Your data matches 7 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000393
(load all 14 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000393: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => 10 => 2
1 => [1,1] => 11 => 2
00 => [3] => 100 => 3
01 => [2,1] => 101 => 2
10 => [1,2] => 110 => 3
11 => [1,1,1] => 111 => 3
000 => [4] => 1000 => 4
001 => [3,1] => 1001 => 3
010 => [2,2] => 1010 => 3
011 => [2,1,1] => 1011 => 3
100 => [1,3] => 1100 => 4
101 => [1,2,1] => 1101 => 3
110 => [1,1,2] => 1110 => 4
111 => [1,1,1,1] => 1111 => 4
0000 => [5] => 10000 => 5
0001 => [4,1] => 10001 => 4
0010 => [3,2] => 10010 => 4
0011 => [3,1,1] => 10011 => 4
0100 => [2,3] => 10100 => 4
0101 => [2,2,1] => 10101 => 3
0110 => [2,1,2] => 10110 => 4
0111 => [2,1,1,1] => 10111 => 4
1000 => [1,4] => 11000 => 5
1001 => [1,3,1] => 11001 => 4
1010 => [1,2,2] => 11010 => 4
1011 => [1,2,1,1] => 11011 => 4
1100 => [1,1,3] => 11100 => 5
1101 => [1,1,2,1] => 11101 => 4
1110 => [1,1,1,2] => 11110 => 5
1111 => [1,1,1,1,1] => 11111 => 5
00000 => [6] => 100000 => 6
00001 => [5,1] => 100001 => 5
00010 => [4,2] => 100010 => 5
00011 => [4,1,1] => 100011 => 5
00100 => [3,3] => 100100 => 5
00101 => [3,2,1] => 100101 => 4
00110 => [3,1,2] => 100110 => 5
00111 => [3,1,1,1] => 100111 => 5
01000 => [2,4] => 101000 => 5
01001 => [2,3,1] => 101001 => 4
01010 => [2,2,2] => 101010 => 4
01011 => [2,2,1,1] => 101011 => 4
01100 => [2,1,3] => 101100 => 5
01101 => [2,1,2,1] => 101101 => 4
01110 => [2,1,1,2] => 101110 => 5
01111 => [2,1,1,1,1] => 101111 => 5
10000 => [1,5] => 110000 => 6
10001 => [1,4,1] => 110001 => 5
10010 => [1,3,2] => 110010 => 5
10011 => [1,3,1,1] => 110011 => 5
Description
The number of strictly increasing runs in a binary word.
Matching statistic: St000863
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000863: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => [1,2] => 2
1 => [1,1] => [1,0,1,0]
 => [2,1] => 2
00 => [3] => [1,1,1,0,0,0]
 => [1,2,3] => 3
01 => [2,1] => [1,1,0,0,1,0]
 => [1,3,2] => 2
10 => [1,2] => [1,0,1,1,0,0]
 => [2,1,3] => 3
11 => [1,1,1] => [1,0,1,0,1,0]
 => [2,3,1] => 3
000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 4
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 3
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 3
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => 3
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 4
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 3
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 4
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => 4
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 5
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => 4
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => 4
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => 4
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => 4
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => 3
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => 4
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => 4
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => 5
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => 4
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => 4
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => 4
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => 5
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => 4
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => 5
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => 5
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => 6
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,2,3,4,6,5] => 5
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,2,3,5,4,6] => 5
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,2,3,5,6,4] => 5
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,2,4,3,5,6] => 5
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,2,4,3,6,5] => 4
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,2,4,5,3,6] => 5
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,2,4,5,6,3] => 5
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,3,2,4,5,6] => 5
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,3,2,4,6,5] => 4
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4,6] => 4
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,3,2,5,6,4] => 4
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,3,4,2,5,6] => 5
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,3,4,2,6,5] => 4
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,3,4,5,2,6] => 5
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,3,4,5,6,2] => 5
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,1,3,4,5,6] => 6
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [2,1,3,4,6,5] => 5
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [2,1,3,5,4,6] => 5
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [2,1,3,5,6,4] => 5
Description
The length of the first row of the shifted shape of a permutation. The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing. The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled. This statistic records the length of the first row of $P$ and $Q$.
