Identifier
Mp00024:
Dyck paths
—to 321-avoiding permutation⟶
Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
Images
[1,0] => [1] => ([],1) => ([],1)
[1,0,1,0] => [2,1] => ([(0,1)],2) => ([(0,1)],2)
[1,1,0,0] => [1,2] => ([],2) => ([],1)
[1,0,1,0,1,0] => [2,1,3] => ([(1,2)],3) => ([(0,1)],2)
[1,0,1,1,0,0] => [2,3,1] => ([(0,2),(1,2)],3) => ([(0,1)],2)
[1,1,0,0,1,0] => [3,1,2] => ([(0,2),(1,2)],3) => ([(0,1)],2)
[1,1,0,1,0,0] => [1,3,2] => ([(1,2)],3) => ([(0,1)],2)
[1,1,1,0,0,0] => [1,2,3] => ([],3) => ([],1)
[1,0,1,0,1,0,1,0] => [2,1,4,3] => ([(0,3),(1,2)],4) => ([(0,1)],2)
[1,0,1,0,1,1,0,0] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => ([(0,1)],2)
[1,0,1,1,0,0,1,0] => [2,1,3,4] => ([(2,3)],4) => ([(0,1)],2)
[1,0,1,1,0,1,0,0] => [2,3,1,4] => ([(1,3),(2,3)],4) => ([(0,1)],2)
[1,0,1,1,1,0,0,0] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4) => ([(0,1)],2)
[1,1,0,0,1,0,1,0] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4) => ([(0,1)],2)
[1,1,0,0,1,1,0,0] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,1)],2)
[1,1,0,1,0,0,1,0] => [3,1,2,4] => ([(1,3),(2,3)],4) => ([(0,1)],2)
[1,1,0,1,0,1,0,0] => [1,3,2,4] => ([(2,3)],4) => ([(0,1)],2)
[1,1,0,1,1,0,0,0] => [1,3,4,2] => ([(1,3),(2,3)],4) => ([(0,1)],2)
[1,1,1,0,0,0,1,0] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4) => ([(0,1)],2)
[1,1,1,0,0,1,0,0] => [1,4,2,3] => ([(1,3),(2,3)],4) => ([(0,1)],2)
[1,1,1,0,1,0,0,0] => [1,2,4,3] => ([(2,3)],4) => ([(0,1)],2)
[1,1,1,1,0,0,0,0] => [1,2,3,4] => ([],4) => ([],1)
[1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,5] => ([(1,4),(2,3)],5) => ([(0,1)],2)
[1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5) => ([(0,1)],2)
[1,0,1,0,1,1,0,0,1,0] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5) => ([(0,1)],2)
[1,0,1,0,1,1,0,1,0,0] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,1)],2)
[1,0,1,0,1,1,1,0,0,0] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,1)],2)
[1,0,1,1,0,0,1,0,1,0] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5) => ([(0,1)],2)
[1,0,1,1,0,0,1,1,0,0] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,1)],2)
[1,0,1,1,0,1,0,0,1,0] => [2,1,3,5,4] => ([(1,4),(2,3)],5) => ([(0,1)],2)
[1,0,1,1,0,1,0,1,0,0] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5) => ([(0,1)],2)
[1,0,1,1,0,1,1,0,0,0] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,1)],2)
[1,0,1,1,1,0,0,0,1,0] => [2,1,3,4,5] => ([(3,4)],5) => ([(0,1)],2)
[1,0,1,1,1,0,0,1,0,0] => [2,3,1,4,5] => ([(2,4),(3,4)],5) => ([(0,1)],2)
[1,0,1,1,1,0,1,0,0,0] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5) => ([(0,1)],2)
[1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,1)],2)
[1,1,0,0,1,0,1,0,1,0] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5) => ([(0,1)],2)
[1,1,0,0,1,0,1,1,0,0] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1)],2)
[1,1,0,0,1,1,0,0,1,0] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,1)],2)
[1,1,0,0,1,1,0,1,0,0] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,1)],2)
[1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1)],2)
[1,1,0,1,0,0,1,0,1,0] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,1)],2)
[1,1,0,1,0,0,1,1,0,0] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,1)],2)
[1,1,0,1,0,1,0,0,1,0] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5) => ([(0,1)],2)
[1,1,0,1,0,1,0,1,0,0] => [1,3,2,5,4] => ([(1,4),(2,3)],5) => ([(0,1)],2)
[1,1,0,1,0,1,1,0,0,0] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5) => ([(0,1)],2)
[1,1,0,1,1,0,0,0,1,0] => [3,1,2,4,5] => ([(2,4),(3,4)],5) => ([(0,1)],2)
[1,1,0,1,1,0,0,1,0,0] => [1,3,2,4,5] => ([(3,4)],5) => ([(0,1)],2)
[1,1,0,1,1,0,1,0,0,0] => [1,3,4,2,5] => ([(2,4),(3,4)],5) => ([(0,1)],2)
[1,1,0,1,1,1,0,0,0,0] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5) => ([(0,1)],2)
[1,1,1,0,0,0,1,0,1,0] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,1)],2)
[1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1)],2)
[1,1,1,0,0,1,0,0,1,0] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,1)],2)
[1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5) => ([(0,1)],2)
[1,1,1,0,0,1,1,0,0,0] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1)],2)
[1,1,1,0,1,0,0,0,1,0] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5) => ([(0,1)],2)
[1,1,1,0,1,0,0,1,0,0] => [1,4,2,3,5] => ([(2,4),(3,4)],5) => ([(0,1)],2)
[1,1,1,0,1,0,1,0,0,0] => [1,2,4,3,5] => ([(3,4)],5) => ([(0,1)],2)
[1,1,1,0,1,1,0,0,0,0] => [1,2,4,5,3] => ([(2,4),(3,4)],5) => ([(0,1)],2)
[1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,1)],2)
[1,1,1,1,0,0,0,1,0,0] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5) => ([(0,1)],2)
