Identifier
-
Mp00014:
Binary trees
—to 132-avoiding permutation⟶
Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001649: Graphs ⟶ ℤ
Values
[.,.] => [1] => ([],1) => ([],1) => 0
[.,[.,.]] => [2,1] => ([],2) => ([],2) => 0
[[.,.],.] => [1,2] => ([(0,1)],2) => ([(0,1)],2) => 1
[.,[.,[.,.]]] => [3,2,1] => ([],3) => ([],3) => 0
[.,[[.,.],.]] => [2,3,1] => ([(1,2)],3) => ([(1,2)],3) => 1
[[.,.],[.,.]] => [3,1,2] => ([(1,2)],3) => ([(1,2)],3) => 1
[[.,[.,.]],.] => [2,1,3] => ([(0,2),(1,2)],3) => ([(0,2),(1,2)],3) => 2
[[[.,.],.],.] => [1,2,3] => ([(0,2),(2,1)],3) => ([(0,2),(1,2)],3) => 2
[.,[.,[.,[.,.]]]] => [4,3,2,1] => ([],4) => ([],4) => 0
[.,[.,[[.,.],.]]] => [3,4,2,1] => ([(2,3)],4) => ([(2,3)],4) => 1
[.,[[.,.],[.,.]]] => [4,2,3,1] => ([(2,3)],4) => ([(2,3)],4) => 1
[.,[[.,[.,.]],.]] => [3,2,4,1] => ([(1,3),(2,3)],4) => ([(1,3),(2,3)],4) => 2
[.,[[[.,.],.],.]] => [2,3,4,1] => ([(1,2),(2,3)],4) => ([(1,3),(2,3)],4) => 2
[[.,.],[.,[.,.]]] => [4,3,1,2] => ([(2,3)],4) => ([(2,3)],4) => 1
[[.,.],[[.,.],.]] => [3,4,1,2] => ([(0,3),(1,2)],4) => ([(0,3),(1,2)],4) => 1
[[.,[.,.]],[.,.]] => [4,2,1,3] => ([(1,3),(2,3)],4) => ([(1,3),(2,3)],4) => 2
[[[.,.],.],[.,.]] => [4,1,2,3] => ([(1,2),(2,3)],4) => ([(1,3),(2,3)],4) => 2
[[.,[.,[.,.]]],.] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4) => ([(0,3),(1,3),(2,3)],4) => 2
[[.,[[.,.],.]],.] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(1,2),(2,3)],4) => 3
[[[.,.],[.,.]],.] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(1,2),(2,3)],4) => 3
[[[.,[.,.]],.],.] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4) => ([(0,3),(1,3),(2,3)],4) => 2
[[[[.,.],.],.],.] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => ([(0,3),(1,2),(2,3)],4) => 3
[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => ([],5) => ([],5) => 0
[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => ([(3,4)],5) => ([(3,4)],5) => 1
[.,[.,[[.,.],[.,.]]]] => [5,3,4,2,1] => ([(3,4)],5) => ([(3,4)],5) => 1
[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => ([(2,4),(3,4)],5) => ([(2,4),(3,4)],5) => 2
[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => ([(2,3),(3,4)],5) => ([(2,4),(3,4)],5) => 2
[.,[[.,.],[.,[.,.]]]] => [5,4,2,3,1] => ([(3,4)],5) => ([(3,4)],5) => 1
[.,[[.,.],[[.,.],.]]] => [4,5,2,3,1] => ([(1,4),(2,3)],5) => ([(1,4),(2,3)],5) => 1
[.,[[.,[.,.]],[.,.]]] => [5,3,2,4,1] => ([(2,4),(3,4)],5) => ([(2,4),(3,4)],5) => 2
[.,[[[.,.],.],[.,.]]] => [5,2,3,4,1] => ([(2,3),(3,4)],5) => ([(2,4),(3,4)],5) => 2
[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5) => ([(1,4),(2,4),(3,4)],5) => 2
[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5) => ([(1,4),(2,3),(3,4)],5) => 3
[.,[[[.,.],[.,.]],.]] => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5) => ([(1,4),(2,3),(3,4)],5) => 3
[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5) => ([(1,4),(2,4),(3,4)],5) => 2
[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5) => ([(1,4),(2,3),(3,4)],5) => 3
[[.,.],[.,[.,[.,.]]]] => [5,4,3,1,2] => ([(3,4)],5) => ([(3,4)],5) => 1
[[.,.],[.,[[.,.],.]]] => [4,5,3,1,2] => ([(1,4),(2,3)],5) => ([(1,4),(2,3)],5) => 1
[[.,.],[[.,.],[.,.]]] => [5,3,4,1,2] => ([(1,4),(2,3)],5) => ([(1,4),(2,3)],5) => 1
[[.,.],[[.,[.,.]],.]] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5) => ([(0,1),(2,4),(3,4)],5) => 2
[[.,.],[[[.,.],.],.]] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5) => ([(0,1),(2,4),(3,4)],5) => 2
[[.,[.,.]],[.,[.,.]]] => [5,4,2,1,3] => ([(2,4),(3,4)],5) => ([(2,4),(3,4)],5) => 2
[[.,[.,.]],[[.,.],.]] => [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5) => ([(0,1),(2,4),(3,4)],5) => 2
