Identifier
-
Mp00152:
Graphs
—Laplacian multiplicities⟶
Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000800: Permutations ⟶ ℤ
Values
([],1) => [1] => [1,0] => [2,1] => 0
([],2) => [2] => [1,1,0,0] => [2,3,1] => 1
([(0,1)],2) => [1,1] => [1,0,1,0] => [3,1,2] => 0
([],3) => [3] => [1,1,1,0,0,0] => [2,3,4,1] => 2
([(1,2)],3) => [1,2] => [1,0,1,1,0,0] => [3,1,4,2] => 1
([(0,2),(1,2)],3) => [1,1,1] => [1,0,1,0,1,0] => [4,1,2,3] => 0
([(0,1),(0,2),(1,2)],3) => [2,1] => [1,1,0,0,1,0] => [2,4,1,3] => 1
([],4) => [4] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 3
([(2,3)],4) => [1,3] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 2
([(1,3),(2,3)],4) => [1,1,2] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 1
([(0,3),(1,3),(2,3)],4) => [1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 1
([(0,3),(1,2)],4) => [2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 1
([(0,3),(1,2),(2,3)],4) => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
([(1,2),(1,3),(2,3)],4) => [2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 1
([(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
([(0,2),(0,3),(1,2),(1,3)],4) => [1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [2,1,1] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [3,1] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 2
([],5) => [5] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 4
([(3,4)],5) => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 3
([(2,4),(3,4)],5) => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 2
([(1,4),(2,4),(3,4)],5) => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 1
([(0,4),(1,4),(2,4),(3,4)],5) => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 2
([(1,4),(2,3)],5) => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 1
([(1,4),(2,3),(3,4)],5) => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => 1
([(0,1),(2,4),(3,4)],5) => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => 1
([(2,3),(2,4),(3,4)],5) => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 1
([(0,4),(1,4),(2,3),(3,4)],5) => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => 0
([(1,4),(2,3),(2,4),(3,4)],5) => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => 1
([(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] => [4,1,2,6,3,5] => 1
([(1,3),(1,4),(2,3),(2,4)],5) => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 1
([(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] => [6,1,2,3,4,5] => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 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] => [6,1,2,3,4,5] => 0
([(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] => [6,1,2,3,4,5] => 0
([(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] => [4,1,2,6,3,5] => 1
([(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] => [2,4,1,6,3,5] => 1
([(0,4),(1,3),(2,3),(2,4)],5) => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => 0
([(0,1),(2,3),(2,4),(3,4)],5) => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 1
([(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] => [6,1,2,3,4,5] => 0
([(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] => [3,1,6,2,4,5] => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 1
([(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] => [6,1,2,3,4,5] => 0
([(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] => [6,1,2,3,4,5] => 0
([(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] => [6,1,2,3,4,5] => 0
([(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] => [2,3,5,1,6,4] => 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] => [3,1,6,2,4,5] => 1
([(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] => [2,6,1,3,4,5] => 1
([(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] => [6,1,2,3,4,5] => 0
([(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] => [2,4,1,6,3,5] => 1
([(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] => [2,3,6,1,4,5] => 2
([(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] => [2,3,4,6,1,5] => 3
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Description
The number of occurrences of the vincular pattern |231 in a permutation.
This is the number of occurrences of the pattern $(2,3,1)$, such that the letter matched by $2$ is the first entry of the permutation.
This is the number of occurrences of the pattern $(2,3,1)$, such that the letter matched by $2$ is the first entry of the permutation.
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 $\lambda_i$.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
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