- FindStat

Identifier

Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000234: Permutations ⟶ ℤ (values match St000056The decomposition (or block) number of a permutation.)

Values

[1] => [.,.] => [1] => 0

[1,2] => [.,[.,.]] => [2,1] => 0

[2,1] => [[.,.],.] => [1,2] => 1

[1,2,3] => [.,[.,[.,.]]] => [3,2,1] => 0

[1,3,2] => [.,[[.,.],.]] => [2,3,1] => 0

[2,1,3] => [[.,.],[.,.]] => [1,3,2] => 1

[2,3,1] => [[.,[.,.]],.] => [2,1,3] => 1

[3,1,2] => [[.,.],[.,.]] => [1,3,2] => 1

[3,2,1] => [[[.,.],.],.] => [1,2,3] => 2

[1,2,3,4] => [.,[.,[.,[.,.]]]] => [4,3,2,1] => 0

[1,2,4,3] => [.,[.,[[.,.],.]]] => [3,4,2,1] => 0

[1,3,2,4] => [.,[[.,.],[.,.]]] => [2,4,3,1] => 0

[1,3,4,2] => [.,[[.,[.,.]],.]] => [3,2,4,1] => 0

[1,4,2,3] => [.,[[.,.],[.,.]]] => [2,4,3,1] => 0

[1,4,3,2] => [.,[[[.,.],.],.]] => [2,3,4,1] => 0

[2,1,3,4] => [[.,.],[.,[.,.]]] => [1,4,3,2] => 1

[2,1,4,3] => [[.,.],[[.,.],.]] => [1,3,4,2] => 1

[2,3,1,4] => [[.,[.,.]],[.,.]] => [2,1,4,3] => 1

[2,3,4,1] => [[.,[.,[.,.]]],.] => [3,2,1,4] => 1

[2,4,1,3] => [[.,[.,.]],[.,.]] => [2,1,4,3] => 1

[2,4,3,1] => [[.,[[.,.],.]],.] => [2,3,1,4] => 1

[3,1,2,4] => [[.,.],[.,[.,.]]] => [1,4,3,2] => 1

[3,1,4,2] => [[.,.],[[.,.],.]] => [1,3,4,2] => 1

[3,2,1,4] => [[[.,.],.],[.,.]] => [1,2,4,3] => 2

[3,2,4,1] => [[[.,.],[.,.]],.] => [1,3,2,4] => 2

[3,4,1,2] => [[.,[.,.]],[.,.]] => [2,1,4,3] => 1

[3,4,2,1] => [[[.,[.,.]],.],.] => [2,1,3,4] => 2

[4,1,2,3] => [[.,.],[.,[.,.]]] => [1,4,3,2] => 1

[4,1,3,2] => [[.,.],[[.,.],.]] => [1,3,4,2] => 1

[4,2,1,3] => [[[.,.],.],[.,.]] => [1,2,4,3] => 2

[4,2,3,1] => [[[.,.],[.,.]],.] => [1,3,2,4] => 2

[4,3,1,2] => [[[.,.],.],[.,.]] => [1,2,4,3] => 2

[4,3,2,1] => [[[[.,.],.],.],.] => [1,2,3,4] => 3

[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => 0

[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => 0

[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => 0

[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => 0

[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => 0

[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => 0

[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => 0

[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => 0

[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => 0

[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => 0

[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => 0

[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => 0

[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => 0

[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => 0

[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => 0

[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => 0

[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => 0

[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => 0

[1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => 0

[1,5,2,4,3] => [.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => 0

[1,5,3,2,4] => [.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => 0

[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => 0

[1,5,4,2,3] => [.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => 0

[1,5,4,3,2] => [.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => 0

[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => 1

[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => 1

[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => 1

[2,1,4,5,3] => [[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => 1

[2,1,5,3,4] => [[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => 1

[2,1,5,4,3] => [[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => 1

[2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => 1

[2,3,1,5,4] => [[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => 1

[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => 1

[2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => 1

[2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => 1

[2,3,5,4,1] => [[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => 1

[2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => 1

[2,4,1,5,3] => [[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => 1

[2,4,3,1,5] => [[.,[[.,.],.]],[.,.]] => [2,3,1,5,4] => 1

[2,4,3,5,1] => [[.,[[.,.],[.,.]]],.] => [2,4,3,1,5] => 1

[2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => 1

[2,4,5,3,1] => [[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => 1

[2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => 1

[2,5,1,4,3] => [[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => 1

[2,5,3,1,4] => [[.,[[.,.],.]],[.,.]] => [2,3,1,5,4] => 1

[2,5,3,4,1] => [[.,[[.,.],[.,.]]],.] => [2,4,3,1,5] => 1

[2,5,4,1,3] => [[.,[[.,.],.]],[.,.]] => [2,3,1,5,4] => 1

[2,5,4,3,1] => [[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => 1

[3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => 1

[3,1,2,5,4] => [[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => 1

[3,1,4,2,5] => [[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => 1

[3,1,4,5,2] => [[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => 1

[3,1,5,2,4] => [[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => 1

[3,1,5,4,2] => [[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => 1

[3,2,1,4,5] => [[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => 2

[3,2,1,5,4] => [[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => 2

[3,2,4,1,5] => [[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => 2

[3,2,4,5,1] => [[[.,.],[.,[.,.]]],.] => [1,4,3,2,5] => 2

[3,2,5,1,4] => [[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => 2

[3,2,5,4,1] => [[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => 2

[3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => 1

[3,4,1,5,2] => [[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => 1

[3,4,2,1,5] => [[[.,[.,.]],.],[.,.]] => [2,1,3,5,4] => 2

[3,4,2,5,1] => [[[.,[.,.]],[.,.]],.] => [2,1,4,3,5] => 2

[3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => 1

[3,4,5,2,1] => [[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => 2

[3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => 1

[3,5,1,4,2] => [[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => 1

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Description

The number of global ascents of a permutation.
The global ascents are the integers $i$ such that
$$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i < k \leq n: \pi(j) < \pi(k)\}.$$
Equivalently, by the pigeonhole principle,
$$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i: \pi(j) \leq i \}.$$
For $n > 1$ it can also be described as an occurrence of the mesh pattern
$$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$$
or equivalently
$$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$$
see [3].
According to [2], this is also the cardinality of the connectivity set of a permutation. The permutation is connected, when the connectivity set is empty. This gives oeis:A003319.

Map

to increasing tree

Description

Sends a permutation to its associated increasing tree.
This tree is recursively obtained by sending the unique permutation of length $0$ to the empty tree, and sending a permutation $\sigma$ of length $n \geq 1$ to a root node with two subtrees $L$ and $R$ by splitting $\sigma$ at the index $\sigma^{-1}(1)$, normalizing both sides again to permutations and sending the permutations on the left and on the right of $\sigma^{-1}(1)$ to the trees $L$ and $R$, respectively.

Map

to 312-avoiding permutation

Description

Return a 312-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 minimal element of this Sylvester class.