- FindStat

Identifier

Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000141: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 4015 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + 2\ q + 3\ q^{2}$

$F_{4} = 1 + 3\ q + 8\ q^{2} + 12\ q^{3}$

$F_{5} = 1 + 4\ q + 15\ q^{2} + 40\ q^{3} + 60\ q^{4}$

$F_{6} = 1 + 5\ q + 24\ q^{2} + 90\ q^{3} + 240\ q^{4} + 360\ q^{5}$

Description

The maximum drop size of a permutation.
The maximum drop size of a permutation

$\pi$ of

$[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of

$i-\pi(i)$ .

Map

binary search tree: left to right

Description

Return the shape of the binary search tree of the permutation as a non labelled binary tree.

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.