- FindStat

Identifier

Mp00114: Permutations —connectivity set⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤ

Values

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

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

[1,2,3] => 11 => [2] => ([],2) => 1

[1,3,2] => 10 => [1,1] => ([(0,1)],2) => 2

[2,1,3] => 01 => [1,1] => ([(0,1)],2) => 2

[2,3,1] => 00 => [2] => ([],2) => 1

[3,1,2] => 00 => [2] => ([],2) => 1

[3,2,1] => 00 => [2] => ([],2) => 1

[1,2,3,4] => 111 => [3] => ([],3) => 1

[1,2,4,3] => 110 => [2,1] => ([(0,2),(1,2)],3) => 2

[1,3,2,4] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3

[1,3,4,2] => 100 => [1,2] => ([(1,2)],3) => 2

[1,4,2,3] => 100 => [1,2] => ([(1,2)],3) => 2

[1,4,3,2] => 100 => [1,2] => ([(1,2)],3) => 2

[2,1,3,4] => 011 => [1,2] => ([(1,2)],3) => 2

[2,1,4,3] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3

[2,3,1,4] => 001 => [2,1] => ([(0,2),(1,2)],3) => 2

[2,3,4,1] => 000 => [3] => ([],3) => 1

[2,4,1,3] => 000 => [3] => ([],3) => 1

[2,4,3,1] => 000 => [3] => ([],3) => 1

[3,1,2,4] => 001 => [2,1] => ([(0,2),(1,2)],3) => 2

[3,1,4,2] => 000 => [3] => ([],3) => 1

[3,2,1,4] => 001 => [2,1] => ([(0,2),(1,2)],3) => 2

[3,2,4,1] => 000 => [3] => ([],3) => 1

[3,4,1,2] => 000 => [3] => ([],3) => 1

[3,4,2,1] => 000 => [3] => ([],3) => 1

[4,1,2,3] => 000 => [3] => ([],3) => 1

[4,1,3,2] => 000 => [3] => ([],3) => 1

[4,2,1,3] => 000 => [3] => ([],3) => 1

[4,2,3,1] => 000 => [3] => ([],3) => 1

[4,3,1,2] => 000 => [3] => ([],3) => 1

[4,3,2,1] => 000 => [3] => ([],3) => 1

[1,2,3,4,5] => 1111 => [4] => ([],4) => 1

[1,2,3,5,4] => 1110 => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2

[1,2,4,5,3] => 1100 => [2,2] => ([(1,3),(2,3)],4) => 2

[1,2,5,3,4] => 1100 => [2,2] => ([(1,3),(2,3)],4) => 2

[1,2,5,4,3] => 1100 => [2,2] => ([(1,3),(2,3)],4) => 2

[1,3,2,4,5] => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => 3

[1,3,2,5,4] => 1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4

[1,3,4,5,2] => 1000 => [1,3] => ([(2,3)],4) => 2

[1,3,5,2,4] => 1000 => [1,3] => ([(2,3)],4) => 2

[1,3,5,4,2] => 1000 => [1,3] => ([(2,3)],4) => 2

[1,4,2,5,3] => 1000 => [1,3] => ([(2,3)],4) => 2

[1,4,3,5,2] => 1000 => [1,3] => ([(2,3)],4) => 2

[1,4,5,2,3] => 1000 => [1,3] => ([(2,3)],4) => 2

[1,4,5,3,2] => 1000 => [1,3] => ([(2,3)],4) => 2

[1,5,2,3,4] => 1000 => [1,3] => ([(2,3)],4) => 2

[1,5,2,4,3] => 1000 => [1,3] => ([(2,3)],4) => 2

[1,5,3,2,4] => 1000 => [1,3] => ([(2,3)],4) => 2

[1,5,3,4,2] => 1000 => [1,3] => ([(2,3)],4) => 2

[1,5,4,2,3] => 1000 => [1,3] => ([(2,3)],4) => 2

[1,5,4,3,2] => 1000 => [1,3] => ([(2,3)],4) => 2

[2,1,3,4,5] => 0111 => [1,3] => ([(2,3)],4) => 2

[2,1,4,3,5] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4

[2,1,4,5,3] => 0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => 3

[2,1,5,3,4] => 0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => 3

[2,1,5,4,3] => 0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => 3