Matching statistic: St001237
(load all 8 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001237: Dyck paths ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 86%
Values
0 => [2] => [1,1,0,0]
 => 3 = 2 + 1
1 => [1,1] => [1,0,1,0]
 => 3 = 2 + 1
00 => [3] => [1,1,1,0,0,0]
 => 4 = 3 + 1
01 => [2,1] => [1,1,0,0,1,0]
 => 3 = 2 + 1
10 => [1,2] => [1,0,1,1,0,0]
 => 4 = 3 + 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => 4 = 3 + 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 4 = 3 + 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 5 = 4 + 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 5 = 4 + 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 5 = 4 + 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 5 = 4 + 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 5 = 4 + 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 4 = 3 + 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 5 = 4 + 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 5 = 4 + 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 6 = 5 + 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 5 = 4 + 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 5 = 4 + 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 5 = 4 + 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 6 = 5 + 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 5 = 4 + 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 6 = 5 + 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 7 = 6 + 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 6 = 5 + 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 6 = 5 + 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 6 = 5 + 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 6 = 5 + 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 5 = 4 + 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 6 = 5 + 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 6 = 5 + 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 6 = 5 + 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 5 = 4 + 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 5 = 4 + 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 5 = 4 + 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 6 = 5 + 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 5 = 4 + 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 6 = 5 + 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 7 = 6 + 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 6 = 5 + 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 6 = 5 + 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 6 = 5 + 1
000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 7 + 1
000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 6 + 1
000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 6 + 1
000011 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 6 + 1
000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 6 + 1
000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => ? = 5 + 1
000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => ? = 6 + 1
000111 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 6 + 1
001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 6 + 1
001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 5 + 1
001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 5 + 1
001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 5 + 1
001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 6 + 1
001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 5 + 1
001110 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 6 + 1
001111 => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 6 + 1
010000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 6 + 1
010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 5 + 1
010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 5 + 1
010011 => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 5 + 1
010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 5 + 1
010101 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 4 + 1
010110 => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 5 + 1
010111 => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 5 + 1
011000 => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 6 + 1
011001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 5 + 1
011010 => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 5 + 1
011011 => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 5 + 1
011100 => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 6 + 1
011101 => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 5 + 1
011110 => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6 + 1
011111 => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6 + 1
Description
The number of simple modules with injective dimension at most one or dominant dimension at least one.
Matching statistic: St001330
Mapping
Mp00262: Binary words —poset of factors⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 86%
Values
0 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 2
1 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 2
00 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3
11 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
000 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
111 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,7),(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)
 => ? = 4
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)
 => ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
 => ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4
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)
 => ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
 => ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4
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)
 => ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
 => ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4