[[[.,.],.],[.,[.,.]]] => [5,4,1,2,3] => ([(2,3),(3,4)],5) => ([(2,4),(3,4)],5) => 2
[[[.,.],.],[[.,.],.]] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5) => ([(0,1),(2,4),(3,4)],5) => 2
[[.,[.,[.,.]]],[.,.]] => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5) => ([(1,4),(2,4),(3,4)],5) => 2
[[.,[[.,.],.]],[.,.]] => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5) => ([(1,4),(2,3),(3,4)],5) => 3
[[[.,.],[.,.]],[.,.]] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5) => ([(1,4),(2,3),(3,4)],5) => 3
[[[.,[.,.]],.],[.,.]] => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5) => ([(1,4),(2,4),(3,4)],5) => 2
[[[[.,.],.],.],[.,.]] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5) => ([(1,4),(2,3),(3,4)],5) => 3
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(1,4),(2,3),(3,4)],5) => 3
[[.,[[.,.],[.,.]]],.] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(1,4),(2,3),(3,4)],5) => 3
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5) => ([(0,4),(1,4),(2,3),(3,4)],5) => 3
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 4
[[[.,.],[.,[.,.]]],.] => [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(1,4),(2,3),(3,4)],5) => 3
[[[.,.],[[.,.],.]],.] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 4
[[[.,[.,.]],[.,.]],.] => [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5) => ([(0,4),(1,4),(2,3),(3,4)],5) => 3
[[[[.,.],.],[.,.]],.] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 4
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5) => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5) => ([(0,4),(1,4),(2,3),(3,4)],5) => 3
[[[[.,.],[.,.]],.],.] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5) => ([(0,4),(1,4),(2,3),(3,4)],5) => 3
[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5) => ([(0,4),(1,4),(2,3),(3,4)],5) => 3
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 4
[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => ([],6) => ([],6) => 0
[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => ([(4,5)],6) => ([(4,5)],6) => 1
[.,[.,[.,[[.,.],[.,.]]]]] => [6,4,5,3,2,1] => ([(4,5)],6) => ([(4,5)],6) => 1
[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => ([(3,5),(4,5)],6) => ([(3,5),(4,5)],6) => 2
[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => ([(3,4),(4,5)],6) => ([(3,5),(4,5)],6) => 2
[.,[.,[[.,.],[.,[.,.]]]]] => [6,5,3,4,2,1] => ([(4,5)],6) => ([(4,5)],6) => 1
[.,[.,[[.,.],[[.,.],.]]]] => [5,6,3,4,2,1] => ([(2,5),(3,4)],6) => ([(2,5),(3,4)],6) => 1
[.,[.,[[.,[.,.]],[.,.]]]] => [6,4,3,5,2,1] => ([(3,5),(4,5)],6) => ([(3,5),(4,5)],6) => 2
[.,[.,[[[.,.],.],[.,.]]]] => [6,3,4,5,2,1] => ([(3,4),(4,5)],6) => ([(3,5),(4,5)],6) => 2
[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => ([(2,5),(3,5),(4,5)],6) => ([(2,5),(3,5),(4,5)],6) => 2
[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => ([(2,5),(3,4),(4,5)],6) => ([(2,5),(3,4),(4,5)],6) => 3
[.,[.,[[[.,.],[.,.]],.]]] => [5,3,4,6,2,1] => ([(2,5),(3,4),(4,5)],6) => ([(2,5),(3,4),(4,5)],6) => 3
[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => ([(2,5),(3,5),(5,4)],6) => ([(2,5),(3,5),(4,5)],6) => 2
[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => ([(2,3),(3,5),(5,4)],6) => ([(2,5),(3,4),(4,5)],6) => 3
[.,[[.,.],[.,[.,[.,.]]]]] => [6,5,4,2,3,1] => ([(4,5)],6) => ([(4,5)],6) => 1
[.,[[.,.],[.,[[.,.],.]]]] => [5,6,4,2,3,1] => ([(2,5),(3,4)],6) => ([(2,5),(3,4)],6) => 1
[.,[[.,.],[[.,.],[.,.]]]] => [6,4,5,2,3,1] => ([(2,5),(3,4)],6) => ([(2,5),(3,4)],6) => 1
[.,[[.,.],[[.,[.,.]],.]]] => [5,4,6,2,3,1] => ([(1,5),(2,5),(3,4)],6) => ([(1,2),(3,5),(4,5)],6) => 2
