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Your data matches 5 different statistics following compositions of up to 3 maps.
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Matching statistic: St001416
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St001416: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1
1 => 1
00 => 2
01 => 1
10 => 1
11 => 2
000 => 3
001 => 2
010 => 3
011 => 2
100 => 2
101 => 3
110 => 2
111 => 3
0000 => 4
0001 => 3
0010 => 3
0011 => 2
0100 => 3
0101 => 3
0110 => 4
0111 => 3
1000 => 3
1001 => 4
1010 => 3
1011 => 3
1100 => 2
1101 => 3
1110 => 3
1111 => 4
00000 => 5
00001 => 4
00010 => 3
00011 => 3
00100 => 5
00101 => 3
00110 => 4
00111 => 3
01000 => 3
01001 => 4
01010 => 5
01011 => 3
01100 => 4
01101 => 4
01110 => 5
01111 => 4
10000 => 4
10001 => 5
10010 => 4
10011 => 4
Description
The length of a longest palindromic factor of a binary word.
A factor of a word is a sequence of consecutive letters. This statistic records the maximal length of a palindromic factor.
Matching statistic: St001232
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 78%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 78%
Values
0 => [1] => [1,0]
=> [1,0]
=> 0 = 1 - 1
1 => [1] => [1,0]
=> [1,0]
=> 0 = 1 - 1
00 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
01 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
10 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
11 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
000 => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2} - 1
001 => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
010 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
011 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
100 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
101 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
110 => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
111 => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2} - 1
0000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4} - 1
0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4} - 1
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4} - 1
0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4} - 1
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4} - 1
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4} - 1
1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4} - 1
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4} - 1
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4} - 1
1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4} - 1
1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4} - 1
1111 => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4} - 1
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4 = 5 - 1
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
10100 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
10110 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
10111 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4 = 5 - 1
11010 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
11011 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
11100 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
11101 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
11110 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
11111 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5} - 1
000000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
000001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
000010 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
000011 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
000100 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
000101 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
000110 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
000111 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6} - 1
001100 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 6 - 1
011001 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4 = 5 - 1
100110 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4 = 5 - 1
110011 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 6 - 1
0011001 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> 6 = 7 - 1
0110011 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 6 - 1
1001100 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 6 - 1
1100110 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> 6 = 7 - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001879
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 56%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 56%
Values
0 => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {1,1}