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)
 => ([(2,7),(3,6),(4,5)],8)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
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)
 => ([(2,5),(3,4),(3,7),(4,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ([(0,3),(0,4),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,7),(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)
 => ? = 4
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,7),(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)
 => ? = 5
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)
 => ([(2,5),(3,4),(3,7),(4,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ([(0,3),(0,4),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4
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)
 => ([(2,7),(3,6),(4,5)],8)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
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)
 => ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
 => ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4
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)
 => ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
 => ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 5
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)
 => ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
 => ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,7),(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)
 => ? = 5
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ([(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
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)
 => ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
 => ([(0,3),(0,5),(0,7),(0,8),(0,9),(1,2),(1,4),(1,6),(1,8),(1,9),(2,4),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 5
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)
 => ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
 => ([(0,3),(0,7),(0,9),(0,10),(0,11),(1,2),(1,6),(1,8),(1,9),(1,10),(1,11),(2,5),(2,6),(2,8),(2,9),(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),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(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)
 => ? = 5
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)
 => ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
 => ([(0,3),(0,7),(0,9),(0,10),(0,11),(1,2),(1,6),(1,8),(1,10),(1,11),(2,4),(2,6),(2,8),(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),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(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)
 => ? = 5
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)
 => ([(2,5),(3,4),(3,10),(4,9),(5,11),(6,9),(6,10),(6,11),(7,8),(7,9),(7,11),(8,10),(8,11),(9,10)],12)
 => ([(0,1),(0,2),(0,6),(0,7),(0,9),(0,10),(0,11),(1,4),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,6),(2,8),(2,9),(2,10),(2,11),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(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),(10,11)],12)
 => ? = 5
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)
 => ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
 => ([(0,2),(0,8),(0,9),(0,10),(0,11),(1,4),(1,5),(1,6),(1,7),(1,9),(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),(3,10),(3,11),(4,5),(4,6),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,9),(5,10),(5,11),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
 => ? = 4
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)
 => ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
 => ([(0,4),(0,8),(0,10),(0,11),(0,12),(1,3),(1,7),(1,9),(1,10),(1,11),(1,12),(2,3),(2,5),(2,7),(2,8),(2,9),(2,11),(2,12),(3,7),(3,9),(3,10),(3,11),(3,12),(4,6),(4,8),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
 => ? = 5
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)
 => ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
 => ([(0,3),(0,7),(0,9),(0,10),(0,11),(1,2),(1,6),(1,8),(1,10),(1,11),(2,4),(2,6),(2,8),(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),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(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)
 => ? = 5
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)
 => ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
 => ([(0,3),(0,7),(0,9),(0,10),(0,11),(1,2),(1,6),(1,8),(1,9),(1,10),(1,11),(2,5),(2,6),(2,8),(2,9),(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),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(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)
 => ? = 5
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)
 => ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
 => ([(0,1),(0,4),(0,6),(0,7),(0,9),(0,10),(0,11),(1,3),(1,6),(1,8),(1,9),(1,10),(1,11),(2,3),(2,5),(2,7),(2,8),(2,9),(2,10),(2,11),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(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),(10,11)],12)
 => ? = 4
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)
 => ([(2,9),(3,8),(4,7),(5,6)],10)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 4
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)
 => ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
 => ([(0,2),(0,8),(0,9),(0,10),(0,11),(1,4),(1,5),(1,6),(1,7),(1,9),(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),(3,10),(3,11),(4,5),(4,6),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,9),(5,10),(5,11),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
 => ? = 4
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)
 => ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
 => ([(0,4),(0,8),(0,10),(0,11),(0,12),(1,3),(1,7),(1,9),(1,10),(1,11),(1,12),(2,3),(2,5),(2,7),(2,8),(2,9),(2,11),(2,12),(3,7),(3,9),(3,10),(3,11),(3,12),(4,6),(4,8),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
 => ? = 5
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)
 => ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
