- FindStat

Identifier

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000629: Binary words ⟶ ℤ

Values

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

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

[.,[.,[.,.]]] => [3,2,1] => 00 => 0

[.,[[.,.],.]] => [2,3,1] => 00 => 0

[[.,.],[.,.]] => [1,3,2] => 10 => 0

[[.,[.,.]],.] => [2,1,3] => 01 => 0

[[[.,.],.],.] => [1,2,3] => 11 => 0

[.,[.,[.,[.,.]]]] => [4,3,2,1] => 000 => 0

[.,[.,[[.,.],.]]] => [3,4,2,1] => 000 => 0

[.,[[.,.],[.,.]]] => [2,4,3,1] => 000 => 0

[.,[[.,[.,.]],.]] => [3,2,4,1] => 000 => 0

[.,[[[.,.],.],.]] => [2,3,4,1] => 000 => 0

[[.,.],[.,[.,.]]] => [1,4,3,2] => 100 => 0

[[.,.],[[.,.],.]] => [1,3,4,2] => 100 => 0

[[.,[.,.]],[.,.]] => [2,1,4,3] => 010 => 0

[[[.,.],.],[.,.]] => [1,2,4,3] => 110 => 0

[[.,[.,[.,.]]],.] => [3,2,1,4] => 001 => 0

[[.,[[.,.],.]],.] => [2,3,1,4] => 001 => 0

[[[.,.],[.,.]],.] => [1,3,2,4] => 101 => 0

[[[.,[.,.]],.],.] => [2,1,3,4] => 011 => 0

[[[[.,.],.],.],.] => [1,2,3,4] => 111 => 0

[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => 0000 => 0

[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => 0000 => 0

[.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => 0000 => 0

[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => 0000 => 0

[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => 0000 => 0

[.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => 0000 => 0

[.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => 0000 => 0

[.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => 0000 => 0

[.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => 0000 => 0

[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => 0000 => 0

[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => 0000 => 0

[.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => 0000 => 0

[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => 0000 => 0

[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => 0000 => 0

[[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => 1000 => 0

[[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => 1000 => 0

[[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => 1000 => 0

[[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => 1000 => 0

[[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => 1000 => 0

[[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => 0100 => 0

[[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => 0100 => 0

[[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => 1100 => 0

[[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => 1100 => 0

[[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => 0010 => 0

[[.,[[.,.],.]],[.,.]] => [2,3,1,5,4] => 0010 => 0

[[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => 1010 => 0

[[[.,[.,.]],.],[.,.]] => [2,1,3,5,4] => 0110 => 0

[[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => 1110 => 0

[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => 0001 => 0

[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => 0001 => 0

[[.,[[.,.],[.,.]]],.] => [2,4,3,1,5] => 0001 => 0

[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => 0001 => 0

[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => 0001 => 0

[[[.,.],[.,[.,.]]],.] => [1,4,3,2,5] => 1001 => 0

[[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => 1001 => 0

[[[.,[.,.]],[.,.]],.] => [2,1,4,3,5] => 0101 => 0

[[[[.,.],.],[.,.]],.] => [1,2,4,3,5] => 1101 => 0

[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => 0011 => 0

[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => 0011 => 0

[[[[.,.],[.,.]],.],.] => [1,3,2,4,5] => 1011 => 0

[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => 0111 => 0

[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => 1111 => 0

[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => 00000 => 0

[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => 00000 => 0

[.,[.,[.,[[.,.],[.,.]]]]] => [4,6,5,3,2,1] => 00000 => 0

[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => 00000 => 0

[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => 00000 => 0

[.,[.,[[.,.],[.,[.,.]]]]] => [3,6,5,4,2,1] => 00000 => 0

[.,[.,[[.,.],[[.,.],.]]]] => [3,5,6,4,2,1] => 00000 => 0

[.,[.,[[.,[.,.]],[.,.]]]] => [4,3,6,5,2,1] => 00000 => 0

[.,[.,[[[.,.],.],[.,.]]]] => [3,4,6,5,2,1] => 00000 => 0

[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => 00000 => 0

[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => 00000 => 0

[.,[.,[[[.,.],[.,.]],.]]] => [3,5,4,6,2,1] => 00000 => 0

[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => 00000 => 0

[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => 00000 => 0

[.,[[.,.],[.,[.,[.,.]]]]] => [2,6,5,4,3,1] => 00000 => 0

[.,[[.,.],[.,[[.,.],.]]]] => [2,5,6,4,3,1] => 00000 => 0

[.,[[.,.],[[.,.],[.,.]]]] => [2,4,6,5,3,1] => 00000 => 0

[.,[[.,.],[[.,[.,.]],.]]] => [2,5,4,6,3,1] => 00000 => 0

[.,[[.,.],[[[.,.],.],.]]] => [2,4,5,6,3,1] => 00000 => 0

[.,[[.,[.,.]],[.,[.,.]]]] => [3,2,6,5,4,1] => 00000 => 0

[.,[[.,[.,.]],[[.,.],.]]] => [3,2,5,6,4,1] => 00000 => 0

[.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => 00000 => 0

[.,[[[.,.],.],[[.,.],.]]] => [2,3,5,6,4,1] => 00000 => 0

[.,[[.,[.,[.,.]]],[.,.]]] => [4,3,2,6,5,1] => 00000 => 0

[.,[[.,[[.,.],.]],[.,.]]] => [3,4,2,6,5,1] => 00000 => 0

[.,[[[.,.],[.,.]],[.,.]]] => [2,4,3,6,5,1] => 00000 => 0

[.,[[[.,[.,.]],.],[.,.]]] => [3,2,4,6,5,1] => 00000 => 0

[.,[[[[.,.],.],.],[.,.]]] => [2,3,4,6,5,1] => 00000 => 0

[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => 00000 => 0

[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => 00000 => 0

[.,[[.,[[.,.],[.,.]]],.]] => [3,5,4,2,6,1] => 00000 => 0

[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => 00000 => 0

[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => 00000 => 0

[.,[[[.,.],[.,[.,.]]],.]] => [2,5,4,3,6,1] => 00000 => 0

[.,[[[.,.],[[.,.],.]],.]] => [2,4,5,3,6,1] => 00000 => 0

[.,[[[.,[.,.]],[.,.]],.]] => [3,2,5,4,6,1] => 00000 => 0

[.,[[[[.,.],.],[.,.]],.]] => [2,3,5,4,6,1] => 00000 => 0

[.,[[[.,[.,[.,.]]],.],.]] => [4,3,2,5,6,1] => 00000 => 0

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Description

The defect of a binary word.
The defect of a finite word $w$ is given by the difference between the maximum possible number and the actual number of palindromic factors contained in $w$. The maximum possible number of palindromic factors in a word $w$ is $|w|+1$.

Map

connectivity set

Description

The connectivity set of a permutation as a binary word.
According to [2], also known as the global ascent set.
The connectivity set is
$$C(\pi)=\{i\in [n-1] | \forall 1 \leq j \leq i < k \leq n : \pi(j) < \pi(k)\}.$$
For $n > 1$ it can also be described as the set of occurrences 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].
The permutation is connected, when the connectivity set is empty.

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.