Identifier
Values
{{1}} => [1] => [1] => ([],1) => 1
{{1,2}} => [2] => [1,1] => ([(0,1)],2) => 2
{{1},{2}} => [1,1] => [2] => ([],2) => 1
{{1,2,3}} => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
{{1,2},{3}} => [2,1] => [1,2] => ([(1,2)],3) => 2
{{1,3},{2}} => [2,1] => [1,2] => ([(1,2)],3) => 2
{{1},{2,3}} => [1,2] => [2,1] => ([(0,2),(1,2)],3) => 2
{{1},{2},{3}} => [1,1,1] => [3] => ([],3) => 1
{{1,2,3,4}} => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
{{1,2,3},{4}} => [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => 3
{{1,2,4},{3}} => [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => 3
{{1,2},{3},{4}} => [2,1,1] => [1,3] => ([(2,3)],4) => 2
{{1,3,4},{2}} => [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => 3
{{1,3},{2},{4}} => [2,1,1] => [1,3] => ([(2,3)],4) => 2
{{1},{2,3},{4}} => [1,2,1] => [2,2] => ([(1,3),(2,3)],4) => 2
{{1,4},{2},{3}} => [2,1,1] => [1,3] => ([(2,3)],4) => 2
{{1},{2,4},{3}} => [1,2,1] => [2,2] => ([(1,3),(2,3)],4) => 2
{{1},{2},{3,4}} => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2
{{1},{2},{3},{4}} => [1,1,1,1] => [4] => ([],4) => 1
{{1,2,3,4,5}} => [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
{{1,2,3,4},{5}} => [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
{{1,2,3,5},{4}} => [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
{{1,2,3},{4},{5}} => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 3
{{1,2,4,5},{3}} => [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
{{1,2,4},{3},{5}} => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 3
{{1,2,5},{3},{4}} => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 3
{{1,2},{3},{4},{5}} => [2,1,1,1] => [1,4] => ([(3,4)],5) => 2
{{1,3,4,5},{2}} => [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
{{1,3,4},{2},{5}} => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 3
{{1,3,5},{2},{4}} => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 3
{{1,3},{2},{4},{5}} => [2,1,1,1] => [1,4] => ([(3,4)],5) => 2
{{1},{2,3},{4},{5}} => [1,2,1,1] => [2,3] => ([(2,4),(3,4)],5) => 2
{{1,4,5},{2},{3}} => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 3
{{1,4},{2},{3},{5}} => [2,1,1,1] => [1,4] => ([(3,4)],5) => 2
{{1},{2,4},{3},{5}} => [1,2,1,1] => [2,3] => ([(2,4),(3,4)],5) => 2
{{1},{2},{3,4},{5}} => [1,1,2,1] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 2
{{1,5},{2},{3},{4}} => [2,1,1,1] => [1,4] => ([(3,4)],5) => 2
{{1},{2,5},{3},{4}} => [1,2,1,1] => [2,3] => ([(2,4),(3,4)],5) => 2
{{1},{2},{3,5},{4}} => [1,1,2,1] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 2
{{1},{2},{3},{4,5}} => [1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
{{1},{2},{3},{4},{5}} => [1,1,1,1,1] => [5] => ([],5) => 1
{{1,2,3,4,5,6}} => [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
{{1,2,3,4,5},{6}} => [5,1] => [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
{{1,2,3,4,6},{5}} => [5,1] => [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
{{1,2,3,4},{5},{6}} => [4,1,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
{{1,2,3,5,6},{4}} => [5,1] => [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
{{1,2,3,5},{4},{6}} => [4,1,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
{{1,2,3,6},{4},{5}} => [4,1,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
{{1,2,3},{4},{5},{6}} => [3,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 3
{{1,2,4,5,6},{3}} => [5,1] => [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
{{1,2,4,5},{3},{6}} => [4,1,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
{{1,2,4,6},{3},{5}} => [4,1,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
{{1,2,4},{3},{5},{6}} => [3,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 3
{{1,2,5,6},{3},{4}} => [4,1,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
{{1,2,5},{3},{4},{6}} => [3,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 3
{{1,2,6},{3},{4},{5}} => [3,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 3
{{1,2},{3},{4},{5},{6}} => [2,1,1,1,1] => [1,5] => ([(4,5)],6) => 2