 => ([(0,1),(0,4),(0,6),(0,7),(0,9),(0,10),(0,11),(1,3),(1,6),(1,8),(1,9),(1,10),(1,11),(2,3),(2,5),(2,7),(2,8),(2,9),(2,10),(2,11),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(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),(10,11)],12)
 => ? = 4
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)
 => ([(2,6),(3,6),(3,9),(3,10),(4,5),(4,8),(4,10),(5,7),(5,9),(6,11),(7,8),(7,10),(7,11),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
 => ([(0,5),(0,6),(0,7),(0,9),(0,10),(0,11),(1,4),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,6),(2,8),(2,9),(2,10),(2,11),(3,5),(3,6),(3,8),(3,9),(3,10),(3,11),(4,5),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(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),(10,11)],12)
 => ? = 5
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)
 => ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
 => ([(0,3),(0,5),(0,7),(0,8),(0,9),(1,2),(1,4),(1,6),(1,8),(1,9),(2,4),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 5
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)
 => ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
 => ([(0,3),(0,5),(0,7),(0,8),(0,9),(1,2),(1,4),(1,6),(1,8),(1,9),(2,4),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 6
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)
 => ([(2,6),(3,6),(3,9),(3,10),(4,5),(4,8),(4,10),(5,7),(5,9),(6,11),(7,8),(7,10),(7,11),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
 => ([(0,5),(0,6),(0,7),(0,9),(0,10),(0,11),(1,4),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,6),(2,8),(2,9),(2,10),(2,11),(3,5),(3,6),(3,8),(3,9),(3,10),(3,11),(4,5),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(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),(10,11)],12)
 => ? = 5
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)
 => ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
 => ([(0,1),(0,4),(0,6),(0,7),(0,9),(0,10),(0,11),(1,3),(1,6),(1,8),(1,9),(1,10),(1,11),(2,3),(2,5),(2,7),(2,8),(2,9),(2,10),(2,11),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(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),(10,11)],12)
 => ? = 5
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)
 => ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
 => ([(0,4),(0,8),(0,10),(0,11),(0,12),(1,3),(1,7),(1,9),(1,10),(1,11),(1,12),(2,3),(2,5),(2,7),(2,8),(2,9),(2,11),(2,12),(3,7),(3,9),(3,10),(3,11),(3,12),(4,6),(4,8),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
 => ? = 5
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)
 => ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
 => ([(0,2),(0,8),(0,9),(0,10),(0,11),(1,4),(1,5),(1,6),(1,7),(1,9),(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),(3,10),(3,11),(4,5),(4,6),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,9),(5,10),(5,11),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
 => ? = 5
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)
 => ([(2,9),(3,8),(4,7),(5,6)],10)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 4
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)
 => ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
 => ([(0,1),(0,4),(0,6),(0,7),(0,9),(0,10),(0,11),(1,3),(1,6),(1,8),(1,9),(1,10),(1,11),(2,3),(2,5),(2,7),(2,8),(2,9),(2,10),(2,11),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(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),(10,11)],12)
 => ? = 5
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)
 => ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
 => ([(0,3),(0,7),(0,9),(0,10),(0,11),(1,2),(1,6),(1,8),(1,9),(1,10),(1,11),(2,5),(2,6),(2,8),(2,9),(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),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(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)
 => ? = 5
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)
 => ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
 => ([(0,3),(0,7),(0,9),(0,10),(0,11),(1,2),(1,6),(1,8),(1,10),(1,11),(2,4),(2,6),(2,8),(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),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(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)
 => ? = 6
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)
 => ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
 => ([(0,4),(0,8),(0,10),(0,11),(0,12),(1,3),(1,7),(1,9),(1,10),(1,11),(1,12),(2,3),(2,5),(2,7),(2,8),(2,9),(2,11),(2,12),(3,7),(3,9),(3,10),(3,11),(3,12),(4,6),(4,8),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
 => ? = 5
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)
 => ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
 => ([(0,2),(0,8),(0,9),(0,10),(0,11),(1,4),(1,5),(1,6),(1,7),(1,9),(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),(3,10),(3,11),(4,5),(4,6),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,9),(5,10),(5,11),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
 => ? = 5
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)
 => ([(2,5),(3,4),(3,10),(4,9),(5,11),(6,9),(6,10),(6,11),(7,8),(7,9),(7,11),(8,10),(8,11),(9,10)],12)
 => ([(0,1),(0,2),(0,6),(0,7),(0,9),(0,10),(0,11),(1,4),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,6),(2,8),(2,9),(2,10),(2,11),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(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),(10,11)],12)
 => ? = 5
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)
 => ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
 => ([(0,3),(0,7),(0,9),(0,10),(0,11),(1,2),(1,6),(1,8),(1,10),(1,11),(2,4),(2,6),(2,8),(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),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(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)
 => ? = 6
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ([(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
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => ([(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
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.
