Identifier
-
Mp00154:
Graphs
—core⟶
Graphs
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001629: Integer compositions ⟶ ℤ
Values
([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(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) => [1,1,1,1] => [4] => 1
([(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,1),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(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] => [2,1] => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => [4] => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => [4] => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => [4] => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 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),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => [4] => 1
([(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) => [1,1,1,1,1] => [5] => 1
([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(1,2),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => [2,2,1] => [2,1] => 0
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => [2,2,1] => [2,1] => 0
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => [4] => 1
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => [4] => 1
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => [4] => 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => [2,2,1] => [2,1] => 0
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => [4] => 1
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => [4] => 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => [4] => 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => [4] => 1
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => [4] => 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => [4] => 1
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => [4] => 1
([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => [3] => 1
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Description
The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.
Map
chromatic difference sequence
Description
The chromatic difference sequence of a graph.
Let $G$ be a simple graph with chromatic number $\kappa$. Let $\alpha_m$ be the maximum number of vertices in a $m$-colorable subgraph of $G$. Set $\delta_m=\alpha_m-\alpha_{m-1}$. The sequence $\delta_1,\delta_2,\dots\delta_\kappa$ is the chromatic difference sequence of $G$.
All entries of the chromatic difference sequence are positive: $\alpha_m > \alpha_{m-1}$ for $m < \kappa$, because we can assign any uncolored vertex of a partial coloring with $m-1$ colors the color $m$. Therefore, the chromatic difference sequence is a composition of the number of vertices of $G$ into $\kappa$ parts.
Let $G$ be a simple graph with chromatic number $\kappa$. Let $\alpha_m$ be the maximum number of vertices in a $m$-colorable subgraph of $G$. Set $\delta_m=\alpha_m-\alpha_{m-1}$. The sequence $\delta_1,\delta_2,\dots\delta_\kappa$ is the chromatic difference sequence of $G$.
All entries of the chromatic difference sequence are positive: $\alpha_m > \alpha_{m-1}$ for $m < \kappa$, because we can assign any uncolored vertex of a partial coloring with $m-1$ colors the color $m$. Therefore, the chromatic difference sequence is a composition of the number of vertices of $G$ into $\kappa$ parts.
Map
delta morphism
Description
Apply the delta morphism to an integer composition.
The delta morphism of a composition $C$ is the compositions composed of the lengths of consecutive runs of the same integer in $C$.
The delta morphism of a composition $C$ is the compositions composed of the lengths of consecutive runs of the same integer in $C$.
Map
core
Description
The core of a graph.
The core of a graph $G$ is the smallest graph $C$ such that there is a homomorphism from $G$ to $C$ and a homomorphism from $C$ to $G$.
Note that the core of a graph is not necessarily connected, see [2].
The core of a graph $G$ is the smallest graph $C$ such that there is a homomorphism from $G$ to $C$ and a homomorphism from $C$ to $G$.
Note that the core of a graph is not necessarily connected, see [2].
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