{{1,3,4,5,6},{2}} => [5,1] => [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
{{1,3,4,5},{2},{6}} => [4,1,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
{{1,3,4,6},{2},{5}} => [4,1,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
{{1,3,4},{2},{5},{6}} => [3,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 3
{{1,3,5,6},{2},{4}} => [4,1,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
{{1,3,5},{2},{4},{6}} => [3,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 3
{{1,3,6},{2},{4},{5}} => [3,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 3
{{1,3},{2},{4},{5},{6}} => [2,1,1,1,1] => [1,5] => ([(4,5)],6) => 2
{{1},{2,3},{4},{5},{6}} => [1,2,1,1,1] => [2,4] => ([(3,5),(4,5)],6) => 2
{{1,4,5,6},{2},{3}} => [4,1,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
{{1,4,5},{2},{3},{6}} => [3,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 3
{{1,4,6},{2},{3},{5}} => [3,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 3
{{1,4},{2},{3},{5},{6}} => [2,1,1,1,1] => [1,5] => ([(4,5)],6) => 2
{{1},{2,4},{3},{5},{6}} => [1,2,1,1,1] => [2,4] => ([(3,5),(4,5)],6) => 2
{{1},{2},{3,4},{5},{6}} => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 2
{{1,5,6},{2},{3},{4}} => [3,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 3
{{1,5},{2},{3},{4},{6}} => [2,1,1,1,1] => [1,5] => ([(4,5)],6) => 2
{{1},{2,5},{3},{4},{6}} => [1,2,1,1,1] => [2,4] => ([(3,5),(4,5)],6) => 2
{{1},{2},{3,5},{4},{6}} => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 2
{{1},{2},{3},{4,5},{6}} => [1,1,1,2,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
{{1,6},{2},{3},{4},{5}} => [2,1,1,1,1] => [1,5] => ([(4,5)],6) => 2
{{1},{2,6},{3},{4},{5}} => [1,2,1,1,1] => [2,4] => ([(3,5),(4,5)],6) => 2
{{1},{2},{3,6},{4},{5}} => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 2
{{1},{2},{3},{4,6},{5}} => [1,1,1,2,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
{{1},{2},{3},{4},{5,6}} => [1,1,1,1,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 2
{{1},{2},{3},{4},{5},{6}} => [1,1,1,1,1,1] => [6] => ([],6) => 1
{{1,2,3,4,5,6,7}} => [7] => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 7
{{1,2,3,4,5,6},{7}} => [6,1] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6
{{1,2,3,4,5,7},{6}} => [6,1] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6
{{1,2,3,4,6,7},{5}} => [6,1] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6
{{1,2,3,5,6,7},{4}} => [6,1] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6
{{1,2,4,5,6,7},{3}} => [6,1] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6
{{1,3,4,5,6,7},{2}} => [6,1] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6
{{1},{2},{3},{4},{5},{6,7}} => [1,1,1,1,1,2] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
{{1},{2},{3},{4},{5},{6},{7,8}} => [1,1,1,1,1,1,2] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2
{{1,2,3,4,5,6,7,8}} => [8] => [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => 8
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Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
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
to composition
Description
The integer composition of block sizes of a set partition.
For a set partition of $\{1,2,\dots,n\}$, this is the integer composition of $n$ obtained by sorting the blocks by their minimal element and then taking the block sizes.
Map
complement
Description
The complement of a composition.
The complement of a composition $I$ is defined as follows:
If $I$ is the empty composition, then the complement is also the empty composition. Otherwise, let $S$ be the descent set corresponding to $I=(i_1,\dots,i_k)$, that is, the subset
$$\{ i_1, i_1 + i_2, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$$
of $\{ 1, 2, \ldots, |I|-1 \}$. Then, the complement of $I$ is the composition of the same size as $I$, whose descent set is $\{ 1, 2, \ldots, |I|-1 \} \setminus S$.
The complement of a composition $I$ coincides with the reversal (Mp00038reverse) of the composition conjugate (Mp00041conjugate) to $I$.