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) => 2
0011 => [2,2] => ([(1,3),(2,3)],4) => 0
0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => 1
0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
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) => 4
1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => 1
1100 => [2,2] => ([(1,3),(2,3)],4) => 0
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
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) => 3
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) => 2
00101 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 7
00110 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
00111 => [2,3] => ([(2,4),(3,4)],5) => 0
01000 => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 1
01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
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) => 10
01011 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
01100 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 1
01101 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 5
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) => 5
10011 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 1
10100 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
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) => 10
10110 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 1
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) => 2
11010 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 7
11011 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
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) => 3
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) => 4
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) => 3
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) => 10
000110 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
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) => 2
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) => 7
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) => 16
001011 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 7
001100 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
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) => 8
001110 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
001111 => [2,4] => ([(3,5),(4,5)],6) => 0
010000 => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 1
010001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
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) => 11
010011 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
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) => 10
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) => 20
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) => 10
010111 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
011000 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6) => 1
011001 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
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) => 13
011011 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
011100 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 1
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) => 6
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) => 6
100011 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 1
100100 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
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) => 13
100110 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
>>> Load all 126 entries. <<<
100111 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6) => 1
101000 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
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) => 10
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) => 20
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) => 10
101100 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
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) => 11
101110 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
101111 => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 1
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) => 2
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) => 8
110011 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
110100 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 7
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) => 16
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) => 7
110111 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
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) => 3
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) => 10
111011 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
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) => 4
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 number of triangles of a graph.
A triangle $T$ of a graph $G$ is a collection of three vertices $\{u,v,w\} \in G$ such that they form $K_3$, the complete graph on three vertices.
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 $w$.