Identifier
-
Mp00152:
Graphs
—Laplacian multiplicities⟶
Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St001235: Integer compositions ⟶ ℤ
Values
([],1) => [1] => [1] => 1
([],2) => [2] => [1,1] => 2
([(0,1)],2) => [1,1] => [2] => 1
([],3) => [3] => [1,1,1] => 3
([(1,2)],3) => [1,2] => [2,1] => 2
([(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,1),(0,2),(1,2)],3) => [2,1] => [1,2] => 2
([],4) => [4] => [1,1,1,1] => 4
([(2,3)],4) => [1,3] => [2,1,1] => 3
([(1,3),(2,3)],4) => [1,1,2] => [3,1] => 2
([(0,3),(1,3),(2,3)],4) => [1,2,1] => [2,2] => 2
([(0,3),(1,2)],4) => [2,2] => [1,2,1] => 2
([(0,3),(1,2),(2,3)],4) => [1,1,1,1] => [4] => 1
([(1,2),(1,3),(2,3)],4) => [2,2] => [1,2,1] => 2
([(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => [4] => 1
([(0,2),(0,3),(1,2),(1,3)],4) => [1,2,1] => [2,2] => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [2,1,1] => [1,3] => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [3,1] => [1,1,2] => 3
([],5) => [5] => [1,1,1,1,1] => 5
([(3,4)],5) => [1,4] => [2,1,1,1] => 4
([(2,4),(3,4)],5) => [1,1,3] => [3,1,1] => 3
([(1,4),(2,4),(3,4)],5) => [1,2,2] => [2,2,1] => 2
([(0,4),(1,4),(2,4),(3,4)],5) => [1,3,1] => [2,1,2] => 3
([(1,4),(2,3)],5) => [2,3] => [1,2,1,1] => 3
([(1,4),(2,3),(3,4)],5) => [1,1,1,2] => [4,1] => 2
([(0,1),(2,4),(3,4)],5) => [1,1,1,2] => [4,1] => 2
([(2,3),(2,4),(3,4)],5) => [2,3] => [1,2,1,1] => 3
([(0,4),(1,4),(2,3),(3,4)],5) => [1,1,1,1,1] => [5] => 1
([(1,4),(2,3),(2,4),(3,4)],5) => [1,1,1,2] => [4,1] => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => [1,1,2,1] => [3,2] => 2
([(1,3),(1,4),(2,3),(2,4)],5) => [1,2,2] => [2,2,1] => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => [1,1,1,1,1] => [5] => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [2,1,2] => [1,3,1] => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [1,1,1,1,1] => [5] => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [1,1,1,1,1] => [5] => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => [1,1,2,1] => [3,2] => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [2,2,1] => [1,2,2] => 2
([(0,4),(1,3),(2,3),(2,4)],5) => [1,1,1,1,1] => [5] => 1
([(0,1),(2,3),(2,4),(3,4)],5) => [2,1,2] => [1,3,1] => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => [1,1,1,1,1] => [5] => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => [1,2,1,1] => [2,3] => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => [2,2,1] => [1,2,2] => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [1,1,1,1,1] => [5] => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => [1,1,1,1,1] => [5] => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => [1,1,1,1,1] => [5] => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [3,2] => [1,1,2,1] => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [1,2,1,1] => [2,3] => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [2,1,1,1] => [1,4] => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => [1,1,1,1,1] => [5] => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => [2,2,1] => [1,2,2] => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,1,3] => 3
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => [1,1,1,2] => 4
([],6) => [6] => [1,1,1,1,1,1] => 6
([(4,5)],6) => [1,5] => [2,1,1,1,1] => 5
([(3,5),(4,5)],6) => [1,1,4] => [3,1,1,1] => 4
([(2,5),(3,5),(4,5)],6) => [1,2,3] => [2,2,1,1] => 3
([(1,5),(2,5),(3,5),(4,5)],6) => [1,3,2] => [2,1,2,1] => 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => [1,4,1] => [2,1,1,2] => 4
([(2,5),(3,4)],6) => [2,4] => [1,2,1,1,1] => 4
([(2,5),(3,4),(4,5)],6) => [1,1,1,3] => [4,1,1] => 3
([(1,2),(3,5),(4,5)],6) => [1,1,1,3] => [4,1,1] => 3
([(3,4),(3,5),(4,5)],6) => [2,4] => [1,2,1,1,1] => 4
([(1,5),(2,5),(3,4),(4,5)],6) => [1,1,1,1,2] => [5,1] => 2
([(0,1),(2,5),(3,5),(4,5)],6) => [1,1,2,2] => [3,2,1] => 2
([(2,5),(3,4),(3,5),(4,5)],6) => [1,1,1,3] => [4,1,1] => 3
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => [1,1,2,1,1] => [3,3] => 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [1,1,2,2] => [3,2,1] => 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [1,1,3,1] => [3,1,2] => 3