1 => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {1,1}
00 => [2] => [[2],[]]
=> ([(0,1)],2)
=> ? ∊ {1,1,2,2}
01 => [1,1] => [[1,1],[]]
=> ([(0,1)],2)
=> ? ∊ {1,1,2,2}
10 => [1,1] => [[1,1],[]]
=> ([(0,1)],2)
=> ? ∊ {1,1,2,2}
11 => [2] => [[2],[]]
=> ([(0,1)],2)
=> ? ∊ {1,1,2,2}
000 => [3] => [[3],[]]
=> ([(0,2),(2,1)],3)
=> 2
001 => [2,1] => [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {3,3,3,3}
010 => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 2
011 => [1,2] => [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? ∊ {3,3,3,3}
100 => [1,2] => [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? ∊ {3,3,3,3}
101 => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 2
110 => [2,1] => [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {3,3,3,3}
111 => [3] => [[3],[]]
=> ([(0,2),(2,1)],3)
=> 2
0000 => [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
0001 => [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4}
0010 => [2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4}
0011 => [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4}
0100 => [1,1,2] => [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4}
0101 => [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
0110 => [1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4}
0111 => [1,3] => [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4}
1000 => [1,3] => [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4}
1001 => [1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4}
1010 => [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
1011 => [1,1,2] => [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4}
1100 => [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4}
1101 => [2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4}
1110 => [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,4,4}
1111 => [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
00000 => [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
00001 => [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
00010 => [3,1,1] => [[3,3,3],[2,2]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
00011 => [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
00100 => [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
00101 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
00110 => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
00111 => [2,3] => [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
01000 => [1,1,3] => [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
01001 => [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
01010 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
01011 => [1,1,1,2] => [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
01100 => [1,2,2] => [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
01101 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
01110 => [1,3,1] => [[3,3,1],[2]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
01111 => [1,4] => [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
10000 => [1,4] => [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
10001 => [1,3,1] => [[3,3,1],[2]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
10010 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
10011 => [1,2,2] => [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
10100 => [1,1,1,2] => [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
10101 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
10110 => [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
10111 => [1,1,3] => [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
11000 => [2,3] => [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
11001 => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
11010 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
11011 => [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
11100 => [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
11101 => [3,1,1] => [[3,3,3],[2,2]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
11110 => [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
11111 => [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
000000 => [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
010101 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
101010 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
111111 => [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
0000000 => [7] => [[7],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
0101010 => [1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
1010101 => [1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
1111111 => [7] => [[7],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Matching statistic: St001880
