- FindStat

Identifier

Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001389: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

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

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

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

$F_{5} = q + q^{2} + 3\ q^{3} + 8\ q^{4} + 21\ q^{5}$

$F_{6} = q + q^{2} + 3\ q^{3} + 9\ q^{4} + 30\ q^{5} + 112\ q^{6}$

$F_{7} = q + q^{2} + 3\ q^{3} + 9\ q^{4} + 32\ q^{5} + 145\ q^{6} + 853\ q^{7}$

Description

The number of partitions of the same length below the given integer partition.
For a partition

$\lambda_1 \geq \dots \lambda_k > 0$ , this number is

$\det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$

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

Laplacian multiplicities

Description

The composition of multiplicities of the Laplacian eigenvalues.
Let

$\lambda_1 > \lambda_2 > \dots$ be the eigenvalues of the Laplacian matrix of a graph on

$n$ vertices. Then this map returns the composition

$a_1,\dots,a_k$ of

$n$ where

$a_i$ is the multiplicity of

$\lambda_i$ .

Map

to partition of connected components

Description

Return the partition of the sizes of the connected components of the graph.