- FindStat

Identifier

Mp00287: Ordered set partitions —to composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤ

Values

[{1}] => [1] => [1] => ([],1) => 1

[{1},{2}] => [1,1] => [2] => ([],2) => 1

[{2},{1}] => [1,1] => [2] => ([],2) => 1

[{1,2}] => [2] => [1,1] => ([(0,1)],2) => 2

[{1},{2},{3}] => [1,1,1] => [3] => ([],3) => 1

[{1},{3},{2}] => [1,1,1] => [3] => ([],3) => 1

[{2},{1},{3}] => [1,1,1] => [3] => ([],3) => 1

[{3},{1},{2}] => [1,1,1] => [3] => ([],3) => 1

[{2},{3},{1}] => [1,1,1] => [3] => ([],3) => 1

[{3},{2},{1}] => [1,1,1] => [3] => ([],3) => 1

[{1},{2,3}] => [1,2] => [2,1] => ([(0,2),(1,2)],3) => 2

[{2},{1,3}] => [1,2] => [2,1] => ([(0,2),(1,2)],3) => 2

[{3},{1,2}] => [1,2] => [2,1] => ([(0,2),(1,2)],3) => 2

[{1,2},{3}] => [2,1] => [1,2] => ([(1,2)],3) => 2

[{1,3},{2}] => [2,1] => [1,2] => ([(1,2)],3) => 2

[{2,3},{1}] => [2,1] => [1,2] => ([(1,2)],3) => 2

[{1,2,3}] => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3

[{1},{2},{3},{4}] => [1,1,1,1] => [4] => ([],4) => 1

[{1},{2},{4},{3}] => [1,1,1,1] => [4] => ([],4) => 1

[{1},{3},{2},{4}] => [1,1,1,1] => [4] => ([],4) => 1

[{1},{4},{2},{3}] => [1,1,1,1] => [4] => ([],4) => 1

[{1},{3},{4},{2}] => [1,1,1,1] => [4] => ([],4) => 1

[{1},{4},{3},{2}] => [1,1,1,1] => [4] => ([],4) => 1

[{2},{1},{3},{4}] => [1,1,1,1] => [4] => ([],4) => 1

[{2},{1},{4},{3}] => [1,1,1,1] => [4] => ([],4) => 1

[{3},{1},{2},{4}] => [1,1,1,1] => [4] => ([],4) => 1

[{4},{1},{2},{3}] => [1,1,1,1] => [4] => ([],4) => 1

[{3},{1},{4},{2}] => [1,1,1,1] => [4] => ([],4) => 1

[{4},{1},{3},{2}] => [1,1,1,1] => [4] => ([],4) => 1

[{2},{3},{1},{4}] => [1,1,1,1] => [4] => ([],4) => 1

[{2},{4},{1},{3}] => [1,1,1,1] => [4] => ([],4) => 1

[{3},{2},{1},{4}] => [1,1,1,1] => [4] => ([],4) => 1

[{4},{2},{1},{3}] => [1,1,1,1] => [4] => ([],4) => 1

[{3},{4},{1},{2}] => [1,1,1,1] => [4] => ([],4) => 1

[{4},{3},{1},{2}] => [1,1,1,1] => [4] => ([],4) => 1

[{2},{3},{4},{1}] => [1,1,1,1] => [4] => ([],4) => 1

[{2},{4},{3},{1}] => [1,1,1,1] => [4] => ([],4) => 1

[{3},{2},{4},{1}] => [1,1,1,1] => [4] => ([],4) => 1

[{4},{2},{3},{1}] => [1,1,1,1] => [4] => ([],4) => 1

[{3},{4},{2},{1}] => [1,1,1,1] => [4] => ([],4) => 1

[{4},{3},{2},{1}] => [1,1,1,1] => [4] => ([],4) => 1

[{1},{2},{3,4}] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2

[{1},{3},{2,4}] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2

[{1},{4},{2,3}] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2

[{2},{1},{3,4}] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2

[{3},{1},{2,4}] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2

[{4},{1},{2,3}] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2

[{2},{3},{1,4}] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2

[{2},{4},{1,3}] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2

[{3},{2},{1,4}] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2

[{4},{2},{1,3}] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2

[{3},{4},{1,2}] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2

[{4},{3},{1,2}] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2

[{1},{2,3},{4}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4) => 2