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 56%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 56%
Values
0 => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {1,1}
1 => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {1,1}
00 => [2] => [[2],[]]
=> ([(0,1)],2)
=> ? ∊ {1,1,2,2}
01 => [1,1] => [[1,1],[]]
=> ([(0,1)],2)
=> ? ∊ {1,1,2,2}
10 => [1,1] => [[1,1],[]]
=> ([(0,1)],2)
=> ? ∊ {1,1,2,2}
11 => [2] => [[2],[]]
=> ([(0,1)],2)
=> ? ∊ {1,1,2,2}
000 => [3] => [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
001 => [2,1] => [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2,2}
010 => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 3
011 => [1,2] => [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? ∊ {2,2,2,2}
100 => [1,2] => [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? ∊ {2,2,2,2}
101 => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 3
110 => [2,1] => [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2,2}
111 => [3] => [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
0000 => [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
0001 => [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3}
0010 => [2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3}
0011 => [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3}
0100 => [1,1,2] => [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3}
0101 => [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
0110 => [1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3}
0111 => [1,3] => [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3}
1000 => [1,3] => [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3}
1001 => [1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3}
1010 => [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
1011 => [1,1,2] => [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3}
1100 => [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3}
1101 => [2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3}
1110 => [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3}
1111 => [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
00000 => [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
00001 => [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
00010 => [3,1,1] => [[3,3,3],[2,2]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
00011 => [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
00100 => [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
00101 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
00110 => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
00111 => [2,3] => [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
01000 => [1,1,3] => [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
01001 => [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
01010 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
01011 => [1,1,1,2] => [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
01100 => [1,2,2] => [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
01101 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
01110 => [1,3,1] => [[3,3,1],[2]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
01111 => [1,4] => [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
10000 => [1,4] => [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
10001 => [1,3,1] => [[3,3,1],[2]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
10010 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
10011 => [1,2,2] => [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
10100 => [1,1,1,2] => [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
10101 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
10110 => [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
10111 => [1,1,3] => [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
11000 => [2,3] => [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
11001 => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
11010 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
11011 => [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
11100 => [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
11101 => [3,1,1] => [[3,3,3],[2,2]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
11110 => [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5}
11111 => [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
000000 => [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
010101 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
101010 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
111111 => [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
