6] => [4,2,1] => [1,4,2] => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,2,3,5,6,4,7] => [5,2] => [2,5] => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,5,6,7,4] => [6,1] => [1,6] => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,2,3,5,7,6,4] => [5,1,1] => [1,5,1] => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,3,6,5,4,7] => [4,1,2] => [2,4,1] => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,3,6,5,7,4] => [4,2,1] => [1,4,2] => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,3,7,5,6,4] => [4,2,1] => [1,4,2] => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,3,7,6,5,4] => [4,1,1,1] => [1,4,1,1] => ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,4,3,5,6,7] => [3,4] => [4,3] => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,2,4,3,5,7,6] => [3,3,1] => [1,3,3] => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,2,4,3,6,5,7] => [3,2,2] => [2,3,2] => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,2,4,3,6,7,5] => [3,3,1] => [1,3,3] => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,2,4,3,7,6,5] => [3,2,1,1] => [1,3,2,1] => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,2,4,5,3,6,7] => [4,3] => [3,4] => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,2,4,5,3,7,6] => [4,2,1] => [1,4,2] => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,2,4,5,6,3,7] => [5,2] => [2,5] => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,2,4,5,6,7,3] => [6,1] => [1,6] => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,2,4,5,7,6,3] => [5,1,1] => [1,5,1] => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,2,4,6,5,3,7] => [4,1,2] => [2,4,1] => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,2,4,6,5,7,3] => [4,2,1] => [1,4,2] => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,2,4,7,5,6,3] => [4,2,1] => [1,4,2] => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,2,4,7,6,5,3] => [4,1,1,1] => [1,4,1,1] => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,2,5,4,3,6,7] => [3,1,3] => [3,3,1] => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,2,5,4,3,7,6] => [3,1,2,1] => [1,3,1,2] => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,2,5,4,6,3,7] => [3,2,2] => [2,3,2] => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,2,5,4,6,7,3] => [3,3,1] => [1,3,3] => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,2,5,4,7,6,3] => [3,2,1,1] => [1,3,2,1] => ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,2,6,4,5,3,7] => [3,2,2] => [2,3,2] => ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,2,6,4,5,7,3] => [3,3,1] => [1,3,3] => ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,2,7,4,5,6,3] => [3,3,1] => [1,3,3] => ? = 1 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,2,7,4,6,5,3] => [3,2,1,1] => [1,3,2,1] => ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,2,6,5,4,3,7] => [3,1,1,2] => [2,3,1,1] => ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,2,6,5,4,7,3] => [3,1,2,1] => [1,3,1,2] => ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,2,7,5,4,6,3] => [3,1,2,1] => [1,3,1,2] => ? = 2 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,2,7,5,6,4,3] => [3,2,1,1] => [1,3,2,1] => ? = 1 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,2,7,6,5,4,3] => [3,1,1,1,1] => [1,3,1,1,1] => ? = 3 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,3,2,4,5,6,7] => [2,5] => [5,2] => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,3,2,4,5,7,6] => [2,4,1] => [1,2,4] => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,3,2,4,6,5,7] => [2,3,2] => [2,2,3] => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,3,2,4,6,7,5] => [2,4,1] => [1,2,4] => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,3,2,4,7,6,5] => [2,3,1,1] => [1,2,3,1] => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,3,2,5,4,6,7] => [2,2,3] => [3,2,2] => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4,7,6] => [2,2,2,1] => [1,2,2,2] => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,3,2,5,6,4,7] => [2,3,2] => [2,2,3] => ? = 1 + 1
Description
The global dimension of the corresponding Comp-Nakayama algebra. We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
Matching statistic: St001330
(load all 6 compositions to match this statistic)
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 73%
Values
[1,0,1,0]
 => [1,2] => [2] => ([],2)
 => 1 = 0 + 1
[1,1,0,0]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,2,3] => [3] => ([],3)
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [2,1,3] => [1,2] => ([(1,2)],3)
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [4] => ([],4)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [5] => ([],5)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [1,4] => ([(3,4)],5)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [3,4,2,1,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [3,4,2,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [3,5,4,2,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [4,3,5,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [4,5,3,2,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => [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
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => [6] => ([],6)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,5,4] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,5,6] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,2,4,5,3,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,4,5,6,3] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,2,4,6,5,3] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,5,4,3,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,5,4,6,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,5,6,4,3] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,6,5,4,3] => [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)
 => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,3,2,4,5,6] => [2,4] => ([(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,3,2,4,6,5] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,3,2,5,6,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,6,5,4] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,3,4,2,5,6] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,3,4,2,6,5] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,3,4,5,2,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,3,4,6,5,2] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,3,5,4,2,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,3,5,4,6,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,3,5,6,4,2] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,3,6,5,4,2] => [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)
 => ? = 2 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6] => [1,5] => ([(4,5)],6)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [2,3,1,4,5,6] => [2,4] => ([(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0,1,0]
 => [2,3,4,1,5,6] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [2,3,4,5,1,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0,1,0]
 => [3,2,1,4,5,6] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
[1,1,1,1,0,0,0,0,1,0,1,0]
 => [4,3,2,1,5,6] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 3 + 1
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,4,3,2,1,6] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,2,1] => [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
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.