Identifier
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Mp00324:
Graphs
—chromatic difference sequence⟶
Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000443: Dyck paths ⟶ ℤ (values match St000024The number of double up and double down steps of a Dyck path., St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path., St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra., St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra.)
Values
([],1) => [1] => [1,0] => 1
([],2) => [2] => [1,1,0,0] => 2
([(0,1)],2) => [1,1] => [1,0,1,0] => 1
([],3) => [3] => [1,1,1,0,0,0] => 3
([(1,2)],3) => [2,1] => [1,1,0,0,1,0] => 2
([(0,2),(1,2)],3) => [2,1] => [1,1,0,0,1,0] => 2
([(0,1),(0,2),(1,2)],3) => [1,1,1] => [1,0,1,0,1,0] => 1
([],4) => [4] => [1,1,1,1,0,0,0,0] => 4
([(2,3)],4) => [3,1] => [1,1,1,0,0,0,1,0] => 3
([(1,3),(2,3)],4) => [3,1] => [1,1,1,0,0,0,1,0] => 3
([(0,3),(1,3),(2,3)],4) => [3,1] => [1,1,1,0,0,0,1,0] => 3
([(0,3),(1,2)],4) => [2,2] => [1,1,0,0,1,1,0,0] => 3
([(0,3),(1,2),(2,3)],4) => [2,2] => [1,1,0,0,1,1,0,0] => 3
([(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] => 3
([(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] => 1
([],5) => [5] => [1,1,1,1,1,0,0,0,0,0] => 5
([(3,4)],5) => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 4
([(2,4),(3,4)],5) => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 4
([(1,4),(2,4),(3,4)],5) => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 4
([(0,4),(1,4),(2,4),(3,4)],5) => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 4
([(1,4),(2,3)],5) => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 4
([(1,4),(2,3),(3,4)],5) => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 4
([(0,1),(2,4),(3,4)],5) => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 4
([(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 3
([(0,4),(1,4),(2,3),(3,4)],5) => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 4
([(1,4),(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 3
([(1,3),(1,4),(2,3),(2,4)],5) => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 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] => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 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] => 3
([(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] => 4
([(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] => 3
([(0,4),(1,3),(2,3),(2,4)],5) => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 4
([(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] => 2
([(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] => 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] => 2
([(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] => 2
([(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] => 1
([],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => 6
([(4,5)],6) => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 5
([(3,5),(4,5)],6) => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 5
([(2,5),(3,5),(4,5)],6) => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 5
([(1,5),(2,5),(3,5),(4,5)],6) => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 5
([(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] => 5
([(2,5),(3,4)],6) => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 5
([(2,5),(3,4),(4,5)],6) => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 5
([(1,2),(3,5),(4,5)],6) => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 5
([(3,4),(3,5),(4,5)],6) => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 4
([(1,5),(2,5),(3,4),(4,5)],6) => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 5
([(0,1),(2,5),(3,5),(4,5)],6) => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 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] => 4
([(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] => 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] => 4
([(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] => 4
([(2,4),(2,5),(3,4),(3,5)],6) => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 5
([(0,5),(1,5),(2,4),(3,4)],6) => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 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] => 5
([(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] => 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] => 4
([(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] => 4
([(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] => 5
([(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] => 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] => 4
([(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] => 4
([(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] => 4
([(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] => 5
([(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] => 5
([(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] => 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] => 4
([(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] => 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] => 4
([(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] => 5
([(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] => 4
([(0,5),(1,4),(2,3)],6) => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 5
([(1,5),(2,4),(3,4),(3,5)],6) => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 5
([(0,1),(2,5),(3,4),(4,5)],6) => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 5
([(1,2),(3,4),(3,5),(4,5)],6) => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 4
([(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] => 5
([(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] => 4
([(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] => 4
([(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] => 4
([(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] => 4
([(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] => 4
([(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] => 4
([(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] => 5
([(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] => 4
([(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] => 4
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Description
The number of long tunnels of a Dyck path.
A long tunnel of a Dyck path is a longest sequence of consecutive usual tunnels, i.e., a longest sequence of tunnels where the end point of one is the starting point of the next. See [1] for the definition of tunnels.
A long tunnel of a Dyck path is a longest sequence of consecutive usual tunnels, i.e., a longest sequence of tunnels where the end point of one is the starting point of the next. See [1] for the definition of tunnels.
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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