[2,3,1,4,5] => 0011 => [2,2] => ([(1,3),(2,3)],4) => 2

[2,3,4,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2

[2,3,4,5,1] => 0000 => [4] => ([],4) => 1

[2,3,5,1,4] => 0000 => [4] => ([],4) => 1

[2,3,5,4,1] => 0000 => [4] => ([],4) => 1

[2,4,1,3,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2

[2,4,1,5,3] => 0000 => [4] => ([],4) => 1

[2,4,3,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2

[2,4,3,5,1] => 0000 => [4] => ([],4) => 1

[2,4,5,1,3] => 0000 => [4] => ([],4) => 1

[2,4,5,3,1] => 0000 => [4] => ([],4) => 1

[2,5,1,3,4] => 0000 => [4] => ([],4) => 1

[2,5,1,4,3] => 0000 => [4] => ([],4) => 1

[2,5,3,1,4] => 0000 => [4] => ([],4) => 1

[2,5,3,4,1] => 0000 => [4] => ([],4) => 1

[2,5,4,1,3] => 0000 => [4] => ([],4) => 1

[2,5,4,3,1] => 0000 => [4] => ([],4) => 1

[3,1,2,4,5] => 0011 => [2,2] => ([(1,3),(2,3)],4) => 2

[3,1,4,2,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2

[3,1,4,5,2] => 0000 => [4] => ([],4) => 1

[3,1,5,2,4] => 0000 => [4] => ([],4) => 1

[3,1,5,4,2] => 0000 => [4] => ([],4) => 1

[3,2,1,4,5] => 0011 => [2,2] => ([(1,3),(2,3)],4) => 2

[3,2,4,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2

[3,2,4,5,1] => 0000 => [4] => ([],4) => 1

[3,2,5,1,4] => 0000 => [4] => ([],4) => 1

[3,2,5,4,1] => 0000 => [4] => ([],4) => 1

[3,4,1,2,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2

[3,4,1,5,2] => 0000 => [4] => ([],4) => 1

[3,4,2,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2

[3,4,2,5,1] => 0000 => [4] => ([],4) => 1

[3,4,5,1,2] => 0000 => [4] => ([],4) => 1

[3,4,5,2,1] => 0000 => [4] => ([],4) => 1

[3,5,1,2,4] => 0000 => [4] => ([],4) => 1

[3,5,1,4,2] => 0000 => [4] => ([],4) => 1

[3,5,2,1,4] => 0000 => [4] => ([],4) => 1

[3,5,2,4,1] => 0000 => [4] => ([],4) => 1

[3,5,4,1,2] => 0000 => [4] => ([],4) => 1

[3,5,4,2,1] => 0000 => [4] => ([],4) => 1

[4,1,2,3,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2

[4,1,2,5,3] => 0000 => [4] => ([],4) => 1

[4,1,3,2,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2

[4,1,3,5,2] => 0000 => [4] => ([],4) => 1

[4,1,5,2,3] => 0000 => [4] => ([],4) => 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

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

connectivity set

Description

The connectivity set of a permutation as a binary word.
According to [2], also known as the global ascent set.
The connectivity set is
$$C(\pi)=\{i\in [n-1] | \forall 1 \leq j \leq i < k \leq n : \pi(j) < \pi(k)\}.$$
For $n > 1$ it can also be described as the set of occurrences of the mesh pattern
$$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$$
or equivalently
$$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$$
see [3].
The permutation is connected, when the connectivity set is empty.

Map

delta morphism

Description

Applies the delta morphism to a binary word.
The delta morphism of a finite word $w$ is the integer compositions composed of the lengths of consecutive runs of the same letter in $w$.