Your data matches 4 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001437
Mapping
St001437: 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 => 2
11 => 2
000 => 3
001 => 1
010 => 2
011 => 1
100 => 3
101 => 2
110 => 3
111 => 3
0000 => 4
0001 => 1
0010 => 2
0011 => 1
0100 => 3
0101 => 2
0110 => 2
0111 => 1
1000 => 4
1001 => 3
1010 => 4
1011 => 2
1100 => 4
1101 => 3
1110 => 4
1111 => 4
00000 => 5
00001 => 1
00010 => 2
00011 => 1
00100 => 3
00101 => 1
00110 => 2
00111 => 1
01000 => 4
01001 => 2
01010 => 3
01011 => 1
01100 => 3
01101 => 2
01110 => 2
01111 => 1
10000 => 5
10001 => 4
10010 => 4
10011 => 3
Description
The flex of a binary word. This is the product of the lex statistic ([[St001436]], augmented by 1) and its frequency ([[St000627]]), see [1, §8].
Matching statistic: St000771
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00117: Graphs —Ore closure⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 67%
Values
0 => [2] => ([],2)
 => ([],2)
 => ? = 1
1 => [1,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
00 => [3] => ([],3)
 => ([],3)
 => ? ∊ {2,2}
01 => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
10 => [1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2}
11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
000 => [4] => ([],4)
 => ([],4)
 => ? ∊ {1,2,3,3}
001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 2
010 => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? ∊ {1,2,3,3}
011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
100 => [1,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => ? ∊ {1,2,3,3}
101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,2,3,3}
111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
0000 => [5] => ([],5)
 => ([],5)
 => ? ∊ {1,1,2,3,3,4,4,4}
0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,2,3,3,4,4,4}
0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
0100 => [2,3] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,2,3,3,4,4,4}
0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,2,3,3,4,4,4}
0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
1000 => [1,4] => ([(3,4)],5)
 => ([(3,4)],5)
 => ? ∊ {1,1,2,3,3,4,4,4}
1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,2,3,3,4,4,4}
1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,2,3,3,4,4,4}
1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,2,3,3,4,4,4}
1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
00000 => [6] => ([],6)
 => ([],6)
 => ? ∊ {1,1,1,1,1,2,2,3,3,4,4,4,4,4,5,5}
00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 4
00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,2,2,3,3,4,4,4,4,4,5,5}
00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
00100 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,2,2,3,3,4,4,4,4,4,5,5}
00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
00110 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,2,2,3,3,4,4,4,4,4,5,5}
00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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
01000 => [2,4] => ([(3,5),(4,5)],6)
 => ([(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,2,2,3,3,4,4,4,4,4,5,5}
01001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
01010 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,2,2,3,3,4,4,4,4,4,5,5}
01011 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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
01100 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,2,2,3,3,4,4,4,4,4,5,5}
01101 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
01110 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,2,2,3,3,4,4,4,4,4,5,5}
01111 => [2,1,1,1,1] => ([(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)
 => ([(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
10000 => [1,5] => ([(4,5)],6)
 => ([(4,5)],6)
 => ? ∊ {1,1,1,1,1,2,2,3,3,4,4,4,4,4,5,5}
10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,2,2,3,3,4,4,4,4,4,5,5}
10011 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,2,2,3,3,4,4,4,4,4,5,5}
10101 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
10110 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,2,2,3,3,4,4,4,4,4,5,5}
10111 => [1,2,1,1,1] => ([(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)
 => ([(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
11000 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,2,2,3,3,4,4,4,4,4,5,5}
11001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
11010 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,2,2,3,3,4,4,4,4,4,5,5}
11011 => [1,1,2,1,1] => ([(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)
 => ([(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
11100 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,2,2,3,3,4,4,4,4,4,5,5}
11101 => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
11110 => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,2,2,3,3,4,4,4,4,4,5,5}
11111 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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
000000 => [7] => ([],7)
 => ([],7)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 5
000010 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
000011 => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 4
000100 => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
000101 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3
000110 => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
000111 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3
001000 => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ([(3,6),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
001001 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
001010 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
001011 => [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3
001100 => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
001101 => [3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
001110 => [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(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)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
001111 => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(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)
 => ([(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
010000 => [2,5] => ([(4,6),(5,6)],7)
 => ([(4,6),(5,6)],7)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
010001 => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3
010010 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
010011 => [2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
010100 => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
010101 => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
010110 => [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(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)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
010111 => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(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)
