Identifier
-
Mp00201:
Dyck paths
—Ringel⟶
Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤ
Values
[1,0] => [2,1] => [2,1] => ([(0,1)],2) => 2
[1,1,0,0,1,0] => [2,4,1,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 5
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 5
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 5
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [5,6,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 6
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [4,6,2,5,1,3] => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [5,6,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 6
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [6,5,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [6,4,2,5,1,3] => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [5,3,6,2,4,1] => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [4,5,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 6
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [6,5,3,2,4,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [6,5,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
[1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => [6,7,2,3,4,5,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 7
[1,0,1,0,1,0,1,0,1,1,0,0] => [6,1,2,3,4,7,5] => [5,7,2,3,6,1,4] => ([(0,3),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => 7
[1,0,1,0,1,0,1,1,0,1,0,0] => [7,1,2,3,6,4,5] => [6,7,2,5,1,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 7
[1,0,1,0,1,0,1,1,1,0,0,0] => [5,1,2,3,6,7,4] => [4,7,2,5,6,1,3] => ([(0,3),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => 7
[1,0,1,0,1,1,0,1,0,1,0,0] => [7,1,2,6,3,4,5] => [6,7,5,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 7
[1,0,1,0,1,1,1,0,1,0,0,0] => [7,1,2,5,6,3,4] => [6,7,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 7
[1,0,1,1,0,1,1,0,0,1,0,0] => [7,1,4,2,6,3,5] => [6,3,7,1,5,2,4] => ([(0,1),(0,4),(0,6),(1,3),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => 7
[1,0,1,1,0,1,1,0,1,0,0,0] => [7,1,5,2,6,3,4] => [6,4,7,1,5,2,3] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7) => 7
[1,0,1,1,1,0,1,0,0,1,0,0] => [7,1,4,6,2,3,5] => [6,3,5,7,1,2,4] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 7
[1,1,0,0,1,0,1,0,1,0,1,0] => [2,7,1,3,4,5,6] => [7,6,2,3,4,5,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 7
[1,1,0,0,1,0,1,0,1,1,0,0] => [2,6,1,3,4,7,5] => [7,5,2,3,6,1,4] => ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 7
[1,1,0,0,1,0,1,1,0,1,0,0] => [2,7,1,3,6,4,5] => [7,6,2,5,1,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 7
[1,1,0,0,1,0,1,1,1,0,0,0] => [2,5,1,3,6,7,4] => [7,4,2,5,6,1,3] => ([(0,3),(0,5),(0,6),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => 7
[1,1,0,0,1,1,0,1,0,1,0,0] => [2,7,1,6,3,4,5] => [7,6,5,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 7
[1,1,0,0,1,1,1,0,1,0,0,0] => [2,7,1,5,6,3,4] => [7,6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 7
[1,1,0,1,0,1,0,0,1,0,1,0] => [7,4,1,2,3,5,6] => [6,3,7,2,4,5,1] => ([(0,3),(0,5),(0,6),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => 7
[1,1,0,1,0,1,0,0,1,1,0,0] => [6,4,1,2,3,7,5] => [5,3,7,2,6,1,4] => ([(0,1),(0,4),(0,6),(1,3),(1,4),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 7
[1,1,0,1,0,1,0,1,0,0,1,0] => [5,7,1,2,3,4,6] => [4,6,7,2,3,5,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 7
[1,1,0,1,0,1,0,1,0,1,0,0] => [7,6,1,2,3,4,5] => [6,5,7,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 7
[1,1,0,1,0,1,0,1,1,0,0,0] => [5,6,1,2,3,7,4] => [4,5,7,2,6,1,3] => ([(0,3),(0,4),(0,6),(1,3),(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => 7
[1,1,0,1,0,1,1,0,0,1,0,0] => [7,4,1,2,6,3,5] => [6,3,7,5,1,2,4] => ([(0,3),(0,4),(0,6),(1,2),(1,5),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 7
[1,1,0,1,0,1,1,0,1,0,0,0] => [5,7,1,2,6,3,4] => [4,6,7,5,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7) => 7
[1,1,0,1,1,0,1,0,1,0,0,0] => [6,7,1,5,2,3,4] => [5,6,4,7,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7) => 7
[1,1,1,0,0,1,0,0,1,0,1,0] => [2,7,4,1,3,5,6] => [7,6,3,2,4,5,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 7
[1,1,1,0,0,1,0,0,1,1,0,0] => [2,6,4,1,3,7,5] => [7,5,3,2,6,1,4] => ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 7
[1,1,1,0,0,1,0,1,0,0,1,0] => [2,7,5,1,3,4,6] => [7,6,4,2,3,5,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 7
[1,1,1,0,0,1,0,1,0,1,0,0] => [2,6,7,1,3,4,5] => [7,5,6,2,3,4,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 7
[1,1,1,0,0,1,0,1,1,0,0,0] => [2,6,5,1,3,7,4] => [7,5,4,2,6,1,3] => ([(0,2),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 7
[1,1,1,0,1,0,1,0,0,0,1,0] => [7,5,4,1,2,3,6] => [6,4,3,7,2,5,1] => ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 7
[1,1,1,0,1,0,1,0,0,1,0,0] => [6,7,4,1,2,3,5] => [5,6,3,7,2,4,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 7
[1,1,1,0,1,0,1,0,1,0,0,0] => [6,7,5,1,2,3,4] => [5,6,4,7,2,3,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 7
[1,1,1,1,0,0,1,0,0,0,1,0] => [2,7,4,5,1,3,6] => [7,6,3,4,2,5,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 7
[1,1,1,1,0,0,1,0,0,1,0,0] => [2,7,4,6,1,3,5] => [7,6,3,5,2,4,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 7
[1,1,1,1,0,0,1,0,1,0,0,0] => [2,7,6,5,1,3,4] => [7,6,5,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 7
[1,1,1,1,1,0,0,1,0,0,0,0] => [2,7,4,5,6,1,3] => [7,6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 7
[] => [1] => [1] => ([],1) => 1
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Description
The pebbling number of a connected graph.
Map
toric promotion
Description
Toric promotion of a permutation.
Let $\sigma\in\mathfrak S_n$ be a permutation and let
$ \tau_{i, j}(\sigma) = \begin{cases} \sigma & \text{if $|\sigma^{-1}(i) - \sigma^{-1}(j)| = 1$}\\ (i, j)\circ\sigma & \text{otherwise}. \end{cases} $
The toric promotion operator is the product $\tau_{n,1}\tau_{n-1,n}\dots\tau_{1,2}$.
This is the special case of toric promotion on graphs for the path graph. Its order is $n-1$.
Let $\sigma\in\mathfrak S_n$ be a permutation and let
$ \tau_{i, j}(\sigma) = \begin{cases} \sigma & \text{if $|\sigma^{-1}(i) - \sigma^{-1}(j)| = 1$}\\ (i, j)\circ\sigma & \text{otherwise}. \end{cases} $
The toric promotion operator is the product $\tau_{n,1}\tau_{n-1,n}\dots\tau_{1,2}$.
This is the special case of toric promotion on graphs for the path graph. Its order is $n-1$.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
graph of inversions
Description
The graph of inversions of a permutation.
For a permutation of $\{1,\dots,n\}$, this is the graph with vertices $\{1,\dots,n\}$, where $(i,j)$ is an edge if and only if it is an inversion of the permutation.
For a permutation of $\{1,\dots,n\}$, this is the graph with vertices $\{1,\dots,n\}$, where $(i,j)$ is an edge if and only if it is an inversion of the permutation.
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