0000000 => [7] => [[7],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
0101010 => [1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
1010101 => [1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
1111111 => [7] => [[7],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001116
Values
0 => ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
1 => ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
00 => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 3 = 2 + 1
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 3 = 2 + 1
11 => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
000 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3 = 2 + 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> 3 = 2 + 1
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> 3 = 2 + 1
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
111 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3 = 2 + 1
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4} + 1
0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(0,8),(1,4),(1,6),(2,6),(2,8),(3,4),(3,5),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
=> ([(0,2),(0,3),(0,4),(0,8),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,4),(3,7),(3,8),(4,6),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4} + 1
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,8),(5,8),(6,8),(7,8)],9)
=> ([(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(3,8),(4,5),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4} + 1
0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(0,8),(1,4),(1,6),(2,6),(2,8),(3,4),(3,5),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
=> ([(0,2),(0,3),(0,4),(0,8),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,4),(3,7),(3,8),(4,6),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4} + 1
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,1),(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,7),(3,4),(3,5),(3,7),(4,5),(4,6),(5,6),(6,7)],8)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4} + 1
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(0,5),(0,6),(1,7),(1,8),(2,5),(2,6),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8)],9)
=> ([(0,4),(0,5),(0,6),(0,8),(1,2),(1,3),(1,4),(1,7),(1,8),(2,3),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,7),(6,7),(7,8)],9)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4} + 1
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4} + 1
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4} + 1
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(0,5),(0,6),(1,7),(1,8),(2,5),(2,6),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8)],9)
=> ([(0,4),(0,5),(0,6),(0,8),(1,2),(1,3),(1,4),(1,7),(1,8),(2,3),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,7),(6,7),(7,8)],9)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4} + 1
1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,1),(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,7),(3,4),(3,5),(3,7),(4,5),(4,6),(5,6),(6,7)],8)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4} + 1
1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(0,8),(1,4),(1,6),(2,6),(2,8),(3,4),(3,5),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
=> ([(0,2),(0,3),(0,4),(0,8),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,4),(3,7),(3,8),(4,6),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4} + 1
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,8),(5,8),(6,8),(7,8)],9)
=> ([(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(3,8),(4,5),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4} + 1
1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(0,8),(1,4),(1,6),(2,6),(2,8),(3,4),(3,5),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
=> ([(0,2),(0,3),(0,4),(0,8),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,4),(3,7),(3,8),(4,6),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4} + 1
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4} + 1
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,3),(0,7),(1,2),(1,6),(2,8),(3,9),(4,5),(4,8),(4,9),(5,6),(5,7),(6,8),(7,9)],10)
=> ([(0,1),(0,3),(0,5),(0,6),(0,7),(0,9),(1,2),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,6),(2,8),(2,9),(3,4),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} + 1
00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,7),(0,11),(1,5),(1,8),(2,6),(2,8),(3,4),(3,7),(3,11),(4,5),(4,9),(5,10),(6,10),(6,11),(7,9),(8,10),(9,10),(9,11)],12)
=> ([(0,3),(0,4),(0,5),(0,6),(0,9),(0,10),(0,11),(1,2),(1,4),(1,5),(1,7),(1,8),(1,10),(1,11),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,9),(5,10),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} + 1
