Identifier
Values
[[1]] => [1] => ([],1) => 0
[[1,2]] => [2] => ([],2) => 0
[[1],[2]] => [2] => ([],2) => 0
[[1,2,3]] => [3] => ([],3) => 0
[[1,3],[2]] => [2,1] => ([(0,2),(1,2)],3) => 2
[[1,2],[3]] => [3] => ([],3) => 0
[[1],[2],[3]] => [3] => ([],3) => 0
[[1,2,3,4]] => [4] => ([],4) => 0
[[1,3,4],[2]] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,2,4],[3]] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2
[[1,2,3],[4]] => [4] => ([],4) => 0
[[1,3],[2,4]] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,2],[3,4]] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2
[[1,4],[2],[3]] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2
[[1,3],[2],[4]] => [2,2] => ([(1,3),(2,3)],4) => 2
[[1,2],[3],[4]] => [4] => ([],4) => 0
[[1],[2],[3],[4]] => [4] => ([],4) => 0
[[1,2,3,4,5]] => [5] => ([],5) => 0
[[1,3,4,5],[2]] => [2,3] => ([(2,4),(3,4)],5) => 2
[[1,2,4,5],[3]] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 4
[[1,2,3,5],[4]] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[[1,2,3,4],[5]] => [5] => ([],5) => 0
[[1,3,5],[2,4]] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
[[1,2,5],[3,4]] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 4
[[1,3,4],[2,5]] => [2,3] => ([(2,4),(3,4)],5) => 2
[[1,2,4],[3,5]] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 4
[[1,2,3],[4,5]] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[[1,4,5],[2],[3]] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 4
[[1,3,5],[2],[4]] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
[[1,2,5],[3],[4]] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[[1,3,4],[2],[5]] => [2,3] => ([(2,4),(3,4)],5) => 2
[[1,2,4],[3],[5]] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 4
[[1,2,3],[4],[5]] => [5] => ([],5) => 0
[[1,4],[2,5],[3]] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 4
[[1,3],[2,5],[4]] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
[[1,2],[3,5],[4]] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[[1,3],[2,4],[5]] => [2,3] => ([(2,4),(3,4)],5) => 2
[[1,2],[3,4],[5]] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 4
[[1,5],[2],[3],[4]] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[[1,4],[2],[3],[5]] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 4
[[1,3],[2],[4],[5]] => [2,3] => ([(2,4),(3,4)],5) => 2
[[1,2],[3],[4],[5]] => [5] => ([],5) => 0
[[1],[2],[3],[4],[5]] => [5] => ([],5) => 0
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Description
The villainy of a graph.
The villainy of a permutation of a proper coloring $c$ of a graph is the minimal Hamming distance between $c$ and a proper coloring.
The villainy of a graph is the maximal villainy of a permutation of a proper coloring.
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 $T$.