Matching statistic: St000454
Mapping
Mp00262: Binary words —poset of factors⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 86%
Values
0 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
1 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
00 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 - 1
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 - 1
11 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
000 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 1
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 1
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 1
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 1
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 1
111 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,7),(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)
 => ? = 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)
 => ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
 => ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 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)
 => ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
 => ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 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)
 => ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
 => ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 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)
 => ([(2,7),(3,6),(4,5)],8)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 - 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)
 => ([(2,5),(3,4),(3,7),(4,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ([(0,3),(0,4),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 - 1
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,7),(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)
 => ? = 4 - 1
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,7),(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)
 => ? = 5 - 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)
 => ([(2,5),(3,4),(3,7),(4,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ([(0,3),(0,4),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 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)
 => ([(2,7),(3,6),(4,5)],8)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 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)
 => ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
 => ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 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)
 => ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
 => ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 5 - 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)
 => ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
 => ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 - 1
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,7),(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)
 => ? = 5 - 1
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ([(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)
 => 5 = 6 - 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)
 => ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
 => ([(0,3),(0,5),(0,7),(0,8),(0,9),(1,2),(1,4),(1,6),(1,8),(1,9),(2,4),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 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)
 => ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
 => ([(0,3),(0,7),(0,9),(0,10),(0,11),(1,2),(1,6),(1,8),(1,9),(1,10),(1,11),(2,5),(2,6),(2,8),(2,9),(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),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(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)
 => ? = 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)
 => ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
 => ([(0,3),(0,7),(0,9),(0,10),(0,11),(1,2),(1,6),(1,8),(1,10),(1,11),(2,4),(2,6),(2,8),(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),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(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)
 => ? = 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)
 => ([(2,5),(3,4),(3,10),(4,9),(5,11),(6,9),(6,10),(6,11),(7,8),(7,9),(7,11),(8,10),(8,11),(9,10)],12)
 => ([(0,1),(0,2),(0,6),(0,7),(0,9),(0,10),(0,11),(1,4),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,6),(2,8),(2,9),(2,10),(2,11),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(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),(10,11)],12)
 => ? = 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)
 => ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
 => ([(0,2),(0,8),(0,9),(0,10),(0,11),(1,4),(1,5),(1,6),(1,7),(1,9),(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),(3,10),(3,11),(4,5),(4,6),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,9),(5,10),(5,11),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
 => ? = 4 - 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)
 => ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
 => ([(0,4),(0,8),(0,10),(0,11),(0,12),(1,3),(1,7),(1,9),(1,10),(1,11),(1,12),(2,3),(2,5),(2,7),(2,8),(2,9),(2,11),(2,12),(3,7),(3,9),(3,10),(3,11),(3,12),(4,6),(4,8),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
 => ? = 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)
 => ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
 => ([(0,3),(0,7),(0,9),(0,10),(0,11),(1,2),(1,6),(1,8),(1,10),(1,11),(2,4),(2,6),(2,8),(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),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(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)
 => ? = 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)
 => ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
