Identifier
Values
0 => [2] => ([],2) => 2
1 => [1,1] => ([(0,1)],2) => 2
00 => [3] => ([],3) => 3
01 => [2,1] => ([(0,2),(1,2)],3) => 4
10 => [1,2] => ([(1,2)],3) => 4
11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
000 => [4] => ([],4) => 4
001 => [3,1] => ([(0,3),(1,3),(2,3)],4) => 6
010 => [2,2] => ([(1,3),(2,3)],4) => 7
011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 6
100 => [1,3] => ([(2,3)],4) => 6
101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 7
110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => 6
111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
0000 => [5] => ([],5) => 5
0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 8
0010 => [3,2] => ([(1,4),(2,4),(3,4)],5) => 10
0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 9
0100 => [2,3] => ([(2,4),(3,4)],5) => 10
0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 12
0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 11
0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 8
1000 => [1,4] => ([(3,4)],5) => 8
1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 11
1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 12
1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 10
1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 9
1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 10
1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 8
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) => 5
00000 => [6] => ([],6) => 6
00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 10
00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 13
00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 12
00100 => [3,3] => ([(2,5),(3,5),(4,5)],6) => 14
00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 17
00110 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 16
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) => 12
01000 => [2,4] => ([(3,5),(4,5)],6) => 13
01001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 18
01010 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 20
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) => 17
01100 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 16
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) => 18
01110 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 15
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) => 10
10000 => [1,5] => ([(4,5)],6) => 10
10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 15
10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 18
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) => 16
10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6) => 17
10101 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 20
10110 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 18
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) => 13
11000 => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 12
11001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 16
11010 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 17
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) => 14
11100 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 12
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) => 13
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) => 10
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) => 6
=> [1] => ([],1) => 1
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Description
The number of induced subgraphs.
A subgraph $H \subseteq G$ is induced if $E(H)$ consists of all edges in $E(G)$ that connect the vertices of $H$.
Map
to composition
Description
The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.
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.