 => ([(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
011000 => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
011001 => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
011010 => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
011011 => [2,1,2,1,1] => ([(0,5),(0,6),(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)
 => ([(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
011100 => [2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
011101 => [2,1,1,2,1] => ([(0,6),(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)
 => ([(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)
 => 4
011110 => [2,1,1,1,2] => ([(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)
 => ([(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)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
011111 => [2,1,1,1,1,1] => ([(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)
 => ([(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
100000 => [1,6] => ([(5,6)],7)
 => ([(5,6)],7)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
100001 => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => 4
100010 => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
100011 => [1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3
100100 => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
100101 => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian $$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$ Its eigenvalues are $0,4,4,6$, so the statistic is $2$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St000772
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00117: Graphs —Ore closure⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 67%
Values
0 => [2] => ([],2)
 => ([],2)
 => ? = 1
1 => [1,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
00 => [3] => ([],3)
 => ([],3)
 => ? ∊ {2,2}
01 => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
10 => [1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2}
11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
000 => [4] => ([],4)
 => ([],4)
 => ? ∊ {1,2,3,3}
001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 2
010 => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? ∊ {1,2,3,3}
011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
100 => [1,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => ? ∊ {1,2,3,3}
101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,2,3,3}
111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
0000 => [5] => ([],5)
 => ([],5)
 => ? ∊ {1,2,2,3,3,4,4,4}
0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,2,2,3,3,4,4,4}
0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
0100 => [2,3] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? ∊ {1,2,2,3,3,4,4,4}
0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,2,2,3,3,4,4,4}
0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
1000 => [1,4] => ([(3,4)],5)
 => ([(3,4)],5)
 => ? ∊ {1,2,2,3,3,4,4,4}
1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,2,2,3,3,4,4,4}
1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,2,2,3,3,4,4,4}
1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,2,2,3,3,4,4,4}
1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
00000 => [6] => ([],6)
 => ([],6)
 => ? ∊ {1,1,2,2,2,3,3,3,3,4,4,4,4,4,5,5}
00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 4
00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,1,2,2,2,3,3,3,3,4,4,4,4,4,5,5}
00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
00100 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,1,2,2,2,3,3,3,3,4,4,4,4,4,5,5}
00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
00110 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,2,2,2,3,3,3,3,4,4,4,4,4,5,5}
00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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
01000 => [2,4] => ([(3,5),(4,5)],6)
 => ([(3,5),(4,5)],6)
 => ? ∊ {1,1,2,2,2,3,3,3,3,4,4,4,4,4,5,5}
01001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
01010 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,2,2,2,3,3,3,3,4,4,4,4,4,5,5}
01011 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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
01100 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,2,2,2,3,3,3,3,4,4,4,4,4,5,5}
01101 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
01110 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,2,2,2,3,3,3,3,4,4,4,4,4,5,5}
01111 => [2,1,1,1,1] => ([(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)
 => ([(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
10000 => [1,5] => ([(4,5)],6)
 => ([(4,5)],6)
 => ? ∊ {1,1,2,2,2,3,3,3,3,4,4,4,4,4,5,5}
10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,2,2,2,3,3,3,3,4,4,4,4,4,5,5}
10011 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,2,2,2,3,3,3,3,4,4,4,4,4,5,5}
10101 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
10110 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,2,2,2,3,3,3,3,4,4,4,4,4,5,5}
10111 => [1,2,1,1,1] => ([(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)
 => ([(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
11000 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,2,2,2,3,3,3,3,4,4,4,4,4,5,5}
11001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
11010 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,2,2,2,3,3,3,3,4,4,4,4,4,5,5}
11011 => [1,1,2,1,1] => ([(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)
 => ([(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
11100 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,2,2,2,3,3,3,3,4,4,4,4,4,5,5}
11101 => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
11110 => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,2,2,2,3,3,3,3,4,4,4,4,4,5,5}
11111 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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
000000 => [7] => ([],7)
 => ([],7)
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 5
000010 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
000011 => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 4
000100 => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
000101 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
000110 => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
000111 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3
001000 => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ([(3,6),(4,6),(5,6)],7)
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
001001 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
001010 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
001011 => [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3
001100 => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
001101 => [3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
001110 => [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(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)