([(2,4),(2,5),(3,4),(3,5)],6) => [1,2,3] => [2,2,1,1] => 3
([(0,5),(1,5),(2,4),(3,4)],6) => [2,2,2] => [1,2,2,1] => 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [1,1,1,1,2] => [5,1] => 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => [1,1,1,1,1,1] => [6] => 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [2,1,3] => [1,3,1,1] => 3
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [1,1,1,1,2] => [5,1] => 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => [1,1,2,1,1] => [3,3] => 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [1,1,1,1,1,1] => [6] => 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [1,1,1,1,2] => [5,1] => 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [1,1,1,1,1,1] => [6] => 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [1,1,1,2,1] => [4,2] => 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [1,1,2,2] => [3,2,1] => 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => [1,1,1,1,1,1] => [6] => 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [1,1,2,1,1] => [3,3] => 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [2,2,2] => [1,2,2,1] => 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [1,1,1,1,1,1] => [6] => 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [1,1,2,1,1] => [3,3] => 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [1,1,3,1] => [3,1,2] => 3
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [2,3,1] => [1,2,1,2] => 3
([(0,5),(1,4),(2,3)],6) => [3,3] => [1,1,2,1,1] => 3
([(1,5),(2,4),(3,4),(3,5)],6) => [1,1,1,1,2] => [5,1] => 2
([(0,1),(2,5),(3,4),(4,5)],6) => [1,2,1,2] => [2,3,1] => 2
([(1,2),(3,4),(3,5),(4,5)],6) => [2,1,3] => [1,3,1,1] => 3
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => [1,1,1,1,1,1] => [6] => 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [1,1,1,1,2] => [5,1] => 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => [1,1,1,1,2] => [5,1] => 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [1,1,1,1,1,1] => [6] => 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [1,2,1,2] => [2,3,1] => 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [1,2,2,1] => [2,2,2] => 2
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => [2,2,2] => [1,2,2,1] => 2
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => [1,1,1,1,1,1] => [6] => 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [1,1,1,1,2] => [5,1] => 2
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => [1,1,1,1,1,1] => [6] => 1
>>> Load all 208 entries. <<<
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
Description
The global dimension of the corresponding Comp-Nakayama algebra.
We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
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
complement
Description
The complement of a composition.
The complement of a composition $I$ is defined as follows:
If $I$ is the empty composition, then the complement is also the empty composition. Otherwise, let $S$ be the descent set corresponding to $I=(i_1,\dots,i_k)$, that is, the subset
$$\{ i_1, i_1 + i_2, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$$
of $\{ 1, 2, \ldots, |I|-1 \}$. Then, the complement of $I$ is the composition of the same size as $I$, whose descent set is $\{ 1, 2, \ldots, |I|-1 \} \setminus S$.
The complement of a composition $I$ coincides with the reversal (Mp00038reverse) of the composition conjugate (Mp00041conjugate) to $I$.
The complement of a composition $I$ is defined as follows:
If $I$ is the empty composition, then the complement is also the empty composition. Otherwise, let $S$ be the descent set corresponding to $I=(i_1,\dots,i_k)$, that is, the subset
$$\{ i_1, i_1 + i_2, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$$
of $\{ 1, 2, \ldots, |I|-1 \}$. Then, the complement of $I$ is the composition of the same size as $I$, whose descent set is $\{ 1, 2, \ldots, |I|-1 \} \setminus S$.
The complement of a composition $I$ coincides with the reversal (Mp00038reverse) of the composition conjugate (Mp00041conjugate) to $I$.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!