00011 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,5),(0,9),(1,4),(1,9),(2,6),(2,8),(3,7),(3,8),(4,6),(4,10),(5,7),(5,10),(6,11),(7,11),(8,11),(9,10),(10,11)],12)
=> ([(0,3),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,2),(1,4),(1,5),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,10),(3,11),(4,5),(4,7),(4,8),(4,9),(4,11),(5,6),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} + 1
00100 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(0,7),(0,11),(1,5),(1,6),(2,3),(2,4),(2,11),(3,6),(3,9),(4,5),(4,8),(5,10),(6,10),(7,8),(7,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ([(0,3),(0,4),(0,5),(0,6),(0,8),(0,9),(0,11),(1,2),(1,5),(1,6),(1,7),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(2,10),(2,11),(3,4),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,11),(5,6),(5,7),(5,8),(5,10),(6,7),(6,9),(6,10),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} + 1
00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,5),(0,6),(1,7),(1,10),(2,4),(2,10),(3,6),(3,8),(3,11),(4,5),(4,11),(5,9),(6,9),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ([(0,3),(0,7),(0,8),(0,9),(0,10),(0,11),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(1,11),(2,4),(2,5),(2,6),(2,9),(2,10),(2,11),(3,5),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,11),(5,6),(5,7),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10),(9,11),(10,11)],12)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} + 1
00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,11),(0,12),(1,8),(1,12),(2,6),(2,7),(3,4),(3,11),(3,12),(4,5),(4,6),(5,9),(5,11),(6,9),(7,8),(7,9),(8,10),(9,10),(10,11),(10,12)],13)
=> ([(0,4),(0,5),(0,6),(0,7),(0,9),(0,10),(0,11),(0,12),(1,2),(1,3),(1,4),(1,6),(1,7),(1,8),(1,9),(1,12),(2,3),(2,6),(2,7),(2,8),(2,9),(2,11),(2,12),(3,5),(3,8),(3,9),(3,10),(3,11),(3,12),(4,5),(4,6),(4,7),(4,8),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,10),(5,11),(5,12),(6,7),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} + 1
00111 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,5),(0,9),(1,4),(1,9),(2,6),(2,8),(3,7),(3,8),(4,6),(4,10),(5,7),(5,10),(6,11),(7,11),(8,11),(9,10),(10,11)],12)
=> ([(0,3),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,2),(1,4),(1,5),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,10),(3,11),(4,5),(4,7),(4,8),(4,9),(4,11),(5,6),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} + 1
01000 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,7),(0,11),(1,5),(1,8),(2,6),(2,8),(3,4),(3,7),(3,11),(4,5),(4,9),(5,10),(6,10),(6,11),(7,9),(8,10),(9,10),(9,11)],12)
=> ([(0,3),(0,4),(0,5),(0,6),(0,9),(0,10),(0,11),(1,2),(1,4),(1,5),(1,7),(1,8),(1,10),(1,11),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,9),(5,10),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} + 1
01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,7),(0,11),(1,5),(1,6),(2,3),(2,10),(2,11),(3,5),(3,8),(4,6),(4,8),(4,9),(5,10),(6,10),(7,8),(7,9),(8,11),(9,10),(9,11)],12)
=> ([(0,2),(0,4),(0,5),(0,7),(0,8),(0,9),(0,11),(1,3),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,8),(2,10),(2,11),(3,6),(3,7),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,11),(5,6),(5,7),(5,8),(5,10),(5,11),(6,7),(6,9),(6,10),(6,11),(7,9),(7,10),(8,9),(8,10),(8,11),(9,10),(10,11)],12)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} + 1
01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ([(0,1),(0,4),(0,5),(0,8),(0,9),(1,4),(1,5),(1,8),(1,9),(2,3),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,7),(5,9),(6,7),(6,8),(7,8),(8,9)],10)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} + 1
01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,5),(0,6),(1,7),(1,10),(2,4),(2,10),(3,6),(3,8),(3,11),(4,5),(4,11),(5,9),(6,9),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ([(0,3),(0,7),(0,8),(0,9),(0,10),(0,11),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(1,11),(2,4),(2,5),(2,6),(2,9),(2,10),(2,11),(3,5),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,11),(5,6),(5,7),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10),(9,11),(10,11)],12)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} + 1
01100 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,11),(0,12),(1,8),(1,12),(2,6),(2,7),(3,4),(3,11),(3,12),(4,5),(4,6),(5,9),(5,11),(6,9),(7,8),(7,9),(8,10),(9,10),(10,11),(10,12)],13)
=> ([(0,4),(0,5),(0,6),(0,7),(0,9),(0,10),(0,11),(0,12),(1,2),(1,3),(1,4),(1,6),(1,7),(1,8),(1,9),(1,12),(2,3),(2,6),(2,7),(2,8),(2,9),(2,11),(2,12),(3,5),(3,8),(3,9),(3,10),(3,11),(3,12),(4,5),(4,6),(4,7),(4,8),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,10),(5,11),(5,12),(6,7),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} + 1
