Identifier
-
Mp00247:
Graphs
—de-duplicate⟶
Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001571: Integer partitions ⟶ ℤ
Values
([(1,2)],3) => ([(1,2)],3) => [2,1] => [1] => 1
([(2,3)],4) => ([(1,2)],3) => [2,1] => [1] => 1
([(1,3),(2,3)],4) => ([(1,2)],3) => [2,1] => [1] => 1
([(0,3),(1,2)],4) => ([(0,3),(1,2)],4) => [2,2] => [2] => 2
([(1,2),(1,3),(2,3)],4) => ([(1,2),(1,3),(2,3)],4) => [3,1] => [1] => 1
([(3,4)],5) => ([(1,2)],3) => [2,1] => [1] => 1
([(2,4),(3,4)],5) => ([(1,2)],3) => [2,1] => [1] => 1
([(1,4),(2,4),(3,4)],5) => ([(1,2)],3) => [2,1] => [1] => 1
([(1,4),(2,3)],5) => ([(1,4),(2,3)],5) => [2,2,1] => [2,1] => 1
([(1,4),(2,3),(3,4)],5) => ([(1,4),(2,3),(3,4)],5) => [4,1] => [1] => 1
([(0,1),(2,4),(3,4)],5) => ([(0,3),(1,2)],4) => [2,2] => [2] => 2
([(2,3),(2,4),(3,4)],5) => ([(1,2),(1,3),(2,3)],4) => [3,1] => [1] => 1
([(1,4),(2,3),(2,4),(3,4)],5) => ([(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => [1] => 1
([(1,3),(1,4),(2,3),(2,4)],5) => ([(1,2)],3) => [2,1] => [1] => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,2),(1,3),(2,3)],4) => [3,1] => [1] => 1
([(0,1),(2,3),(2,4),(3,4)],5) => ([(0,1),(2,3),(2,4),(3,4)],5) => [3,2] => [2] => 2
([(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) => [4,1] => [1] => 1
([(4,5)],6) => ([(1,2)],3) => [2,1] => [1] => 1
([(3,5),(4,5)],6) => ([(1,2)],3) => [2,1] => [1] => 1
([(2,5),(3,5),(4,5)],6) => ([(1,2)],3) => [2,1] => [1] => 1
([(1,5),(2,5),(3,5),(4,5)],6) => ([(1,2)],3) => [2,1] => [1] => 1
([(2,5),(3,4)],6) => ([(1,4),(2,3)],5) => [2,2,1] => [2,1] => 1
([(2,5),(3,4),(4,5)],6) => ([(1,4),(2,3),(3,4)],5) => [4,1] => [1] => 1
([(1,2),(3,5),(4,5)],6) => ([(1,4),(2,3)],5) => [2,2,1] => [2,1] => 1
([(3,4),(3,5),(4,5)],6) => ([(1,2),(1,3),(2,3)],4) => [3,1] => [1] => 1
([(1,5),(2,5),(3,4),(4,5)],6) => ([(1,4),(2,3),(3,4)],5) => [4,1] => [1] => 1
([(0,1),(2,5),(3,5),(4,5)],6) => ([(0,3),(1,2)],4) => [2,2] => [2] => 2
([(2,5),(3,4),(3,5),(4,5)],6) => ([(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => [1] => 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => [1] => 1
([(2,4),(2,5),(3,4),(3,5)],6) => ([(1,2)],3) => [2,1] => [1] => 1
([(0,5),(1,5),(2,4),(3,4)],6) => ([(0,3),(1,2)],4) => [2,2] => [2] => 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(1,4),(2,3),(3,4)],5) => [4,1] => [1] => 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,2),(1,3),(2,3)],4) => [3,1] => [1] => 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [5,1] => [1] => 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => [1] => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(1,2)],3) => [2,1] => [1] => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,2),(1,3),(2,3)],4) => [3,1] => [1] => 1
([(0,5),(1,4),(2,3)],6) => ([(0,5),(1,4),(2,3)],6) => [2,2,2] => [2,2] => 2
([(1,5),(2,4),(3,4),(3,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => [5,1] => [1] => 1
([(0,1),(2,5),(3,4),(4,5)],6) => ([(0,1),(2,5),(3,4),(4,5)],6) => [4,2] => [2] => 2
([(1,2),(3,4),(3,5),(4,5)],6) => ([(1,2),(3,4),(3,5),(4,5)],6) => [3,2,1] => [2,1] => 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [5,1] => [1] => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => [4,2] => [2] => 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [5,1] => [1] => 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => [5,1] => [1] => 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [5,1] => [1] => 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => [5,1] => [1] => 1
