Identifier
Values
0 => [2] => [1,1] => ([(0,1)],2) => 1
1 => [1,1] => [2] => ([],2) => 0
00 => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
01 => [2,1] => [2,1] => ([(0,2),(1,2)],3) => 2
10 => [1,2] => [1,2] => ([(1,2)],3) => 1
11 => [1,1,1] => [3] => ([],3) => 0
000 => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 5
001 => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 3
010 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 3
011 => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 3
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) => 2
110 => [1,1,2] => [1,3] => ([(2,3)],4) => 1
111 => [1,1,1,1] => [4] => ([],4) => 0
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) => 6
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) => 4
0100 => [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
0101 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
0110 => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 4
0111 => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 4
1000 => [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 5
1001 => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
1010 => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 3
1011 => [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 3
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) => 2
1110 => [1,1,1,2] => [1,4] => ([(3,4)],5) => 1
1111 => [1,1,1,1,1] => [5] => ([],5) => 0
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) => 5
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) => 5
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) => 5
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) => 5
01101 => [2,1,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
01110 => [2,1,1,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 5
01111 => [2,1,1,1,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 5
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) => 6
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) => 4
10100 => [1,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
10101 => [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
10110 => [1,2,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 4
10111 => [1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
11000 => [1,1,4] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
11001 => [1,1,3,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
11010 => [1,1,2,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6) => 3
11011 => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 3
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) => 2
11110 => [1,1,1,1,2] => [1,5] => ([(4,5)],6) => 1
11111 => [1,1,1,1,1,1] => [6] => ([],6) => 0
=> [1] => [1] => ([],1) => 0
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
click to show known generating functions       
Description
The game chromatic index of a graph.
Two players, Alice and Bob, take turns colouring properly any uncolored edge of the graph. Alice begins. If it is not possible for either player to colour a edge, then Bob wins. If the graph is completely colored, Alice wins.
The game chromatic index is the smallest number of colours such that Alice has a winning strategy.
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.