- FindStat

Identifier

Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000301: Graphs ⟶ ℤ

Values

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

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

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

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

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

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

[1,1,0,1,0,0] => [3] => ([],3) => 6

[1,1,1,0,0,0] => [3] => ([],3) => 6

[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) => 5

[1,0,1,0,1,1,0,0] => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => 6

[1,0,1,1,0,0,1,0] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 6

[1,0,1,1,0,1,0,0] => [1,3] => ([(2,3)],4) => 7

[1,0,1,1,1,0,0,0] => [1,3] => ([(2,3)],4) => 7

[1,1,0,0,1,0,1,0] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 6

[1,1,0,0,1,1,0,0] => [2,2] => ([(1,3),(2,3)],4) => 7

[1,1,0,1,0,0,1,0] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 7

[1,1,0,1,0,1,0,0] => [4] => ([],4) => 8

[1,1,0,1,1,0,0,0] => [4] => ([],4) => 8

[1,1,1,0,0,0,1,0] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 7

[1,1,1,0,0,1,0,0] => [4] => ([],4) => 8

[1,1,1,0,1,0,0,0] => [4] => ([],4) => 8

[1,1,1,1,0,0,0,0] => [4] => ([],4) => 8

[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) => 6

[1,0,1,0,1,0,1,1,0,0] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 7

[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) => 7

[1,0,1,0,1,1,0,1,0,0] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 8

[1,0,1,0,1,1,1,0,0,0] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 8

[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) => 7

[1,0,1,1,0,0,1,1,0,0] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 8

[1,0,1,1,0,1,0,0,1,0] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 8

[1,0,1,1,0,1,0,1,0,0] => [1,4] => ([(3,4)],5) => 9

[1,0,1,1,0,1,1,0,0,0] => [1,4] => ([(3,4)],5) => 9

[1,0,1,1,1,0,0,0,1,0] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 8

[1,0,1,1,1,0,0,1,0,0] => [1,4] => ([(3,4)],5) => 9

[1,0,1,1,1,0,1,0,0,0] => [1,4] => ([(3,4)],5) => 9

[1,0,1,1,1,1,0,0,0,0] => [1,4] => ([(3,4)],5) => 9

[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) => 7

[1,1,0,0,1,0,1,1,0,0] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 8

[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) => 8

[1,1,0,0,1,1,0,1,0,0] => [2,3] => ([(2,4),(3,4)],5) => 9

[1,1,0,0,1,1,1,0,0,0] => [2,3] => ([(2,4),(3,4)],5) => 9

[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) => 8

[1,1,0,1,0,0,1,1,0,0] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 9

[1,1,0,1,0,1,0,0,1,0] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 9

[1,1,0,1,0,1,0,1,0,0] => [5] => ([],5) => 10

[1,1,0,1,0,1,1,0,0,0] => [5] => ([],5) => 10

[1,1,0,1,1,0,0,0,1,0] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 9

[1,1,0,1,1,0,0,1,0,0] => [5] => ([],5) => 10

[1,1,0,1,1,0,1,0,0,0] => [5] => ([],5) => 10

[1,1,0,1,1,1,0,0,0,0] => [5] => ([],5) => 10

[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) => 8

[1,1,1,0,0,0,1,1,0,0] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 9

[1,1,1,0,0,1,0,0,1,0] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 9

[1,1,1,0,0,1,0,1,0,0] => [5] => ([],5) => 10

[1,1,1,0,0,1,1,0,0,0] => [5] => ([],5) => 10

[1,1,1,0,1,0,0,0,1,0] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 9

[1,1,1,0,1,0,0,1,0,0] => [5] => ([],5) => 10

[1,1,1,0,1,0,1,0,0,0] => [5] => ([],5) => 10

[1,1,1,0,1,1,0,0,0,0] => [5] => ([],5) => 10

[1,1,1,1,0,0,0,0,1,0] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 9

[1,1,1,1,0,0,0,1,0,0] => [5] => ([],5) => 10

[1,1,1,1,0,0,1,0,0,0] => [5] => ([],5) => 10

[1,1,1,1,0,1,0,0,0,0] => [5] => ([],5) => 10

[1,1,1,1,1,0,0,0,0,0] => [5] => ([],5) => 10

[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) => 7

[1,0,1,0,1,0,1,0,1,1,0,0] => [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) => 8

[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) => 8

[1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 9

[1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 9

[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) => 8

[1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 9

[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) => 9

[1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 10

[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 10

[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) => 9

[1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 10

[1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 10

[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 10

[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) => 8

[1,0,1,1,0,0,1,0,1,1,0,0] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 9

[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) => 9

[1,0,1,1,0,0,1,1,0,1,0,0] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6) => 10

[1,0,1,1,0,0,1,1,1,0,0,0] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6) => 10

[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) => 9

[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 10

[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) => 10

[1,0,1,1,0,1,0,1,0,1,0,0] => [1,5] => ([(4,5)],6) => 11

[1,0,1,1,0,1,0,1,1,0,0,0] => [1,5] => ([(4,5)],6) => 11

[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) => 10

[1,0,1,1,0,1,1,0,0,1,0,0] => [1,5] => ([(4,5)],6) => 11

[1,0,1,1,0,1,1,0,1,0,0,0] => [1,5] => ([(4,5)],6) => 11

[1,0,1,1,0,1,1,1,0,0,0,0] => [1,5] => ([(4,5)],6) => 11

[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) => 9

[1,0,1,1,1,0,0,0,1,1,0,0] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 10

[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) => 10

[1,0,1,1,1,0,0,1,0,1,0,0] => [1,5] => ([(4,5)],6) => 11

[1,0,1,1,1,0,0,1,1,0,0,0] => [1,5] => ([(4,5)],6) => 11

[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) => 10

[1,0,1,1,1,0,1,0,0,1,0,0] => [1,5] => ([(4,5)],6) => 11

[1,0,1,1,1,0,1,0,1,0,0,0] => [1,5] => ([(4,5)],6) => 11

[1,0,1,1,1,0,1,1,0,0,0,0] => [1,5] => ([(4,5)],6) => 11

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$F_{1} = q^{2}$

$F_{2} = q^{3} + q^{4}$

$F_{3} = q^{4} + 2\ q^{5} + 2\ q^{6}$

$F_{4} = q^{5} + 3\ q^{6} + 5\ q^{7} + 5\ q^{8}$

$F_{5} = q^{6} + 4\ q^{7} + 9\ q^{8} + 14\ q^{9} + 14\ q^{10}$

$F_{6} = q^{7} + 5\ q^{8} + 14\ q^{9} + 28\ q^{10} + 42\ q^{11} + 42\ q^{12}$

Description

The number of facets of the stable set polytope of a graph.
The stable set polytope of a graph

$G$ is the convex hull of the characteristic vectors of stable (or independent) sets of vertices of

$G$ inside

$\mathbb{R}^{V(G)}$ .

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

touch composition

Description

Sends a Dyck path to its touch composition given by the composition of lengths of its touch points.