Identifier
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Mp00152:
Graphs
—Laplacian multiplicities⟶
Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤ (values match St000053The number of valleys of the Dyck path., St001068Number of torsionless simple modules in the corresponding Nakayama algebra., St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra.)
Values
([],1) => [1] => [1,0] => 1
([],2) => [2] => [1,1,0,0] => 1
([(0,1)],2) => [1,1] => [1,0,1,0] => 2
([],3) => [3] => [1,1,1,0,0,0] => 1
([(1,2)],3) => [1,2] => [1,0,1,1,0,0] => 2
([(0,2),(1,2)],3) => [1,1,1] => [1,0,1,0,1,0] => 3
([(0,1),(0,2),(1,2)],3) => [2,1] => [1,1,0,0,1,0] => 2
([],4) => [4] => [1,1,1,1,0,0,0,0] => 1
([(2,3)],4) => [1,3] => [1,0,1,1,1,0,0,0] => 2
([(1,3),(2,3)],4) => [1,1,2] => [1,0,1,0,1,1,0,0] => 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
([(0,3),(1,2),(2,3)],4) => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 4
([(1,2),(1,3),(2,3)],4) => [2,2] => [1,1,0,0,1,1,0,0] => 2
([(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 4
([(0,2),(0,3),(1,2),(1,3)],4) => [1,2,1] => [1,0,1,1,0,0,1,0] => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [2,1,1] => [1,1,0,0,1,0,1,0] => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [3,1] => [1,1,1,0,0,0,1,0] => 2
([],5) => [5] => [1,1,1,1,1,0,0,0,0,0] => 1
([(3,4)],5) => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 2
([(2,4),(3,4)],5) => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 3
([(1,4),(2,4),(3,4)],5) => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 3
([(0,4),(1,4),(2,4),(3,4)],5) => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 3
([(1,4),(2,3)],5) => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2
([(1,4),(2,3),(3,4)],5) => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 4
([(0,1),(2,4),(3,4)],5) => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 4
([(2,3),(2,4),(3,4)],5) => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2
([(0,4),(1,4),(2,3),(3,4)],5) => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 5
([(1,4),(2,3),(2,4),(3,4)],5) => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 4
([(1,3),(1,4),(2,3),(2,4)],5) => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 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
([(0,4),(1,3),(2,3),(2,4)],5) => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 5
([(0,1),(2,3),(2,4),(3,4)],5) => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 4
([(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
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 5
([(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
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 5
([(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
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 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] => 2
([],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
([(4,5)],6) => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
([(3,5),(4,5)],6) => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => 3
([(2,5),(3,5),(4,5)],6) => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 3
([(1,5),(2,5),(3,5),(4,5)],6) => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 3
([(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] => 3
([(2,5),(3,4)],6) => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 2
([(2,5),(3,4),(4,5)],6) => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 4
([(1,2),(3,5),(4,5)],6) => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 4
([(3,4),(3,5),(4,5)],6) => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 2
([(1,5),(2,5),(3,4),(4,5)],6) => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 5
([(0,1),(2,5),(3,5),(4,5)],6) => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 4
([(2,5),(3,4),(3,5),(4,5)],6) => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 4
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 5
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 4
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 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] => 3
([(0,5),(1,5),(2,4),(3,4)],6) => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 5
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 3
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 5
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 5
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 5
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => 5
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 4
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 5
([(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] => 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 5
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 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] => 3
([(0,5),(1,4),(2,3)],6) => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 2
([(1,5),(2,4),(3,4),(3,5)],6) => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 5
([(0,1),(2,5),(3,4),(4,5)],6) => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 4
([(1,2),(3,4),(3,5),(4,5)],6) => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 3
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 5
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 5
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 4
([(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] => 4
([(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] => 3
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 5
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
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Description
The number of peaks of a Dyck path.
Map
bounce path
Description
The bounce path determined by an integer composition.
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 $\lambda_i$.
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 $\lambda_i$.
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