- FindStat

Identifier

Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([],7) => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 0

([(5,6)],7) => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6

([(3,6),(4,6),(5,6)],7) => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => 6

([(2,6),(3,6),(4,6),(5,6)],7) => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => 6

([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => 6

([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 6

([(3,6),(4,5)],7) => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => 5

([(4,5),(4,6),(5,6)],7) => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => 5

([(3,5),(3,6),(4,5),(4,6)],7) => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => 6

([(1,6),(2,6),(3,5),(4,5)],7) => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => 5

([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => 5

([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => 5

([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => 5

([(1,6),(2,5),(3,4)],7) => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => 4

([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 6

([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => 6

([(2,5),(2,6),(3,4),(3,6),(4,5)],7) => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => 5

([(1,2),(3,5),(3,6),(4,5),(4,6)],7) => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => 6

([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7) => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 6

([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => 5

([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7) => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 6

([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => 4

([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 6

([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => 6

([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => 5

([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7) => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 6

([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => 5

([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7) => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 6

([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7) => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => 6

([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => 4

([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7) => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => 6

([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => 4

([(0,3),(1,2),(4,5),(4,6),(5,6)],7) => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => 5

([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7) => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => 6

([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7) => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => 3

([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => 5

([(0,5),(0,6),(1,2),(1,4),(2,3),(3,5),(4,6)],7) => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => 5

([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 6

([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,4),(3,5),(4,5)],7) => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => 4

([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7) => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 6

([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => 3

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

([(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)],7) => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => 6

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

([(1,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 6

([(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),(4,6),(5,6)],7) => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => 5

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$F_{1} = 1$

$F_{2} = 1 + q$

Description

The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.

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

bounce path

Description

The bounce path determined by an integer composition.