01101 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,7),(0,11),(1,5),(1,6),(2,3),(2,10),(2,11),(3,5),(3,8),(4,6),(4,8),(4,9),(5,10),(6,10),(7,8),(7,9),(8,11),(9,10),(9,11)],12)
=> ([(0,2),(0,4),(0,5),(0,7),(0,8),(0,9),(0,11),(1,3),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,8),(2,10),(2,11),(3,6),(3,7),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,11),(5,6),(5,7),(5,8),(5,10),(5,11),(6,7),(6,9),(6,10),(6,11),(7,9),(7,10),(8,9),(8,10),(8,11),(9,10),(10,11)],12)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} + 1
01110 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(0,9),(0,11),(1,7),(1,8),(2,5),(2,9),(2,11),(3,4),(3,9),(3,11),(4,7),(4,10),(5,8),(5,10),(6,7),(6,8),(6,10),(10,11)],12)
=> ([(0,2),(0,3),(0,4),(0,7),(0,8),(0,9),(0,11),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,10),(2,11),(3,4),(3,6),(3,7),(3,9),(3,10),(3,11),(4,5),(4,7),(4,8),(4,10),(4,11),(5,6),(5,8),(5,9),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,11),(8,9),(8,10),(9,10),(10,11)],12)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} + 1
01111 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,3),(0,7),(1,2),(1,6),(2,8),(3,9),(4,5),(4,8),(4,9),(5,6),(5,7),(6,8),(7,9)],10)
=> ([(0,1),(0,3),(0,5),(0,6),(0,7),(0,9),(1,2),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,6),(2,8),(2,9),(3,4),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} + 1
10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,3),(0,7),(1,2),(1,6),(2,8),(3,9),(4,5),(4,8),(4,9),(5,6),(5,7),(6,8),(7,9)],10)
=> ([(0,1),(0,3),(0,5),(0,6),(0,7),(0,9),(1,2),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,6),(2,8),(2,9),(3,4),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} + 1
10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(0,9),(0,11),(1,7),(1,8),(2,5),(2,9),(2,11),(3,4),(3,9),(3,11),(4,7),(4,10),(5,8),(5,10),(6,7),(6,8),(6,10),(10,11)],12)
=> ([(0,2),(0,3),(0,4),(0,7),(0,8),(0,9),(0,11),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,10),(2,11),(3,4),(3,6),(3,7),(3,9),(3,10),(3,11),(4,5),(4,7),(4,8),(4,10),(4,11),(5,6),(5,8),(5,9),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,11),(8,9),(8,10),(9,10),(10,11)],12)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} + 1
10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,7),(0,11),(1,5),(1,6),(2,3),(2,10),(2,11),(3,5),(3,8),(4,6),(4,8),(4,9),(5,10),(6,10),(7,8),(7,9),(8,11),(9,10),(9,11)],12)
=> ([(0,2),(0,4),(0,5),(0,7),(0,8),(0,9),(0,11),(1,3),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,8),(2,10),(2,11),(3,6),(3,7),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,11),(5,6),(5,7),(5,8),(5,10),(5,11),(6,7),(6,9),(6,10),(6,11),(7,9),(7,10),(8,9),(8,10),(8,11),(9,10),(10,11)],12)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} + 1
10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,11),(0,12),(1,8),(1,12),(2,6),(2,7),(3,4),(3,11),(3,12),(4,5),(4,6),(5,9),(5,11),(6,9),(7,8),(7,9),(8,10),(9,10),(10,11),(10,12)],13)
=> ([(0,4),(0,5),(0,6),(0,7),(0,9),(0,10),(0,11),(0,12),(1,2),(1,3),(1,4),(1,6),(1,7),(1,8),(1,9),(1,12),(2,3),(2,6),(2,7),(2,8),(2,9),(2,11),(2,12),(3,5),(3,8),(3,9),(3,10),(3,11),(3,12),(4,5),(4,6),(4,7),(4,8),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,10),(5,11),(5,12),(6,7),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} + 1
10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,5),(0,6),(1,7),(1,10),(2,4),(2,10),(3,6),(3,8),(3,11),(4,5),(4,11),(5,9),(6,9),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ([(0,3),(0,7),(0,8),(0,9),(0,10),(0,11),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(1,11),(2,4),(2,5),(2,6),(2,9),(2,10),(2,11),(3,5),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,11),(5,6),(5,7),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10),(9,11),(10,11)],12)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} + 1
10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ([(0,1),(0,4),(0,5),(0,8),(0,9),(1,4),(1,5),(1,8),(1,9),(2,3),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,7),(5,9),(6,7),(6,8),(7,8),(8,9)],10)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} + 1
10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,7),(0,11),(1,5),(1,6),(2,3),(2,10),(2,11),(3,5),(3,8),(4,6),(4,8),(4,9),(5,10),(6,10),(7,8),(7,9),(8,11),(9,10),(9,11)],12)
=> ([(0,2),(0,4),(0,5),(0,7),(0,8),(0,9),(0,11),(1,3),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,8),(2,10),(2,11),(3,6),(3,7),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,11),(5,6),(5,7),(5,8),(5,10),(5,11),(6,7),(6,9),(6,10),(6,11),(7,9),(7,10),(8,9),(8,10),(8,11),(9,10),(10,11)],12)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} + 1