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => [1] => 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,3),(1,2)],4) => [2,2] => [2] => 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => ([(0,1),(2,3),(2,4),(3,4)],5) => [3,2] => [2] => 2
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(2,3),(2,4),(3,4)],5) => [3,2] => [2] => 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => [1] => 1
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => [1] => 1
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => [1] => 1
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => ([(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => [1] => 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => ([(1,2),(1,3),(2,3)],4) => [3,1] => [1] => 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => [1] => 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => [3,3] => [3] => 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,2] => [2] => 2
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => [1] => 1
([(5,6)],7) => ([(1,2)],3) => [2,1] => [1] => 1
([(4,6),(5,6)],7) => ([(1,2)],3) => [2,1] => [1] => 1
([(3,6),(4,6),(5,6)],7) => ([(1,2)],3) => [2,1] => [1] => 1
([(2,6),(3,6),(4,6),(5,6)],7) => ([(1,2)],3) => [2,1] => [1] => 1
([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => ([(1,2)],3) => [2,1] => [1] => 1
([(3,6),(4,5)],7) => ([(1,4),(2,3)],5) => [2,2,1] => [2,1] => 1
([(3,6),(4,5),(5,6)],7) => ([(1,4),(2,3),(3,4)],5) => [4,1] => [1] => 1
([(2,3),(4,6),(5,6)],7) => ([(1,4),(2,3)],5) => [2,2,1] => [2,1] => 1
([(4,5),(4,6),(5,6)],7) => ([(1,2),(1,3),(2,3)],4) => [3,1] => [1] => 1
([(2,6),(3,6),(4,5),(5,6)],7) => ([(1,4),(2,3),(3,4)],5) => [4,1] => [1] => 1
([(1,2),(3,6),(4,6),(5,6)],7) => ([(1,4),(2,3)],5) => [2,2,1] => [2,1] => 1
([(3,6),(4,5),(4,6),(5,6)],7) => ([(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => [1] => 1
([(1,6),(2,6),(3,6),(4,5),(5,6)],7) => ([(1,4),(2,3),(3,4)],5) => [4,1] => [1] => 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7) => ([(0,3),(1,2)],4) => [2,2] => [2] => 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => [1] => 1
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => [1] => 1
([(3,5),(3,6),(4,5),(4,6)],7) => ([(1,2)],3) => [2,1] => [1] => 1
([(1,6),(2,6),(3,5),(4,5)],7) => ([(1,4),(2,3)],5) => [2,2,1] => [2,1] => 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7) => ([(1,4),(2,3),(3,4)],5) => [4,1] => [1] => 1
([(1,6),(2,6),(3,4),(4,5),(5,6)],7) => ([(1,5),(2,4),(3,4),(3,5)],6) => [5,1] => [1] => 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7) => ([(0,3),(1,2)],4) => [2,2] => [2] => 2
([(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,2),(1,3),(2,3)],4) => [3,1] => [1] => 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7) => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [5,1] => [1] => 1
([(1,6),(2,6),(3,5),(4,5),(5,6)],7) => ([(1,4),(2,3),(3,4)],5) => [4,1] => [1] => 1
([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => ([(1,4),(2,3),(3,4)],5) => [4,1] => [1] => 1
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => [1] => 1
([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [5,1] => [1] => 1
([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => [1] => 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => ([(1,2)],3) => [2,1] => [1] => 1
