Identifier
Values
[[1,3,4],[2]] => [2,2] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,3],[2,4]] => [2,2] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,3],[2],[4]] => [2,2] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,3,4,5],[2]] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 3
[[1,3,5],[2,4]] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[[1,3,4],[2,5]] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 3
[[1,3,5],[2],[4]] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[[1,3,4],[2],[5]] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 3
[[1,3],[2,5],[4]] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[[1,3],[2,4],[5]] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 3
[[1,3],[2],[4],[5]] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 3
[[1,3,4,5,6],[2]] => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,3,5,6],[2,4]] => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,2,4,6],[3,5]] => [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,3,4,5],[2,6]] => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,3,5,6],[2],[4]] => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,2,4,6],[3],[5]] => [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,3,4,5],[2],[6]] => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,3,5],[2,4,6]] => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,2,4],[3,5,6]] => [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,4,6],[2,5],[3]] => [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,3,6],[2,5],[4]] => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,2,6],[3,4],[5]] => [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,3,5],[2,6],[4]] => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,2,4],[3,6],[5]] => [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,3,5],[2,4],[6]] => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,3,4],[2,5],[6]] => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,4,6],[2],[3],[5]] => [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,3,5],[2],[4],[6]] => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,3,4],[2],[5],[6]] => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,3],[2,5],[4,6]] => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,2],[3,4],[5,6]] => [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,4],[2,6],[3],[5]] => [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,3],[2,5],[4],[6]] => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,3],[2,4],[5],[6]] => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,3],[2],[4],[5],[6]] => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
search for individual values
searching the database for the individual values of this statistic
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
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
valley composition
Description
The composition corresponding to the valley set of a standard tableau.
Let $T$ be a standard tableau of size $n$.
An entry $i$ of $T$ is a descent if $i+1$ is in a lower row (in English notation), otherwise $i$ is an ascent.
An entry $2 \leq i \leq n-1$ is a valley if $i-1$ is a descent and $i$ is an ascent.
This map returns the composition $c_1,\dots,c_k$ of $n$ such that $\{c_1, c_1+c_2,\dots, c_1+\dots+c_k\}$ is the valley set of $T$.
Map
rotate back to front
Description
The back to front rotation of an integer composition.