10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,7),(0,11),(1,5),(1,8),(2,6),(2,8),(3,4),(3,7),(3,11),(4,5),(4,9),(5,10),(6,10),(6,11),(7,9),(8,10),(9,10),(9,11)],12)
=> ([(0,3),(0,4),(0,5),(0,6),(0,9),(0,10),(0,11),(1,2),(1,4),(1,5),(1,7),(1,8),(1,10),(1,11),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,9),(5,10),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} + 1
11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,5),(0,9),(1,4),(1,9),(2,6),(2,8),(3,7),(3,8),(4,6),(4,10),(5,7),(5,10),(6,11),(7,11),(8,11),(9,10),(10,11)],12)
=> ([(0,3),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,2),(1,4),(1,5),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,10),(3,11),(4,5),(4,7),(4,8),(4,9),(4,11),(5,6),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} + 1
11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,11),(0,12),(1,8),(1,12),(2,6),(2,7),(3,4),(3,11),(3,12),(4,5),(4,6),(5,9),(5,11),(6,9),(7,8),(7,9),(8,10),(9,10),(10,11),(10,12)],13)
=> ([(0,4),(0,5),(0,6),(0,7),(0,9),(0,10),(0,11),(0,12),(1,2),(1,3),(1,4),(1,6),(1,7),(1,8),(1,9),(1,12),(2,3),(2,6),(2,7),(2,8),(2,9),(2,11),(2,12),(3,5),(3,8),(3,9),(3,10),(3,11),(3,12),(4,5),(4,6),(4,7),(4,8),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,10),(5,11),(5,12),(6,7),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} + 1
11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,5),(0,6),(1,7),(1,10),(2,4),(2,10),(3,6),(3,8),(3,11),(4,5),(4,11),(5,9),(6,9),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ([(0,3),(0,7),(0,8),(0,9),(0,10),(0,11),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(1,11),(2,4),(2,5),(2,6),(2,9),(2,10),(2,11),(3,5),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,11),(5,6),(5,7),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10),(9,11),(10,11)],12)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} + 1
11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(0,7),(0,11),(1,5),(1,6),(2,3),(2,4),(2,11),(3,6),(3,9),(4,5),(4,8),(5,10),(6,10),(7,8),(7,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ([(0,3),(0,4),(0,5),(0,6),(0,8),(0,9),(0,11),(1,2),(1,5),(1,6),(1,7),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(2,10),(2,11),(3,4),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,11),(5,6),(5,7),(5,8),(5,10),(6,7),(6,9),(6,10),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} + 1
11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,5),(0,9),(1,4),(1,9),(2,6),(2,8),(3,7),(3,8),(4,6),(4,10),(5,7),(5,10),(6,11),(7,11),(8,11),(9,10),(10,11)],12)
=> ([(0,3),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,2),(1,4),(1,5),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,10),(3,11),(4,5),(4,7),(4,8),(4,9),(4,11),(5,6),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} + 1
11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,7),(0,11),(1,5),(1,8),(2,6),(2,8),(3,4),(3,7),(3,11),(4,5),(4,9),(5,10),(6,10),(6,11),(7,9),(8,10),(9,10),(9,11)],12)
=> ([(0,3),(0,4),(0,5),(0,6),(0,9),(0,10),(0,11),(1,2),(1,4),(1,5),(1,7),(1,8),(1,10),(1,11),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,9),(5,10),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} + 1
11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,3),(0,7),(1,2),(1,6),(2,8),(3,9),(4,5),(4,8),(4,9),(5,6),(5,7),(6,8),(7,9)],10)
=> ([(0,1),(0,3),(0,5),(0,6),(0,7),(0,9),(1,2),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,6),(2,8),(2,9),(3,4),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} + 1
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ([(0,3),(0,11),(1,2),(1,8),(2,9),(3,10),(4,5),(4,6),(4,8),(5,7),(5,9),(6,7),(6,10),(7,11),(8,9),(10,11)],12)
=> ([(0,2),(0,3),(0,5),(0,6),(0,7),(0,9),(0,10),(0,11),(1,2),(1,3),(1,4),(1,6),(1,7),(1,8),(1,10),(1,11),(2,4),(2,5),(2,6),(2,8),(2,9),(2,10),(3,4),(3,5),(3,7),(3,8),(3,9),(3,11),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,8),(5,9),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11)],12)
=> ? ∊ {3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6} + 1
000010 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ([(0,9),(0,14),(1,7),(1,10),(2,8),(2,10),(3,4),(3,9),(3,14),(4,5),(4,11),(5,7),(5,13),(6,8),(6,13),(6,14),(7,12),(8,12),(9,11),(10,12),(11,13),(11,14),(12,13)],15)
=> ([(0,3),(0,5),(0,6),(0,8),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(1,2),(1,4),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(1,13),(1,14),(2,3),(2,4),(2,6),(2,7),(2,8),(2,9),(2,10),(2,13),(2,14),(3,4),(3,5),(3,6),(3,7),(3,11),(3,12),(3,13),(3,14),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,13),(4,14),(5,6),(5,7),(5,8),(5,10),(5,11),(5,12),(5,13),(5,14),(6,8),(6,9),(6,11),(6,12),(6,13),(6,14),(7,8),(7,9),(7,10),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,14),(11,12),(11,13),(11,14),(12,13),(12,14),(13,14)],15)
