Identifier
-
Mp00324:
Graphs
—chromatic difference sequence⟶
Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤ
Values
([],1) => [1] => [1,0] => 0
([],2) => [2] => [1,1,0,0] => 0
([(0,1)],2) => [1,1] => [1,0,1,0] => 1
([],3) => [3] => [1,1,1,0,0,0] => 0
([(1,2)],3) => [2,1] => [1,1,0,0,1,0] => 1
([(0,2),(1,2)],3) => [2,1] => [1,1,0,0,1,0] => 1
([(0,1),(0,2),(1,2)],3) => [1,1,1] => [1,0,1,0,1,0] => 2
([],4) => [4] => [1,1,1,1,0,0,0,0] => 0
([(2,3)],4) => [3,1] => [1,1,1,0,0,0,1,0] => 1
([(1,3),(2,3)],4) => [3,1] => [1,1,1,0,0,0,1,0] => 1
([(0,3),(1,3),(2,3)],4) => [3,1] => [1,1,1,0,0,0,1,0] => 1
([(0,3),(1,2)],4) => [2,2] => [1,1,0,0,1,1,0,0] => 2
([(0,3),(1,2),(2,3)],4) => [2,2] => [1,1,0,0,1,1,0,0] => 2
([(1,2),(1,3),(2,3)],4) => [2,1,1] => [1,1,0,0,1,0,1,0] => 2
([(0,3),(1,2),(1,3),(2,3)],4) => [2,1,1] => [1,1,0,0,1,0,1,0] => 2
([(0,2),(0,3),(1,2),(1,3)],4) => [2,2] => [1,1,0,0,1,1,0,0] => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [2,1,1] => [1,1,0,0,1,0,1,0] => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 3
([],5) => [5] => [1,1,1,1,1,0,0,0,0,0] => 0
([(3,4)],5) => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 1
([(2,4),(3,4)],5) => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 1
([(1,4),(2,4),(3,4)],5) => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 1
([(0,4),(1,4),(2,4),(3,4)],5) => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 1
([(1,4),(2,3)],5) => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 2
([(1,4),(2,3),(3,4)],5) => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 2
([(0,1),(2,4),(3,4)],5) => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 2
([(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 2
([(0,4),(1,4),(2,3),(3,4)],5) => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 2
([(1,4),(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 2
([(1,3),(1,4),(2,3),(2,4)],5) => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 2
([(0,4),(1,3),(2,3),(2,4)],5) => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 2
([(0,1),(2,3),(2,4),(3,4)],5) => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 3
([(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] => [1,0,1,0,1,0,1,0,1,0] => 4
([],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0
([(4,5)],6) => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
([(3,5),(4,5)],6) => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
([(2,5),(3,5),(4,5)],6) => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
([(1,5),(2,5),(3,5),(4,5)],6) => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
([(2,5),(3,4)],6) => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
([(2,5),(3,4),(4,5)],6) => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
([(1,2),(3,5),(4,5)],6) => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
([(3,4),(3,5),(4,5)],6) => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
([(1,5),(2,5),(3,4),(4,5)],6) => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
([(0,1),(2,5),(3,5),(4,5)],6) => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
([(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
([(2,4),(2,5),(3,4),(3,5)],6) => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
([(0,5),(1,5),(2,4),(3,4)],6) => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
([(0,5),(1,4),(2,3)],6) => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3
([(1,5),(2,4),(3,4),(3,5)],6) => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
([(0,1),(2,5),(3,4),(4,5)],6) => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3
([(1,2),(3,4),(3,5),(4,5)],6) => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3
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Description
The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.
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
bounce path
Description
The bounce path determined by an integer composition.
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