Identifier
-
Mp00031:
Dyck paths
—to 312-avoiding permutation⟶
Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000741: Graphs ⟶ ℤ
Values
[1,0] => [1] => [1] => ([],1) => 0
[1,0,1,0] => [1,2] => [1,2] => ([],2) => 1
[1,1,0,0] => [2,1] => [2,1] => ([(0,1)],2) => 1
[1,0,1,0,1,0] => [1,2,3] => [1,2,3] => ([],3) => 1
[1,1,0,1,0,0] => [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 2
[1,1,1,0,0,0] => [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3) => 2
[1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3,4] => ([],4) => 1
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4) => 2
[1,0,1,1,1,0,0,0] => [1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4) => 1
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4) => 2
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 3
[1,1,0,1,1,0,0,0] => [2,4,3,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
[1,1,1,0,0,0,1,0] => [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4) => 1
[1,1,1,0,0,1,0,0] => [3,2,4,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 2
[1,1,1,0,1,0,0,0] => [3,4,2,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
[1,1,1,1,0,0,0,0] => [4,3,2,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 2
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5) => 1
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5) => 2
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5) => 1
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5) => 2
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5) => 1
[1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[1,0,1,1,1,0,1,0,0,0] => [1,4,5,3,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5) => 2
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5) => 1
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5) => 2
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5) => 2
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [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) => 4
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[1,1,0,1,1,0,1,0,0,0] => [2,4,5,3,1] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5) => 1
[1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5) => 1
[1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[1,1,1,0,1,0,0,0,1,0] => [3,4,2,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[1,1,1,0,1,0,1,0,0,0] => [3,4,5,2,1] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[1,1,1,0,1,1,0,0,0,0] => [3,5,4,2,1] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 2
[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5) => 2
[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => 2
[1,1,1,1,0,0,1,0,0,0] => [4,3,5,2,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[1,1,1,1,0,1,0,0,0,0] => [4,5,3,2,1] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 2
[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6) => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6) => 2
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6) => 1
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => [1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6) => 2
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [1,2,6,5,4,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,5,3] => [1,2,5,6,4,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6) => 1
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => [1,2,4,6,5,3] => ([(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,4,3] => [1,2,6,4,5,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => [1,2,5,4,6,3] => ([(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => [1,3,2,6,5,4] => ([(1,2),(3,4),(3,5),(4,5)],6) => 2
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => [1,3,2,5,6,4] => ([(1,2),(3,5),(4,5)],6) => 1
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => [1,4,3,2,5,6] => ([(3,4),(3,5),(4,5)],6) => 2
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => [1,4,3,2,6,5] => ([(1,2),(3,4),(3,5),(4,5)],6) => 2
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => [1,5,4,3,2,6] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,5,2] => [1,5,6,4,3,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,4,2,6] => [1,4,5,3,2,6] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2] => [1,4,6,5,3,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,4,2] => [1,6,4,5,3,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => [1,5,4,6,3,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => [1,3,4,2,5,6] => ([(3,5),(4,5)],6) => 1
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => [1,3,4,2,6,5] => ([(1,2),(3,5),(4,5)],6) => 1
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,3,5,2,6] => [1,3,5,4,2,6] => ([(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => [1,3,6,5,4,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2] => [1,3,5,6,4,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,3,2,6] => [1,5,3,4,2,6] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,3,6,2] => [1,6,5,3,4,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,3,2] => [1,6,3,5,4,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,5,3,2] => [1,5,3,6,4,2] => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,0,1,1,1,1,0,0,0,0,1,0] => [1,5,4,3,2,6] => [1,4,3,5,2,6] => ([(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,0,1,1,1,1,0,0,0,1,0,0] => [1,5,4,3,6,2] => [1,4,3,6,5,2] => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 2
[1,0,1,1,1,1,0,0,1,0,0,0] => [1,5,4,6,3,2] => [1,6,4,3,5,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,0,1,1,1,1,0,1,0,0,0,0] => [1,5,6,4,3,2] => [1,4,6,3,5,2] => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,5,4,3,2] => [1,4,5,3,6,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,1,0,0,1,0,1,1,0,1,0,0] => [2,1,3,5,6,4] => [2,1,3,6,5,4] => ([(1,2),(3,4),(3,5),(4,5)],6) => 2
[1,1,0,0,1,0,1,1,1,0,0,0] => [2,1,3,6,5,4] => [2,1,3,5,6,4] => ([(1,2),(3,5),(4,5)],6) => 1
[1,1,0,0,1,1,0,1,0,0,1,0] => [2,1,4,5,3,6] => [2,1,5,4,3,6] => ([(1,2),(3,4),(3,5),(4,5)],6) => 2
[1,1,0,0,1,1,0,1,0,1,0,0] => [2,1,4,5,6,3] => [2,1,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,1,0,0,1,1,0,1,1,0,0,0] => [2,1,4,6,5,3] => [2,1,5,6,4,3] => ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,1,0,0,1,1,1,0,0,0,1,0] => [2,1,5,4,3,6] => [2,1,4,5,3,6] => ([(1,2),(3,5),(4,5)],6) => 1
[1,1,0,0,1,1,1,0,0,1,0,0] => [2,1,5,4,6,3] => [2,1,4,6,5,3] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,1,0,0,1,1,1,0,1,0,0,0] => [2,1,5,6,4,3] => [2,1,6,4,5,3] => ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,1,0,0,1,1,1,1,0,0,0,0] => [2,1,6,5,4,3] => [2,1,5,4,6,3] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,1,0,1,0,0,1,0,1,0,1,0] => [2,3,1,4,5,6] => [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6) => 2
[1,1,0,1,0,0,1,0,1,1,0,0] => [2,3,1,4,6,5] => [3,2,1,4,6,5] => ([(1,2),(3,4),(3,5),(4,5)],6) => 2
[1,1,0,1,0,0,1,1,0,0,1,0] => [2,3,1,5,4,6] => [3,2,1,5,4,6] => ([(1,2),(3,4),(3,5),(4,5)],6) => 2
[1,1,0,1,0,0,1,1,0,1,0,0] => [2,3,1,5,6,4] => [3,2,1,6,5,4] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => 2
[1,1,0,1,0,0,1,1,1,0,0,0] => [2,3,1,6,5,4] => [3,2,1,5,6,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => 2
[1,1,0,1,0,1,0,0,1,0,1,0] => [2,3,4,1,5,6] => [4,3,2,1,5,6] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
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Description
The Colin de Verdière graph invariant.
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.
Map
Clarke-Steingrimsson-Zeng inverse
Description
The inverse of the Clarke-Steingrimsson-Zeng map, sending excedances to descents.
This is the inverse of the map $\Phi$ in [1, sec.3].
This is the inverse of the map $\Phi$ in [1, sec.3].
Map
to 312-avoiding permutation
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