- FindStat

Identifier

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000019: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The cardinality of the support of a permutation.
A permutation $\sigma$ may be written as a product $\sigma = s_{i_1}\dots s_{i_k}$ with $k$ minimal, where $s_i = (i,i+1)$ denotes the simple transposition swapping the entries in positions $i$ and $i+1$.
The set of indices $\{i_1,\dots,i_k\}$ is the support of $\sigma$ and independent of the chosen way to write $\sigma$ as such a product.
See [2], Definition 1 and Proposition 10.
The connectivity set of $\sigma$ of length $n$ is the set of indices $1 \leq i < n$ such that $\sigma(k) < i$ for all $k < i$.
Thus, the connectivity set is the complement of the support.

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.