 => ([(0,3),(0,7),(0,9),(0,10),(0,11),(1,2),(1,6),(1,8),(1,9),(1,10),(1,11),(2,5),(2,6),(2,8),(2,9),(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),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(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)
 => ? = 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)
 => ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
 => ([(0,1),(0,4),(0,6),(0,7),(0,9),(0,10),(0,11),(1,3),(1,6),(1,8),(1,9),(1,10),(1,11),(2,3),(2,5),(2,7),(2,8),(2,9),(2,10),(2,11),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(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),(10,11)],12)
 => ? = 4 - 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)
 => ([(2,9),(3,8),(4,7),(5,6)],10)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 4 - 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)
 => ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
 => ([(0,2),(0,8),(0,9),(0,10),(0,11),(1,4),(1,5),(1,6),(1,7),(1,9),(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),(3,10),(3,11),(4,5),(4,6),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,9),(5,10),(5,11),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
 => ? = 4 - 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)
 => ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
 => ([(0,4),(0,8),(0,10),(0,11),(0,12),(1,3),(1,7),(1,9),(1,10),(1,11),(1,12),(2,3),(2,5),(2,7),(2,8),(2,9),(2,11),(2,12),(3,7),(3,9),(3,10),(3,11),(3,12),(4,6),(4,8),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
 => ? = 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)
 => ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
 => ([(0,1),(0,4),(0,6),(0,7),(0,9),(0,10),(0,11),(1,3),(1,6),(1,8),(1,9),(1,10),(1,11),(2,3),(2,5),(2,7),(2,8),(2,9),(2,10),(2,11),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(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),(10,11)],12)
 => ? = 4 - 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)
 => ([(2,6),(3,6),(3,9),(3,10),(4,5),(4,8),(4,10),(5,7),(5,9),(6,11),(7,8),(7,10),(7,11),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
 => ([(0,5),(0,6),(0,7),(0,9),(0,10),(0,11),(1,4),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,6),(2,8),(2,9),(2,10),(2,11),(3,5),(3,6),(3,8),(3,9),(3,10),(3,11),(4,5),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(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),(10,11)],12)
 => ? = 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)
 => ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
 => ([(0,3),(0,5),(0,7),(0,8),(0,9),(1,2),(1,4),(1,6),(1,8),(1,9),(2,4),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 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)
 => ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
 => ([(0,3),(0,5),(0,7),(0,8),(0,9),(1,2),(1,4),(1,6),(1,8),(1,9),(2,4),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 6 - 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)
 => ([(2,6),(3,6),(3,9),(3,10),(4,5),(4,8),(4,10),(5,7),(5,9),(6,11),(7,8),(7,10),(7,11),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
 => ([(0,5),(0,6),(0,7),(0,9),(0,10),(0,11),(1,4),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,6),(2,8),(2,9),(2,10),(2,11),(3,5),(3,6),(3,8),(3,9),(3,10),(3,11),(4,5),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(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),(10,11)],12)
 => ? = 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)
 => ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
 => ([(0,1),(0,4),(0,6),(0,7),(0,9),(0,10),(0,11),(1,3),(1,6),(1,8),(1,9),(1,10),(1,11),(2,3),(2,5),(2,7),(2,8),(2,9),(2,10),(2,11),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(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),(10,11)],12)
 => ? = 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)
 => ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
 => ([(0,4),(0,8),(0,10),(0,11),(0,12),(1,3),(1,7),(1,9),(1,10),(1,11),(1,12),(2,3),(2,5),(2,7),(2,8),(2,9),(2,11),(2,12),(3,7),(3,9),(3,10),(3,11),(3,12),(4,6),(4,8),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
 => ? = 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)
 => ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
 => ([(0,2),(0,8),(0,9),(0,10),(0,11),(1,4),(1,5),(1,6),(1,7),(1,9),(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),(3,10),(3,11),(4,5),(4,6),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,9),(5,10),(5,11),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
 => ? = 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)
 => ([(2,9),(3,8),(4,7),(5,6)],10)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 4 - 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)
 => ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
 => ([(0,1),(0,4),(0,6),(0,7),(0,9),(0,10),(0,11),(1,3),(1,6),(1,8),(1,9),(1,10),(1,11),(2,3),(2,5),(2,7),(2,8),(2,9),(2,10),(2,11),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(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),(10,11)],12)
 => ? = 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)
 => ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
 => ([(0,3),(0,7),(0,9),(0,10),(0,11),(1,2),(1,6),(1,8),(1,9),(1,10),(1,11),(2,5),(2,6),(2,8),(2,9),(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),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(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)
 => ? = 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)
 => ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
 => ([(0,3),(0,7),(0,9),(0,10),(0,11),(1,2),(1,6),(1,8),(1,10),(1,11),(2,4),(2,6),(2,8),(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),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(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)
