Identifier
Values
([],1) => [1] => [[1],[]] => 0
([],2) => [2] => [[2],[]] => 0
([(0,1)],2) => [1,1] => [[1,1],[]] => 0
([],3) => [3] => [[3],[]] => 0
([(1,2)],3) => [1,2] => [[2,1],[]] => 0
([(0,2),(1,2)],3) => [1,1,1] => [[1,1,1],[]] => 0
([(0,1),(0,2),(1,2)],3) => [2,1] => [[2,2],[1]] => 1
([],4) => [4] => [[4],[]] => 0
([(2,3)],4) => [1,3] => [[3,1],[]] => 0
([(1,3),(2,3)],4) => [1,1,2] => [[2,1,1],[]] => 0
([(0,3),(1,3),(2,3)],4) => [1,2,1] => [[2,2,1],[1]] => 1
([(0,3),(1,2)],4) => [2,2] => [[3,2],[1]] => 1
([(0,3),(1,2),(2,3)],4) => [1,1,1,1] => [[1,1,1,1],[]] => 0
([(1,2),(1,3),(2,3)],4) => [2,2] => [[3,2],[1]] => 1
([(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => [[1,1,1,1],[]] => 0
([(0,2),(0,3),(1,2),(1,3)],4) => [1,2,1] => [[2,2,1],[1]] => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [2,1,1] => [[2,2,2],[1,1]] => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [3,1] => [[3,3],[2]] => 2
([],5) => [5] => [[5],[]] => 0
([(3,4)],5) => [1,4] => [[4,1],[]] => 0
([(2,4),(3,4)],5) => [1,1,3] => [[3,1,1],[]] => 0
([(1,4),(2,4),(3,4)],5) => [1,2,2] => [[3,2,1],[1]] => 1
([(0,4),(1,4),(2,4),(3,4)],5) => [1,3,1] => [[3,3,1],[2]] => 2
([(1,4),(2,3)],5) => [2,3] => [[4,2],[1]] => 1
([(1,4),(2,3),(3,4)],5) => [1,1,1,2] => [[2,1,1,1],[]] => 0
([(0,1),(2,4),(3,4)],5) => [1,1,1,2] => [[2,1,1,1],[]] => 0
([(2,3),(2,4),(3,4)],5) => [2,3] => [[4,2],[1]] => 1
([(0,4),(1,4),(2,3),(3,4)],5) => [1,1,1,1,1] => [[1,1,1,1,1],[]] => 0
([(1,4),(2,3),(2,4),(3,4)],5) => [1,1,1,2] => [[2,1,1,1],[]] => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => [1,1,2,1] => [[2,2,1,1],[1]] => 1
([(1,3),(1,4),(2,3),(2,4)],5) => [1,2,2] => [[3,2,1],[1]] => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => [1,1,1,1,1] => [[1,1,1,1,1],[]] => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [2,1,2] => [[3,2,2],[1,1]] => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [1,1,1,1,1] => [[1,1,1,1,1],[]] => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [1,1,1,1,1] => [[1,1,1,1,1],[]] => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => [1,1,2,1] => [[2,2,1,1],[1]] => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [2,2,1] => [[3,3,2],[2,1]] => 2
([(0,4),(1,3),(2,3),(2,4)],5) => [1,1,1,1,1] => [[1,1,1,1,1],[]] => 0
([(0,1),(2,3),(2,4),(3,4)],5) => [2,1,2] => [[3,2,2],[1,1]] => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => [1,1,1,1,1] => [[1,1,1,1,1],[]] => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => [1,2,1,1] => [[2,2,2,1],[1,1]] => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => [2,2,1] => [[3,3,2],[2,1]] => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [1,1,1,1,1] => [[1,1,1,1,1],[]] => 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => [1,1,1,1,1] => [[1,1,1,1,1],[]] => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => [1,1,1,1,1] => [[1,1,1,1,1],[]] => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [3,2] => [[4,3],[2]] => 2
([(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]] => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [2,1,1,1] => [[2,2,2,2],[1,1,1]] => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => [1,1,1,1,1] => [[1,1,1,1,1],[]] => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => [2,2,1] => [[3,3,2],[2,1]] => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [3,1,1] => [[3,3,3],[2,2]] => 2
([(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 missing boxes in the first row.
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 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.