Identifier
-
Mp00100:
Dyck paths
—touch composition⟶
Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001330: Graphs ⟶ ℤ
Values
[1,0] => [1] => ([],1) => ([(0,1)],2) => 2
[1,0,1,0] => [1,1] => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 3
[1,1,0,0] => [2] => ([],2) => ([(0,2),(1,2)],3) => 2
[1,0,1,0,1,0] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
[1,1,0,1,0,0] => [3] => ([],3) => ([(0,3),(1,3),(2,3)],4) => 2
[1,1,1,0,0,0] => [3] => ([],3) => ([(0,3),(1,3),(2,3)],4) => 2
[1,0,1,0,1,0,1,0] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 5
[1,1,0,1,0,1,0,0] => [4] => ([],4) => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[1,1,0,1,1,0,0,0] => [4] => ([],4) => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[1,1,1,0,0,1,0,0] => [4] => ([],4) => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[1,1,1,0,1,0,0,0] => [4] => ([],4) => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[1,1,1,1,0,0,0,0] => [4] => ([],4) => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[1,0,1,0,1,0,1,0,1,0] => [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) => ([(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
[1,1,0,1,0,1,0,1,0,0] => [5] => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 2
[1,1,0,1,0,1,1,0,0,0] => [5] => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 2
[1,1,0,1,1,0,0,1,0,0] => [5] => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 2
[1,1,0,1,1,0,1,0,0,0] => [5] => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 2
[1,1,0,1,1,1,0,0,0,0] => [5] => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 2
[1,1,1,0,0,1,0,1,0,0] => [5] => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 2
[1,1,1,0,0,1,1,0,0,0] => [5] => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 2
[1,1,1,0,1,0,0,1,0,0] => [5] => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 2
[1,1,1,0,1,0,1,0,0,0] => [5] => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 2
[1,1,1,0,1,1,0,0,0,0] => [5] => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 2
[1,1,1,1,0,0,0,1,0,0] => [5] => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 2
[1,1,1,1,0,0,1,0,0,0] => [5] => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 2
[1,1,1,1,0,1,0,0,0,0] => [5] => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 2
[1,1,1,1,1,0,0,0,0,0] => [5] => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 2
[1,0,1,0,1,0,1,0,1,0,1,0] => [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) => ([(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
[1,1,0,1,0,1,0,1,0,1,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,0,1,0,1,0,1,1,0,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,0,1,0,1,1,0,0,1,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,0,1,0,1,1,0,1,0,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,0,1,0,1,1,1,0,0,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,0,1,1,0,0,1,0,1,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,0,1,1,0,0,1,1,0,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,0,1,1,0,1,0,0,1,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,0,1,1,0,1,0,1,0,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,0,1,1,0,1,1,0,0,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,0,1,1,1,0,0,0,1,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,0,1,1,1,0,0,1,0,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,0,1,1,1,0,1,0,0,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,0,1,1,1,1,0,0,0,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,0,0,1,0,1,0,1,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,0,0,1,0,1,1,0,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,0,0,1,1,0,0,1,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,0,0,1,1,0,1,0,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,0,0,1,1,1,0,0,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,0,1,0,0,1,0,1,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,0,1,0,0,1,1,0,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,0,1,0,1,0,0,1,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,0,1,0,1,0,1,0,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,0,1,0,1,1,0,0,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,0,1,1,0,0,0,1,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,0,1,1,0,0,1,0,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,0,1,1,0,1,0,0,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,0,1,1,1,0,0,0,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,1,0,0,0,1,0,1,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,1,0,0,0,1,1,0,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,1,0,0,1,0,0,1,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,1,0,0,1,0,1,0,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,1,0,0,1,1,0,0,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,1,0,1,0,0,0,1,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,1,0,1,0,0,1,0,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,1,0,1,0,1,0,0,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,1,0,1,1,0,0,0,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,1,1,0,0,0,0,1,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,1,1,0,0,0,1,0,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,1,1,0,0,1,0,0,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,1,1,0,1,0,0,0,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,1,1,1,1,1,0,0,0,0,0,0] => [6] => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [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) => ([(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
[1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [7] => ([],7) => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2
[1,1,0,1,0,1,0,1,0,1,1,0,0,0] => [7] => ([],7) => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2
[1,1,0,1,0,1,0,1,1,0,0,1,0,0] => [7] => ([],7) => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2
[1,1,0,1,0,1,0,1,1,0,1,0,0,0] => [7] => ([],7) => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2
[1,1,0,1,0,1,0,1,1,1,0,0,0,0] => [7] => ([],7) => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2
[1,1,0,1,0,1,1,0,0,1,0,1,0,0] => [7] => ([],7) => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2
[1,1,0,1,0,1,1,0,0,1,1,0,0,0] => [7] => ([],7) => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2
[1,1,0,1,0,1,1,0,1,0,0,1,0,0] => [7] => ([],7) => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2
[1,1,0,1,0,1,1,0,1,0,1,0,0,0] => [7] => ([],7) => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2
[1,1,0,1,0,1,1,0,1,1,0,0,0,0] => [7] => ([],7) => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2
[1,1,0,1,0,1,1,1,0,0,0,1,0,0] => [7] => ([],7) => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2
[1,1,0,1,0,1,1,1,0,0,1,0,0,0] => [7] => ([],7) => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2
[1,1,0,1,0,1,1,1,0,1,0,0,0,0] => [7] => ([],7) => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2
[1,1,0,1,0,1,1,1,1,0,0,0,0,0] => [7] => ([],7) => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2
[1,1,0,1,1,0,0,1,0,1,0,1,0,0] => [7] => ([],7) => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2
[1,1,0,1,1,0,0,1,0,1,1,0,0,0] => [7] => ([],7) => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2
[1,1,0,1,1,0,0,1,1,0,0,1,0,0] => [7] => ([],7) => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2
[1,1,0,1,1,0,0,1,1,0,1,0,0,0] => [7] => ([],7) => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2
[1,1,0,1,1,0,0,1,1,1,0,0,0,0] => [7] => ([],7) => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2
[1,1,0,1,1,0,1,0,0,1,0,1,0,0] => [7] => ([],7) => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2
[1,1,0,1,1,0,1,0,0,1,1,0,0,0] => [7] => ([],7) => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2
[1,1,0,1,1,0,1,0,1,0,0,1,0,0] => [7] => ([],7) => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2
[1,1,0,1,1,0,1,0,1,0,1,0,0,0] => [7] => ([],7) => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2
[1,1,0,1,1,0,1,0,1,1,0,0,0,0] => [7] => ([],7) => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2
[1,1,0,1,1,0,1,1,0,0,0,1,0,0] => [7] => ([],7) => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2
[1,1,0,1,1,0,1,1,0,0,1,0,0,0] => [7] => ([],7) => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2
[1,1,0,1,1,0,1,1,0,1,0,0,0,0] => [7] => ([],7) => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2
[1,1,0,1,1,0,1,1,1,0,0,0,0,0] => [7] => ([],7) => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2
[1,1,0,1,1,1,0,0,0,1,0,1,0,0] => [7] => ([],7) => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2
[1,1,0,1,1,1,0,0,0,1,1,0,0,0] => [7] => ([],7) => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2
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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.
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.
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
cone
Description
The cone of a graph.
The cone of a graph is obtained by joining a new vertex to all the vertices of the graph. The added vertex is called a universal vertex or a dominating vertex.
The cone of a graph is obtained by joining a new vertex to all the vertices of the graph. The added vertex is called a universal vertex or a dominating vertex.
Map
touch composition
Description
Sends a Dyck path to its touch composition given by the composition of lengths of its touch points.
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