[1,1,1,1,0,0,1,0,0,0] => [1,2,5,3,4] => ([(2,4),(3,4)],5) => ([(0,1)],2)
[1,1,1,1,0,1,0,0,0,0] => [1,2,3,5,4] => ([(3,4)],5) => ([(0,1)],2)
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([],5) => ([],1)
[1,0,1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6) => ([(0,1)],2)
[1,0,1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6) => ([(0,1)],2)
[1,0,1,0,1,0,1,1,0,0,1,0] => [2,1,4,6,3,5] => ([(0,1),(2,5),(3,4),(4,5)],6) => ([(0,1)],2)
[1,0,1,0,1,0,1,1,0,1,0,0] => [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => ([(0,1)],2)
[1,0,1,0,1,0,1,1,1,0,0,0] => [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2)
[1,0,1,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => ([(2,5),(3,4)],6) => ([(0,1)],2)
[1,0,1,0,1,1,0,0,1,1,0,0] => [2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6) => ([(0,1)],2)
[1,0,1,0,1,1,0,1,0,0,1,0] => [2,1,4,5,3,6] => ([(1,2),(3,5),(4,5)],6) => ([(0,1)],2)
[1,0,1,0,1,1,0,1,0,1,0,0] => [2,4,1,5,3,6] => ([(1,5),(2,4),(3,4),(3,5)],6) => ([(0,1)],2)
[1,0,1,0,1,1,0,1,1,0,0,0] => [2,4,5,1,3,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2)
[1,0,1,0,1,1,1,0,0,0,1,0] => [2,1,4,5,6,3] => ([(0,1),(2,5),(3,5),(4,5)],6) => ([(0,1)],2)
[1,0,1,0,1,1,1,0,0,1,0,0] => [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,1)],2)
[1,0,1,0,1,1,1,0,1,0,0,0] => [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2)
[1,0,1,0,1,1,1,1,0,0,0,0] => [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2)
[1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,5,3,6,4] => ([(0,1),(2,5),(3,4),(4,5)],6) => ([(0,1)],2)
[1,0,1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,1)],2)
[1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,5,6,3,4] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2)
[1,0,1,1,0,0,1,1,0,1,0,0] => [2,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => ([(0,1)],2)
[1,0,1,1,0,0,1,1,1,0,0,0] => [2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2)
[1,0,1,1,0,1,0,0,1,0,1,0] => [2,1,5,3,4,6] => ([(1,2),(3,5),(4,5)],6) => ([(0,1)],2)
[1,0,1,1,0,1,0,0,1,1,0,0] => [2,5,1,3,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6) => ([(0,1)],2)
[1,0,1,1,0,1,0,1,0,0,1,0] => [2,1,3,5,4,6] => ([(2,5),(3,4)],6) => ([(0,1)],2)
[1,0,1,1,0,1,0,1,0,1,0,0] => [2,3,1,5,4,6] => ([(1,2),(3,5),(4,5)],6) => ([(0,1)],2)
[1,0,1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6) => ([(0,1)],2)
[1,0,1,1,0,1,1,0,0,0,1,0] => [2,1,3,5,6,4] => ([(1,2),(3,5),(4,5)],6) => ([(0,1)],2)
[1,0,1,1,0,1,1,0,0,1,0,0] => [2,3,1,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6) => ([(0,1)],2)
[1,0,1,1,0,1,1,0,1,0,0,0] => [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,1)],2)
[1,0,1,1,0,1,1,1,0,0,0,0] => [2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2)
[1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,6,3,4,5] => ([(0,1),(2,5),(3,5),(4,5)],6) => ([(0,1)],2)
[1,0,1,1,1,0,0,0,1,1,0,0] => [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,1)],2)
[1,0,1,1,1,0,0,1,0,0,1,0] => [2,1,3,6,4,5] => ([(1,2),(3,5),(4,5)],6) => ([(0,1)],2)
[1,0,1,1,1,0,0,1,0,1,0,0] => [2,3,1,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6) => ([(0,1)],2)
[1,0,1,1,1,0,0,1,1,0,0,0] => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,1)],2)
[1,0,1,1,1,0,1,0,0,0,1,0] => [2,1,3,4,6,5] => ([(2,5),(3,4)],6) => ([(0,1)],2)
[1,0,1,1,1,0,1,0,0,1,0,0] => [2,3,1,4,6,5] => ([(1,2),(3,5),(4,5)],6) => ([(0,1)],2)
[1,0,1,1,1,0,1,0,1,0,0,0] => [2,3,4,1,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6) => ([(0,1)],2)
[1,0,1,1,1,0,1,1,0,0,0,0] => [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,1)],2)
>>> Load all 197 entries. <<<Map
to 321-avoiding permutation
Description
Sends a Dyck path to a 321-avoiding permutation.
This bijection defined in [3, pp. 60] and in [2, Section 3.1].
It is shown in [1] that it sends the number of centered tunnels to the number of fixed points, the number of right tunnels to the number of exceedences, and the semilength plus the height of the middle point to 2 times the length of the longest increasing subsequence.
This bijection defined in [3, pp. 60] and in [2, Section 3.1].
It is shown in [1] that it sends the number of centered tunnels to the number of fixed points, the number of right tunnels to the number of exceedences, and the semilength plus the height of the middle point to 2 times the length of the longest increasing subsequence.
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
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].
searching the database
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