 => ? = 6 - 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)
 => ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
 => ([(0,4),(0,8),(0,10),(0,11),(0,12),(1,3),(1,7),(1,9),(1,10),(1,11),(1,12),(2,3),(2,5),(2,7),(2,8),(2,9),(2,11),(2,12),(3,7),(3,9),(3,10),(3,11),(3,12),(4,6),(4,8),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
 => ? = 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)
 => ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
 => ([(0,2),(0,8),(0,9),(0,10),(0,11),(1,4),(1,5),(1,6),(1,7),(1,9),(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),(3,10),(3,11),(4,5),(4,6),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,9),(5,10),(5,11),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
 => ? = 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)
 => ([(2,5),(3,4),(3,10),(4,9),(5,11),(6,9),(6,10),(6,11),(7,8),(7,9),(7,11),(8,10),(8,11),(9,10)],12)
 => ([(0,1),(0,2),(0,6),(0,7),(0,9),(0,10),(0,11),(1,4),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,6),(2,8),(2,9),(2,10),(2,11),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(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),(10,11)],12)
 => ? = 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)
 => ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
 => ([(0,3),(0,7),(0,9),(0,10),(0,11),(1,2),(1,6),(1,8),(1,10),(1,11),(2,4),(2,6),(2,8),(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),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(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)
 => ? = 6 - 1
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ([(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)
 => 5 = 6 - 1
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => ([(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)
 => 6 = 7 - 1
Description
The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001880
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 71%
Values
0 => [2] => [[2],[]]
 => ([(0,1)],2)
 => ? = 2
1 => [1,1] => [[1,1],[]]
 => ([(0,1)],2)
 => ? = 2
00 => [3] => [[3],[]]
 => ([(0,2),(2,1)],3)
 => 3
01 => [2,1] => [[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => ? = 2
10 => [1,2] => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => ? = 3
11 => [1,1,1] => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 3
000 => [4] => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
001 => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3
010 => [2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ? = 3
011 => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3
100 => [1,3] => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ? = 4
101 => [1,2,1] => [[2,2,1],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ? = 3
110 => [1,1,2] => [[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ? = 4
111 => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
0000 => [5] => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
0001 => [4,1] => [[4,4],[3]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 4
0010 => [3,2] => [[4,3],[2]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ? = 4
0011 => [3,1,1] => [[3,3,3],[2,2]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 4
0100 => [2,3] => [[4,2],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4
0101 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3
0110 => [2,1,2] => [[3,2,2],[1,1]]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ? = 4
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 4
1000 => [1,4] => [[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ? = 5
1001 => [1,3,1] => [[3,3,1],[2]]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ? = 4
1010 => [1,2,2] => [[3,2,1],[1]]
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => ? = 4
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ? = 4
1100 => [1,1,3] => [[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ? = 5
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4
1110 => [1,1,1,2] => [[2,1,1,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ? = 5
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
00000 => [6] => [[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
00001 => [5,1] => [[5,5],[4]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 5
00010 => [4,2] => [[5,4],[3]]
 => ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
 => ? = 5
00011 => [4,1,1] => [[4,4,4],[3,3]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 5
00100 => [3,3] => [[5,3],[2]]
 => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
 => ? = 5
00101 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => ? = 4
00110 => [3,1,2] => [[4,3,3],[2,2]]
 => ([(0,4),(1,2),(1,3),(3,5),(4,5)],6)
 => ? = 5
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 5
01000 => [2,4] => [[5,2],[1]]
 => ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
 => ? = 5
01001 => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 4
01010 => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
 => ? = 4
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => ? = 4
01100 => [2,1,3] => [[4,2,2],[1,1]]
 => ([(0,5),(1,3),(1,4),(3,5),(4,2)],6)
 => ? = 5
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 4
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => ([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
 => ? = 5
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 5
10000 => [1,5] => [[5,1],[]]
 => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => ? = 6
10001 => [1,4,1] => [[4,4,1],[3]]
 => ([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
 => ? = 5
10010 => [1,3,2] => [[4,3,1],[2]]
 => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
