Identifier
-
Mp00117:
Graphs
—Ore closure⟶
Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001488: Skew partitions ⟶ ℤ
Values
([],1) => ([],1) => [1] => [[1],[]] => 1
([],2) => ([],2) => [2] => [[2],[]] => 2
([(0,1)],2) => ([(0,1)],2) => [1,1] => [[1,1],[]] => 2
([],3) => ([],3) => [3] => [[3],[]] => 2
([(1,2)],3) => ([(1,2)],3) => [1,2] => [[2,1],[]] => 3
([(0,2),(1,2)],3) => ([(0,2),(1,2)],3) => [1,1,1] => [[1,1,1],[]] => 2
([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => [2,1] => [[2,2],[1]] => 3
([],4) => ([],4) => [4] => [[4],[]] => 2
([(2,3)],4) => ([(2,3)],4) => [1,3] => [[3,1],[]] => 3
([(1,3),(2,3)],4) => ([(1,3),(2,3)],4) => [1,1,2] => [[2,1,1],[]] => 3
([(0,3),(1,3),(2,3)],4) => ([(0,3),(1,3),(2,3)],4) => [1,2,1] => [[2,2,1],[1]] => 4
([(0,3),(1,2)],4) => ([(0,3),(1,2)],4) => [2,2] => [[3,2],[1]] => 4
([(0,3),(1,2),(2,3)],4) => ([(0,3),(1,2),(2,3)],4) => [1,1,1,1] => [[1,1,1,1],[]] => 2
([(1,2),(1,3),(2,3)],4) => ([(1,2),(1,3),(2,3)],4) => [2,2] => [[3,2],[1]] => 4
([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => [[1,1,1,1],[]] => 2
([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [3,1] => [[3,3],[2]] => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [3,1] => [[3,3],[2]] => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [3,1] => [[3,3],[2]] => 3
([],5) => ([],5) => [5] => [[5],[]] => 2
([(3,4)],5) => ([(3,4)],5) => [1,4] => [[4,1],[]] => 3
([(2,4),(3,4)],5) => ([(2,4),(3,4)],5) => [1,1,3] => [[3,1,1],[]] => 3
([(1,4),(2,4),(3,4)],5) => ([(1,4),(2,4),(3,4)],5) => [1,2,2] => [[3,2,1],[1]] => 5
([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,4),(1,4),(2,4),(3,4)],5) => [1,3,1] => [[3,3,1],[2]] => 4
([(1,4),(2,3)],5) => ([(1,4),(2,3)],5) => [2,3] => [[4,2],[1]] => 4
([(1,4),(2,3),(3,4)],5) => ([(1,4),(2,3),(3,4)],5) => [1,1,1,2] => [[2,1,1,1],[]] => 3
([(0,1),(2,4),(3,4)],5) => ([(0,1),(2,4),(3,4)],5) => [1,1,1,2] => [[2,1,1,1],[]] => 3
([(2,3),(2,4),(3,4)],5) => ([(2,3),(2,4),(3,4)],5) => [2,3] => [[4,2],[1]] => 4
([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(1,4),(2,3),(3,4)],5) => [1,1,1,1,1] => [[1,1,1,1,1],[]] => 2
([(1,4),(2,3),(2,4),(3,4)],5) => ([(1,4),(2,3),(2,4),(3,4)],5) => [1,1,1,2] => [[2,1,1,1],[]] => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => [1,1,2,1] => [[2,2,1,1],[1]] => 4
([(1,3),(1,4),(2,3),(2,4)],5) => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [3,2] => [[4,3],[2]] => 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [1,1,1,1,1] => [[1,1,1,1,1],[]] => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [3,2] => [[4,3],[2]] => 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [1,1,1,1,1] => [[1,1,1,1,1],[]] => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [1,1,1,1,1] => [[1,1,1,1,1],[]] => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [2,2,1] => [[3,3,2],[2,1]] => 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [2,2,1] => [[3,3,2],[2,1]] => 5
([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => [1,1,1,1,1] => [[1,1,1,1,1],[]] => 2
([(0,1),(2,3),(2,4),(3,4)],5) => ([(0,1),(2,3),(2,4),(3,4)],5) => [2,1,2] => [[3,2,2],[1,1]] => 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => [1,1,1,1,1] => [[1,1,1,1,1],[]] => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => [1,2,1,1] => [[2,2,2,1],[1,1]] => 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => [2,2,1] => [[3,3,2],[2,1]] => 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => [[4,4],[3]] => 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => [[4,4],[3]] => 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [1,2,1,1] => [[2,2,2,1],[1,1]] => 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [3,2] => [[4,3],[2]] => 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [1,2,1,1] => [[2,2,2,1],[1,1]] => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => [[4,4],[3]] => 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => [[4,4],[3]] => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => [[4,4],[3]] => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => [[4,4],[3]] => 3
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => [[4,4],[3]] => 3
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Description
The number of corners of a skew partition.
This is also known as the number of removable cells of the skew partition.
This is also known as the number of removable cells of the skew partition.
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
to ribbon
Description
The ribbon shape corresponding to an integer composition.
For an integer composition $(a_1, \dots, a_n)$, this is the ribbon shape whose $i$th row from the bottom has $a_i$ cells.
For an integer composition $(a_1, \dots, a_n)$, this is the ribbon shape whose $i$th row from the bottom has $a_i$ cells.
Map
Ore closure
Description
The Ore closure of a graph.
The Ore closure of a connected graph $G$ has the same vertices as $G$, and the smallest set of edges containing the edges of $G$ such that for any two vertices $u$ and $v$ whose sum of degrees is at least the number of vertices, then $(u,v)$ is also an edge.
For disconnected graphs, we compute the closure separately for each component.
The Ore closure of a connected graph $G$ has the same vertices as $G$, and the smallest set of edges containing the edges of $G$ such that for any two vertices $u$ and $v$ whose sum of degrees is at least the number of vertices, then $(u,v)$ is also an edge.
For disconnected graphs, we compute the closure separately for each component.
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