- FindStat

Identifier

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001085: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 489 entries. <<<

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 4 + q$

$F_{4} = 10 + 4\ q$

$F_{5} = 28 + 14\ q$

$F_{6} = 84 + 48\ q$

Description

The number of occurrences of the vincular pattern |21-3 in a permutation.
This is the number of occurrences of the pattern

$213$ , where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive.
In other words, this is the number of ascents whose bottom value is strictly smaller and the top value is strictly larger than the first entry of the permutation.

Map

reverse

Description

Sends a permutation to its reverse.
The reverse of a permutation

$\sigma$ of length

$n$ is given by

$\tau$ with

$\tau(i) = \sigma(n+1-i)$ .

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.