 => ? = 5
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
 => ([(0,4),(1,2),(1,3),(3,5),(4,5)],6)
 => ? = 5
10100 => [1,2,3] => [[4,2,1],[1]]
 => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
 => ? = 5
10101 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
 => ? = 4
10110 => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
 => ? = 5
10111 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
 => ? = 5
11000 => [1,1,4] => [[4,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => ? = 6
11001 => [1,1,3,1] => [[3,3,1,1],[2]]
 => ([(0,5),(1,3),(1,4),(3,5),(4,2)],6)
 => ? = 5
11010 => [1,1,2,2] => [[3,2,1,1],[1]]
 => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
 => ? = 5
11111 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
000000 => [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: St001879
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 71%
Values
0 => [2] => [[2],[]]
 => ([(0,1)],2)
 => ? = 2 - 1
1 => [1,1] => [[1,1],[]]
 => ([(0,1)],2)
 => ? = 2 - 1
00 => [3] => [[3],[]]
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
01 => [2,1] => [[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => ? = 2 - 1
10 => [1,2] => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => ? = 3 - 1
11 => [1,1,1] => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
000 => [4] => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
001 => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 - 1
010 => [2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ? = 3 - 1
011 => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 - 1
100 => [1,3] => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ? = 4 - 1
101 => [1,2,1] => [[2,2,1],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ? = 3 - 1
110 => [1,1,2] => [[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ? = 4 - 1
111 => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
0000 => [5] => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
0001 => [4,1] => [[4,4],[3]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 4 - 1
0010 => [3,2] => [[4,3],[2]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ? = 4 - 1
0011 => [3,1,1] => [[3,3,3],[2,2]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 4 - 1
0100 => [2,3] => [[4,2],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 1
0101 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 - 1
0110 => [2,1,2] => [[3,2,2],[1,1]]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ? = 4 - 1
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 4 - 1
1000 => [1,4] => [[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ? = 5 - 1
1001 => [1,3,1] => [[3,3,1],[2]]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ? = 4 - 1
1010 => [1,2,2] => [[3,2,1],[1]]
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => ? = 4 - 1
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ? = 4 - 1
1100 => [1,1,3] => [[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ? = 5 - 1
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 1
1110 => [1,1,1,2] => [[2,1,1,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ? = 5 - 1
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
00000 => [6] => [[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
00001 => [5,1] => [[5,5],[4]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 5 - 1
00010 => [4,2] => [[5,4],[3]]
 => ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
 => ? = 5 - 1
00011 => [4,1,1] => [[4,4,4],[3,3]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 5 - 1
00100 => [3,3] => [[5,3],[2]]
 => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
 => ? = 5 - 1
00101 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => ? = 4 - 1
00110 => [3,1,2] => [[4,3,3],[2,2]]
 => ([(0,4),(1,2),(1,3),(3,5),(4,5)],6)
 => ? = 5 - 1
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 5 - 1
01000 => [2,4] => [[5,2],[1]]
 => ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
 => ? = 5 - 1
01001 => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 4 - 1
01010 => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
 => ? = 4 - 1
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => ? = 4 - 1
01100 => [2,1,3] => [[4,2,2],[1,1]]
 => ([(0,5),(1,3),(1,4),(3,5),(4,2)],6)
 => ? = 5 - 1
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 4 - 1
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => ([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
 => ? = 5 - 1
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 5 - 1
10000 => [1,5] => [[5,1],[]]
 => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => ? = 6 - 1
10001 => [1,4,1] => [[4,4,1],[3]]
 => ([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
 => ? = 5 - 1
10010 => [1,3,2] => [[4,3,1],[2]]
 => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
 => ? = 5 - 1
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
 => ([(0,4),(1,2),(1,3),(3,5),(4,5)],6)
 => ? = 5 - 1
10100 => [1,2,3] => [[4,2,1],[1]]
 => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
 => ? = 5 - 1
10101 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
 => ? = 4 - 1
10110 => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
 => ? = 5 - 1
10111 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
 => ? = 5 - 1
11000 => [1,1,4] => [[4,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => ? = 6 - 1
11001 => [1,1,3,1] => [[3,3,1,1],[2]]
 => ([(0,5),(1,3),(1,4),(3,5),(4,2)],6)
 => ? = 5 - 1
11010 => [1,1,2,2] => [[3,2,1,1],[1]]
 => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
 => ? = 5 - 1
11111 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
000000 => [7] => [[7],[]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 7 - 1
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.