Identifier
-
Mp00100:
Dyck paths
—touch composition⟶
Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤ
Values
[1,0] => [1] => ([],1) => 1
[1,0,1,0] => [1,1] => ([(0,1)],2) => 2
[1,0,1,0,1,0] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 2
[1,1,0,0,1,0] => [2,1] => ([(0,2),(1,2)],3) => 3
[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) => 2
[1,0,1,1,0,0,1,0] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 4
[1,1,0,0,1,0,1,0] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 3
[1,1,0,1,0,0,1,0] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 3
[1,1,1,0,0,0,1,0] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 3
[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) => 2
[1,0,1,0,1,1,0,0,1,0] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
[1,0,1,1,0,0,1,0,1,0] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
[1,0,1,1,0,1,0,0,1,0] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 4
[1,0,1,1,1,0,0,0,1,0] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 4
[1,1,0,0,1,0,1,0,1,0] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[1,1,0,0,1,1,0,0,1,0] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 5
[1,1,0,1,0,0,1,0,1,0] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[1,1,0,1,0,1,0,0,1,0] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 3
[1,1,0,1,1,0,0,0,1,0] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 3
[1,1,1,0,0,0,1,0,1,0] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[1,1,1,0,0,1,0,0,1,0] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 3
[1,1,1,0,1,0,0,0,1,0] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 3
[1,1,1,1,0,0,0,0,1,0] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 3
[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) => 2
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,2,1,1] => ([(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) => 4
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,2,1,1,1] => ([(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) => 4
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 4
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 4
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 4
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 4
[1,0,1,1,1,1,0,0,0,0,1,0] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 4
[1,1,0,0,1,0,1,0,1,0,1,0] => [2,1,1,1,1] => ([(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) => 3
[1,1,0,0,1,0,1,1,0,0,1,0] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
[1,1,0,0,1,1,0,0,1,0,1,0] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
[1,1,0,0,1,1,0,1,0,0,1,0] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
[1,1,0,0,1,1,1,0,0,0,1,0] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
[1,1,0,1,0,0,1,0,1,0,1,0] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,1,0,1,0,0,1,1,0,0,1,0] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
[1,1,0,1,0,1,0,0,1,0,1,0] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,1,0,1,0,1,0,1,0,0,1,0] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 3
[1,1,0,1,0,1,1,0,0,0,1,0] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 3
[1,1,0,1,1,0,0,0,1,0,1,0] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,1,0,1,1,0,0,1,0,0,1,0] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 3
[1,1,0,1,1,0,1,0,0,0,1,0] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 3
[1,1,0,1,1,1,0,0,0,0,1,0] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 3
[1,1,1,0,0,0,1,0,1,0,1,0] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,1,1,0,0,0,1,1,0,0,1,0] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
[1,1,1,0,0,1,0,0,1,0,1,0] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,1,1,0,0,1,0,1,0,0,1,0] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 3
[1,1,1,0,0,1,1,0,0,0,1,0] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 3
[1,1,1,0,1,0,0,0,1,0,1,0] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,1,1,0,1,0,0,1,0,0,1,0] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 3
[1,1,1,0,1,0,1,0,0,0,1,0] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 3
[1,1,1,0,1,1,0,0,0,0,1,0] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 3
[1,1,1,1,0,0,0,0,1,0,1,0] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,1,1,1,0,0,0,1,0,0,1,0] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 3
[1,1,1,1,0,0,1,0,0,0,1,0] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 3
[1,1,1,1,0,1,0,0,0,0,1,0] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 3
[1,1,1,1,1,0,0,0,0,0,1,0] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 3
[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) => 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,1,2,1] => ([(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) => 4
[1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,2,1,1] => ([(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) => 4
[1,0,1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
[1,0,1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
[1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,2,1,1,1] => ([(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) => 4
[1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,2,2,1] => ([(0,6),(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
[1,0,1,0,1,1,0,1,0,0,1,0,1,0] => [1,1,3,1,1] => ([(0,5),(0,6),(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) => 4
[1,0,1,0,1,1,0,1,0,1,0,0,1,0] => [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
[1,0,1,0,1,1,0,1,1,0,0,0,1,0] => [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
[1,0,1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,3,1,1] => ([(0,5),(0,6),(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) => 4
[1,0,1,0,1,1,1,0,0,1,0,0,1,0] => [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
[1,0,1,0,1,1,1,0,1,0,0,0,1,0] => [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
[1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
[1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [1,2,1,1,1,1] => ([(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) => 4
[1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [1,2,1,2,1] => ([(0,6),(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
[1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [1,2,2,1,1] => ([(0,5),(0,6),(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
[1,0,1,1,0,0,1,1,0,1,0,0,1,0] => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6
[1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6
[1,0,1,1,0,1,0,0,1,0,1,0,1,0] => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(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) => 4
[1,0,1,1,0,1,0,0,1,1,0,0,1,0] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6
[1,0,1,1,0,1,0,1,0,0,1,0,1,0] => [1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
[1,0,1,1,0,1,0,1,0,1,0,0,1,0] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 4
[1,0,1,1,0,1,0,1,1,0,0,0,1,0] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 4
[1,0,1,1,0,1,1,0,0,0,1,0,1,0] => [1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
[1,0,1,1,0,1,1,0,0,1,0,0,1,0] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 4
[1,0,1,1,0,1,1,0,1,0,0,0,1,0] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 4
[1,0,1,1,0,1,1,1,0,0,0,0,1,0] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 4
[1,0,1,1,1,0,0,0,1,0,1,0,1,0] => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(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) => 4
[1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6
[1,0,1,1,1,0,0,1,0,0,1,0,1,0] => [1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
[1,0,1,1,1,0,0,1,0,1,0,0,1,0] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 4
[1,0,1,1,1,0,0,1,1,0,0,0,1,0] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 4
[1,0,1,1,1,0,1,0,0,0,1,0,1,0] => [1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
[1,0,1,1,1,0,1,0,0,1,0,0,1,0] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 4
[1,0,1,1,1,0,1,0,1,0,0,0,1,0] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 4
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Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Map
touch composition
Description
Sends a Dyck path to its touch composition given by the composition of lengths of its touch points.
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.
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