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
001111 => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(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)
 => ([(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
010000 => [2,5] => ([(4,6),(5,6)],7)
 => ([(4,6),(5,6)],7)
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
010001 => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3
010010 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
010011 => [2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
010100 => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
010101 => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
010110 => [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(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)
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
010111 => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(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)
 => ([(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
011000 => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
011001 => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
011010 => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
011011 => [2,1,2,1,1] => ([(0,5),(0,6),(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)
 => ([(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
011100 => [2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
011101 => [2,1,1,2,1] => ([(0,6),(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)
 => ([(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)
 => 1
011110 => [2,1,1,1,2] => ([(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)
 => ([(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)
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
011111 => [2,1,1,1,1,1] => ([(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)
 => ([(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
100000 => [1,6] => ([(5,6)],7)
 => ([(5,6)],7)
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
100001 => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => 4
100010 => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
100011 => [1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3
100100 => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6}
100101 => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian $$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$ Its eigenvalues are $0,4,4,6$, so the statistic is $1$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$. The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Matching statistic: St001880
Mapping
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%
Values
0 => [1] => [[1],[]]
 => ([],1)
 => ? ∊ {1,1}
1 => [1] => [[1],[]]
 => ([],1)
 => ? ∊ {1,1}
00 => [2] => [[2],[]]
 => ([(0,1)],2)
 => ? ∊ {1,2,2,2}
01 => [1,1] => [[1,1],[]]
 => ([(0,1)],2)
 => ? ∊ {1,2,2,2}
10 => [1,1] => [[1,1],[]]
 => ([(0,1)],2)
 => ? ∊ {1,2,2,2}
11 => [2] => [[2],[]]
 => ([(0,1)],2)
 => ? ∊ {1,2,2,2}
000 => [3] => [[3],[]]
 => ([(0,2),(2,1)],3)
 => 3
001 => [2,1] => [[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => ? ∊ {1,1,2,2}
010 => [1,1,1] => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 3
011 => [1,2] => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => ? ∊ {1,1,2,2}
100 => [1,2] => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => ? ∊ {1,1,2,2}
101 => [1,1,1] => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 3
110 => [2,1] => [[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => ? ∊ {1,1,2,2}
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)
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,4,4}
0010 => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,4,4}
0011 => [2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,4,4}
0100 => [1,1,2] => [[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,4,4}
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)
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,4,4}
0111 => [1,3] => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,4,4}
1000 => [1,3] => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,4,4}
1001 => [1,2,1] => [[2,2,1],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,4,4}
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)
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,4,4}
1100 => [2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,4,4}
1101 => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,4,4}
1110 => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,4,4}
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)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5}
00010 => [3,1,1] => [[3,3,3],[2,2]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5}
00011 => [3,2] => [[4,3],[2]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5}
00100 => [2,1,2] => [[3,2,2],[1,1]]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5}
00101 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5}
00110 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5}
00111 => [2,3] => [[4,2],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5}
01000 => [1,1,3] => [[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5}
01001 => [1,1,2,1] => [[2,2,1,1],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5}
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)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5}
01100 => [1,2,2] => [[3,2,1],[1]]
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5}
01101 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5}
01110 => [1,3,1] => [[3,3,1],[2]]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5}
01111 => [1,4] => [[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5}
10000 => [1,4] => [[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5}
10001 => [1,3,1] => [[3,3,1],[2]]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5}
10010 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5}
10011 => [1,2,2] => [[3,2,1],[1]]
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5}
10100 => [1,1,1,2] => [[2,1,1,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5}
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)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5}
10111 => [1,1,3] => [[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5}
11000 => [2,3] => [[4,2],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5}
11001 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5}
11010 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5}
11011 => [2,1,2] => [[3,2,2],[1,1]]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5}
11100 => [3,2] => [[4,3],[2]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5}
11101 => [3,1,1] => [[3,3,3],[2,2]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5}
11110 => [4,1] => [[4,4],[3]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5}
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.