[.,[[.,.],[[[.,.],.],.]]] => [4,5,6,2,3,1] => ([(1,3),(2,4),(4,5)],6) => ([(1,2),(3,5),(4,5)],6) => 2
[.,[[.,[.,.]],[.,[.,.]]]] => [6,5,3,2,4,1] => ([(3,5),(4,5)],6) => ([(3,5),(4,5)],6) => 2
[.,[[.,[.,.]],[[.,.],.]]] => [5,6,3,2,4,1] => ([(1,5),(2,5),(3,4)],6) => ([(1,2),(3,5),(4,5)],6) => 2
[.,[[[.,.],.],[.,[.,.]]]] => [6,5,2,3,4,1] => ([(3,4),(4,5)],6) => ([(3,5),(4,5)],6) => 2
[.,[[[.,.],.],[[.,.],.]]] => [5,6,2,3,4,1] => ([(1,3),(2,4),(4,5)],6) => ([(1,2),(3,5),(4,5)],6) => 2
[.,[[.,[.,[.,.]]],[.,.]]] => [6,4,3,2,5,1] => ([(2,5),(3,5),(4,5)],6) => ([(2,5),(3,5),(4,5)],6) => 2
[.,[[.,[[.,.],.]],[.,.]]] => [6,3,4,2,5,1] => ([(2,5),(3,4),(4,5)],6) => ([(2,5),(3,4),(4,5)],6) => 3
[.,[[[.,.],[.,.]],[.,.]]] => [6,4,2,3,5,1] => ([(2,5),(3,4),(4,5)],6) => ([(2,5),(3,4),(4,5)],6) => 3
[.,[[[.,[.,.]],.],[.,.]]] => [6,3,2,4,5,1] => ([(2,5),(3,5),(5,4)],6) => ([(2,5),(3,5),(4,5)],6) => 2
[.,[[[[.,.],.],.],[.,.]]] => [6,2,3,4,5,1] => ([(2,3),(3,5),(5,4)],6) => ([(2,5),(3,4),(4,5)],6) => 3
[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => ([(1,5),(2,5),(3,5),(4,5)],6) => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => ([(1,5),(2,5),(3,4),(4,5)],6) => ([(1,5),(2,5),(3,4),(4,5)],6) => 3
[.,[[.,[[.,.],[.,.]]],.]] => [5,3,4,2,6,1] => ([(1,5),(2,5),(3,4),(4,5)],6) => ([(1,5),(2,5),(3,4),(4,5)],6) => 3
[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => ([(1,5),(2,4),(3,4),(4,5)],6) => ([(1,5),(2,5),(3,4),(4,5)],6) => 3
[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => ([(1,5),(2,3),(3,4),(4,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => 4
[.,[[[.,.],[.,[.,.]]],.]] => [5,4,2,3,6,1] => ([(1,5),(2,5),(3,4),(4,5)],6) => ([(1,5),(2,5),(3,4),(4,5)],6) => 3
[.,[[[.,.],[[.,.],.]],.]] => [4,5,2,3,6,1] => ([(1,4),(2,3),(3,5),(4,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => 4
[.,[[[.,[.,.]],[.,.]],.]] => [5,3,2,4,6,1] => ([(1,5),(2,4),(3,4),(4,5)],6) => ([(1,5),(2,5),(3,4),(4,5)],6) => 3
[.,[[[[.,.],.],[.,.]],.]] => [5,2,3,4,6,1] => ([(1,5),(2,3),(3,4),(4,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => 4
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Description
The length of a longest trail in a graph.
A trail is a sequence of distinct edges, such that two consecutive edges share a vertex.
A trail is a sequence of distinct edges, such that two consecutive edges share a vertex.
Map
to graph
Description
Returns the Hasse diagram of the poset as an undirected graph.
Map
permutation poset
Description
Sends a permutation to its permutation poset.
For a permutation $\pi$ of length $n$, this poset has vertices
$$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$$
and the cover relation is given by $(w, x) \leq (y, z)$ if $w \leq y$ and $x \leq z$.
For example, the permutation $[3,1,5,4,2]$ is mapped to the poset with cover relations
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
For a permutation $\pi$ of length $n$, this poset has vertices
$$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$$
and the cover relation is given by $(w, x) \leq (y, z)$ if $w \leq y$ and $x \leq z$.
For example, the permutation $[3,1,5,4,2]$ is mapped to the poset with cover relations
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
Map
to 132-avoiding permutation
Description
Return a 132-avoiding permutation corresponding to a binary tree.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the maximal element of the Sylvester class.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the maximal element of the Sylvester class.
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