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7) => ([(1,5),(2,4),(3,4),(3,5)],6) => [5,1] => [1] => 1
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => ([(1,4),(2,3),(3,4)],5) => [4,1] => [1] => 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,2),(1,3),(2,3)],4) => [3,1] => [1] => 1
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [5,1] => [1] => 1
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => [1] => 1
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => ([(1,2)],3) => [2,1] => [1] => 1
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,2),(1,3),(2,3)],4) => [3,1] => [1] => 1
([(1,6),(2,5),(3,4)],7) => ([(1,6),(2,5),(3,4)],7) => [2,2,2,1] => [2,2,1] => 1
([(2,6),(3,5),(4,5),(4,6)],7) => ([(1,5),(2,4),(3,4),(3,5)],6) => [5,1] => [1] => 1
([(1,2),(3,6),(4,5),(5,6)],7) => ([(1,2),(3,6),(4,5),(5,6)],7) => [4,2,1] => [2,1] => 1
([(0,3),(1,2),(4,6),(5,6)],7) => ([(0,5),(1,4),(2,3)],6) => [2,2,2] => [2,2] => 2
([(2,3),(4,5),(4,6),(5,6)],7) => ([(1,2),(3,4),(3,5),(4,5)],6) => [3,2,1] => [2,1] => 1
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Description
The Cartan determinant of the integer partition.
Let $p=[p_1,...,p_r]$ be a given integer partition with highest part t. Let $A=K[x]/(x^t)$ be the finite dimensional algebra over the field $K$ and $M$ the direct sum of the indecomposable $A$-modules of vector space dimension $p_i$ for each $i$. Then the Cartan determinant of $p$ is the Cartan determinant of the endomorphism algebra of $M$ over $A$.
Explicitly, this is the determinant of the matrix $\left(\min(\bar p_i, \bar p_j)\right)_{i,j}$, where $\bar p$ is the set of distinct parts of the partition.
Let $p=[p_1,...,p_r]$ be a given integer partition with highest part t. Let $A=K[x]/(x^t)$ be the finite dimensional algebra over the field $K$ and $M$ the direct sum of the indecomposable $A$-modules of vector space dimension $p_i$ for each $i$. Then the Cartan determinant of $p$ is the Cartan determinant of the endomorphism algebra of $M$ over $A$.
Explicitly, this is the determinant of the matrix $\left(\min(\bar p_i, \bar p_j)\right)_{i,j}$, where $\bar p$ is the set of distinct parts of the partition.
Map
de-duplicate
Description
The de-duplicate of a graph.
Let $G = (V, E)$ be a graph. This map yields the graph whose vertex set is the set of (distinct) neighbourhoods $\{N_v | v \in V\}$ of $G$, and has an edge $(N_a, N_b)$ between two vertices if and only if $(a, b)$ is an edge of $G$. This is well-defined, because if $N_a = N_c$ and $N_b = N_d$, then $(a, b)\in E$ if and only if $(c, d)\in E$.
The image of this map is the set of so-called 'mating graphs' or 'point-determining graphs'.
This map preserves the chromatic number.
Let $G = (V, E)$ be a graph. This map yields the graph whose vertex set is the set of (distinct) neighbourhoods $\{N_v | v \in V\}$ of $G$, and has an edge $(N_a, N_b)$ between two vertices if and only if $(a, b)$ is an edge of $G$. This is well-defined, because if $N_a = N_c$ and $N_b = N_d$, then $(a, b)\in E$ if and only if $(c, d)\in E$.
The image of this map is the set of so-called 'mating graphs' or 'point-determining graphs'.
This map preserves the chromatic number.
Map
first row removal
Description
Removes the first entry of an integer partition
Map
to partition of connected components
Description
Return the partition of the sizes of the connected components of the graph.
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