=> ? ∊ {3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6} + 1
000011 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ([(0,7),(0,11),(1,6),(1,10),(2,8),(2,10),(3,9),(3,11),(4,8),(4,9),(4,14),(5,6),(5,7),(5,14),(6,12),(7,13),(8,12),(9,13),(10,12),(11,13),(12,14),(13,14)],15)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,11),(0,12),(0,13),(0,14),(1,2),(1,4),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(1,13),(1,14),(2,3),(2,5),(2,6),(2,9),(2,10),(2,11),(2,12),(2,13),(2,14),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(3,12),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,13),(4,14),(5,6),(5,7),(5,8),(5,10),(5,11),(5,12),(5,14),(6,7),(6,8),(6,9),(6,11),(6,12),(6,13),(7,8),(7,9),(7,11),(7,13),(7,14),(8,10),(8,12),(8,13),(8,14),(9,10),(9,11),(9,12),(9,13),(9,14),(10,11),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14),(12,13),(12,14),(13,14)],15)
=> ? ∊ {3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6} + 1
000100 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ([(0,7),(0,15),(1,5),(1,14),(2,6),(2,8),(3,4),(3,13),(3,15),(4,6),(4,10),(5,12),(5,13),(6,11),(7,8),(7,9),(8,11),(9,11),(9,12),(9,15),(10,11),(10,12),(10,13),(12,14),(13,14),(14,15)],16)
=> ([(0,1),(0,3),(0,4),(0,5),(0,7),(0,8),(0,9),(0,11),(0,13),(0,14),(0,15),(1,4),(1,5),(1,6),(1,7),(1,10),(1,11),(1,12),(1,13),(1,14),(1,15),(2,3),(2,5),(2,6),(2,8),(2,9),(2,10),(2,11),(2,12),(2,13),(2,14),(2,15),(3,6),(3,7),(3,8),(3,9),(3,10),(3,12),(3,13),(3,14),(3,15),(4,5),(4,7),(4,8),(4,9),(4,10),(4,11),(4,13),(4,14),(4,15),(5,7),(5,8),(5,9),(5,11),(5,12),(5,13),(5,15),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(7,9),(7,11),(7,12),(7,13),(7,14),(7,15),(8,9),(8,10),(8,11),(8,12),(8,14),(8,15),(9,10),(9,12),(9,14),(9,15),(10,11),(10,12),(10,13),(10,14),(10,15),(11,12),(11,13),(11,15),(12,13),(12,14),(13,14),(13,15),(14,15)],16)
=> ? ∊ {3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6} + 1
000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ([(0,5),(0,7),(1,6),(1,8),(2,9),(2,14),(3,4),(3,8),(3,13),(4,10),(4,15),(5,6),(5,11),(6,12),(7,11),(7,14),(8,12),(9,10),(9,15),(10,13),(10,14),(11,12),(11,15),(12,13),(13,15),(14,15)],16)
=> ([(0,4),(0,5),(0,8),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(1,11),(1,12),(1,13),(1,14),(1,15),(2,3),(2,5),(2,6),(2,7),(2,9),(2,10),(2,11),(2,12),(2,14),(2,15),(3,4),(3,6),(3,7),(3,10),(3,11),(3,12),(3,13),(3,14),(3,15),(4,5),(4,8),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(4,15),(5,6),(5,7),(5,8),(5,9),(5,10),(5,13),(5,14),(5,15),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(6,14),(6,15),(7,8),(7,9),(7,11),(7,12),(7,13),(7,14),(7,15),(8,9),(8,10),(8,11),(8,12),(8,13),(8,15),(9,10),(9,11),(9,13),(9,15),(10,12),(10,13),(10,14),(10,15),(11,12),(11,13),(11,14),(12,13),(12,14),(13,14),(13,15),(14,15)],16)
=> ? ∊ {3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6} + 1
000110 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ([(0,15),(0,16),(1,7),(1,10),(2,8),(2,9),(3,4),(3,5),(3,6),(4,15),(4,16),(5,11),(5,15),(6,8),(6,11),(7,14),(7,16),(8,13),(9,10),(9,13),(10,14),(11,12),(11,13),(12,14),(12,15),(12,16),(13,14)],17)
=> ([(0,3),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(0,16),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,12),(1,13),(1,14),(1,15),(1,16),(2,4),(2,5),(2,7),(2,8),(2,10),(2,11),(2,12),(2,13),(2,14),(2,15),(2,16),(3,4),(3,5),(3,6),(3,9),(3,10),(3,11),(3,12),(3,13),(3,14),(3,15),(3,16),(4,5),(4,6),(4,7),(4,8),(4,9),(4,11),(4,13),(4,15),(4,16),(5,7),(5,8),(5,9),(5,10),(5,11),(5,13),(5,15),(5,16),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(6,16),(7,8),(7,9),(7,10),(7,12),(7,13),(7,14),(7,15),(8,10),(8,11),(8,12),(8,13),(8,14),(8,15),(9,10),(9,11),(9,12),(9,13),(9,14),(9,16),(10,11),(10,12),(10,14),(10,15),(10,16),(11,12),(11,14),(11,15),(11,16),(12,14),(12,15),(12,16),(13,14),(13,15),(13,16),(14,15),(14,16),(15,16)],17)
=> ? ∊ {3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6} + 1
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
Description
The game chromatic number of a graph.
Two players, Alice and Bob, take turns colouring properly any uncolored vertex of the graph. Alice begins. If it is not possible for either player to colour a vertex, then Bob wins. If the graph is completely colored, Alice wins.
The game chromatic number is the smallest number of colours such that Alice has a winning strategy.
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