Identifier
Values
0 => [2] => [1,1] => ([(0,1)],2) => 2
1 => [1,1] => [2] => ([],2) => 1
00 => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
01 => [2,1] => [2,1] => ([(0,2),(1,2)],3) => 3
10 => [1,2] => [1,2] => ([(1,2)],3) => 2
11 => [1,1,1] => [3] => ([],3) => 1
000 => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
001 => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
010 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 4
011 => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 4
100 => [1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => 3
101 => [1,2,1] => [2,2] => ([(1,3),(2,3)],4) => 3
110 => [1,1,2] => [1,3] => ([(2,3)],4) => 2
111 => [1,1,1,1] => [4] => ([],4) => 1
0000 => [5] => [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
0001 => [4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 5
0010 => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 5
0011 => [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 5
0100 => [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 5
0101 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 5
0110 => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 5
0111 => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 5
1000 => [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
1001 => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
1010 => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 4
1011 => [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 4
1100 => [1,1,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 3
1101 => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5) => 3
1110 => [1,1,1,2] => [1,4] => ([(3,4)],5) => 2
1111 => [1,1,1,1,1] => [5] => ([],5) => 1
00000 => [6] => [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
00001 => [5,1] => [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) => 6
00010 => [4,2] => [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) => 6
00011 => [4,1,1] => [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) => 6
00100 => [3,3] => [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) => 6
00101 => [3,2,1] => [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) => 6
00110 => [3,1,2] => [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
00111 => [3,1,1,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
01000 => [2,4] => [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) => 6
01001 => [2,3,1] => [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) => 6
01010 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
01011 => [2,2,1,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
01100 => [2,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
01101 => [2,1,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
01110 => [2,1,1,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 6
01111 => [2,1,1,1,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 6
10000 => [1,5] => [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) => 5
10001 => [1,4,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
10010 => [1,3,2] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
10011 => [1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
10100 => [1,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
10101 => [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
10110 => [1,2,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 5
10111 => [1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 5
11000 => [1,1,4] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
11001 => [1,1,3,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
11010 => [1,1,2,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6) => 4
11011 => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 4
11100 => [1,1,1,3] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 3
11101 => [1,1,1,2,1] => [2,4] => ([(3,5),(4,5)],6) => 3
11110 => [1,1,1,1,2] => [1,5] => ([(4,5)],6) => 2
11111 => [1,1,1,1,1,1] => [6] => ([],6) => 1
001111 => [3,1,1,1,1] => [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) => 7
010111 => [2,2,1,1,1] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 7
011010 => [2,1,2,2] => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 7
011011 => [2,1,2,1,1] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 7
011100 => [2,1,1,3] => [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 7
011101 => [2,1,1,2,1] => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 7
011110 => [2,1,1,1,2] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 7
011111 => [2,1,1,1,1,1] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 7
100111 => [1,3,1,1,1] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6
101010 => [1,2,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6
101011 => [1,2,2,1,1] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6
101100 => [1,2,1,3] => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6
101101 => [1,2,1,2,1] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6
101110 => [1,2,1,1,2] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 6
101111 => [1,2,1,1,1,1] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 6
110010 => [1,1,3,2] => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5
110011 => [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5
110100 => [1,1,2,3] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5
110101 => [1,1,2,2,1] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 5
110110 => [1,1,2,1,2] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 5
110111 => [1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 5
111000 => [1,1,1,4] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
111001 => [1,1,1,3,1] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
111010 => [1,1,1,2,2] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7) => 4
111011 => [1,1,1,2,1,1] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 4
111100 => [1,1,1,1,3] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 3
111101 => [1,1,1,1,2,1] => [2,5] => ([(4,6),(5,6)],7) => 3
111110 => [1,1,1,1,1,2] => [1,6] => ([(5,6)],7) => 2
111111 => [1,1,1,1,1,1,1] => [7] => ([],7) => 1
=> [1] => [1] => ([],1) => 1
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Description
The harmonious chromatic number of a graph.
A harmonious colouring is a proper vertex colouring such that any pair of colours appears at most once on adjacent 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
conjugate
Description
The conjugate of a composition.
The conjugate of a composition $C$ is defined as the complement (Mp00039complement) of the reversal (Mp00038reverse) of $C$.
Equivalently, the ribbon shape corresponding to the conjugate of $C$ is the conjugate of the ribbon shape of $C$.
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.