Identifier
Values
0 => [1] => ([],1) => 0
1 => [1] => ([],1) => 0
00 => [2] => ([],2) => 0
01 => [1,1] => ([(0,1)],2) => 0
10 => [1,1] => ([(0,1)],2) => 0
11 => [2] => ([],2) => 0
000 => [3] => ([],3) => 0
001 => [2,1] => ([(0,2),(1,2)],3) => 0
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 1
011 => [1,2] => ([(1,2)],3) => 0
100 => [1,2] => ([(1,2)],3) => 0
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 1
110 => [2,1] => ([(0,2),(1,2)],3) => 0
111 => [3] => ([],3) => 0
0000 => [4] => ([],4) => 0
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4) => 0
0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
0011 => [2,2] => ([(1,3),(2,3)],4) => 0
0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => 0
0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 12
0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 1
0111 => [1,3] => ([(2,3)],4) => 0
1000 => [1,3] => ([(2,3)],4) => 0
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 1
1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 12
1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => 0
1100 => [2,2] => ([(1,3),(2,3)],4) => 0
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4) => 0
1111 => [4] => ([],4) => 0
00000 => [5] => ([],5) => 0
00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
00010 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 12
00011 => [3,2] => ([(1,4),(2,4),(3,4)],5) => 0
00100 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
00101 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 81
00110 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
00111 => [2,3] => ([(2,4),(3,4)],5) => 0
01000 => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 0
01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 12
01010 => [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) => 162
01011 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
01100 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 0
01101 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 36
01110 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 1
01111 => [1,4] => ([(3,4)],5) => 0
10000 => [1,4] => ([(3,4)],5) => 0
10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 1
10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 36
10011 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 0
10100 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
10101 => [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) => 162
10110 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 12
10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 0
11000 => [2,3] => ([(2,4),(3,4)],5) => 0
11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
11010 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 81
11011 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
11100 => [3,2] => ([(1,4),(2,4),(3,4)],5) => 0
11101 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 12
11110 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
11111 => [5] => ([],5) => 0
000000 => [6] => ([],6) => 0
000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
000010 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 32
000011 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
000100 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
000101 => [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) => 432
000110 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 12
000111 => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
001000 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
001001 => [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) => 81
001010 => [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) => 1536
001011 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
001100 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
001101 => [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) => 216
001110 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
001111 => [2,4] => ([(3,5),(4,5)],6) => 0
010000 => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 0
010001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 12
010010 => [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) => 432
010011 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
010100 => [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) => 0
010101 => [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) => 2560
010110 => [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) => 162
010111 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
011000 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6) => 0
011001 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 36
011010 => [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) => 864
011011 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
011100 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 0
011101 => [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) => 96
011110 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 1
011111 => [1,5] => ([(4,5)],6) => 0
100000 => [1,5] => ([(4,5)],6) => 0
100001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 1
100010 => [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) => 96
100011 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 0
100100 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
100101 => [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) => 864
100110 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 36
>>> Load all 126 entries. <<<
100111 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6) => 0
101000 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
101001 => [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) => 162
101010 => [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) => 2560
101011 => [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) => 0
101100 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
101101 => [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) => 432
101110 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 12
101111 => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 0
110000 => [2,4] => ([(3,5),(4,5)],6) => 0
110001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
110010 => [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) => 216
110011 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
110100 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
110101 => [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) => 1536
110110 => [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) => 81
110111 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
111000 => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
111001 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 12
111010 => [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) => 432
111011 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
111100 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
111101 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 32
111110 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
111111 => [6] => ([],6) => 0
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Description
The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$.
Map
to threshold graph
Description
The threshold graph corresponding to the composition.
A threshold graph is a graph that can be obtained from the empty graph by adding successively isolated and dominating vertices.
A threshold graph is uniquely determined by its degree sequence.
The Laplacian spectrum of a threshold graph is integral. Interpreting it as an integer partition, it is the conjugate of the partition given by its degree sequence.
Map
delta morphism
Description
Applies the delta morphism to a binary word.
The delta morphism of a finite word $w$ is the integer compositions composed of the lengths of consecutive runs of the same letter in $w$.