[{1},{2,4},{3}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4) => 2

[{1},{3,4},{2}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4) => 2

[{2},{1,3},{4}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4) => 2

[{2},{1,4},{3}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4) => 2

[{3},{1,2},{4}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4) => 2

[{4},{1,2},{3}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4) => 2

[{3},{1,4},{2}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4) => 2

[{4},{1,3},{2}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4) => 2

[{2},{3,4},{1}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4) => 2

[{3},{2,4},{1}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4) => 2

[{4},{2,3},{1}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4) => 2

[{1,2},{3},{4}] => [2,1,1] => [1,3] => ([(2,3)],4) => 2

[{1,2},{4},{3}] => [2,1,1] => [1,3] => ([(2,3)],4) => 2

[{1,3},{2},{4}] => [2,1,1] => [1,3] => ([(2,3)],4) => 2

[{1,4},{2},{3}] => [2,1,1] => [1,3] => ([(2,3)],4) => 2

[{1,3},{4},{2}] => [2,1,1] => [1,3] => ([(2,3)],4) => 2

[{1,4},{3},{2}] => [2,1,1] => [1,3] => ([(2,3)],4) => 2

[{2,3},{1},{4}] => [2,1,1] => [1,3] => ([(2,3)],4) => 2

[{2,4},{1},{3}] => [2,1,1] => [1,3] => ([(2,3)],4) => 2

[{3,4},{1},{2}] => [2,1,1] => [1,3] => ([(2,3)],4) => 2

[{2,3},{4},{1}] => [2,1,1] => [1,3] => ([(2,3)],4) => 2

[{2,4},{3},{1}] => [2,1,1] => [1,3] => ([(2,3)],4) => 2

[{3,4},{2},{1}] => [2,1,1] => [1,3] => ([(2,3)],4) => 2

[{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

[{1,3,4},{2}] => [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => 3

[{2,3,4},{1}] => [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => 3

[{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},{5}] => [1,1,1,1,1] => [5] => ([],5) => 1

[{1},{2},{3},{5},{4}] => [1,1,1,1,1] => [5] => ([],5) => 1

[{1},{2},{4},{3},{5}] => [1,1,1,1,1] => [5] => ([],5) => 1

[{1},{2},{5},{3},{4}] => [1,1,1,1,1] => [5] => ([],5) => 1

[{1},{2},{4},{5},{3}] => [1,1,1,1,1] => [5] => ([],5) => 1

[{1},{2},{5},{4},{3}] => [1,1,1,1,1] => [5] => ([],5) => 1

[{1},{3},{2},{4},{5}] => [1,1,1,1,1] => [5] => ([],5) => 1

[{1},{3},{2},{5},{4}] => [1,1,1,1,1] => [5] => ([],5) => 1

[{1},{4},{2},{3},{5}] => [1,1,1,1,1] => [5] => ([],5) => 1

[{1},{5},{2},{3},{4}] => [1,1,1,1,1] => [5] => ([],5) => 1

[{1},{4},{2},{5},{3}] => [1,1,1,1,1] => [5] => ([],5) => 1

[{1},{5},{2},{4},{3}] => [1,1,1,1,1] => [5] => ([],5) => 1

[{1},{3},{4},{2},{5}] => [1,1,1,1,1] => [5] => ([],5) => 1

[{1},{3},{5},{2},{4}] => [1,1,1,1,1] => [5] => ([],5) => 1

[{1},{4},{3},{2},{5}] => [1,1,1,1,1] => [5] => ([],5) => 1

[{1},{5},{3},{2},{4}] => [1,1,1,1,1] => [5] => ([],5) => 1

[{1},{4},{5},{2},{3}] => [1,1,1,1,1] => [5] => ([],5) => 1

[{1},{5},{4},{2},{3}] => [1,1,1,1,1] => [5] => ([],5) => 1

[{1},{3},{4},{5},{2}] => [1,1,1,1,1] => [5] => ([],5) => 1

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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

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 $I$.

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

Return the sizes of the blocks as a composition.