- FindStat

Identifier

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000314: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 196 entries. <<<

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searching the database for the individual values of this statistic

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$F_{1} = q$

$F_{2} = q + q^{2}$

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

$F_{4} = 5\ q + 5\ q^{2} + 3\ q^{3} + q^{4}$

$F_{5} = 14\ q + 14\ q^{2} + 9\ q^{3} + 4\ q^{4} + q^{5}$

$F_{6} = 42\ q + 42\ q^{2} + 28\ q^{3} + 14\ q^{4} + 5\ q^{5} + q^{6}$

Description

The number of left-to-right-maxima of a permutation.
An integer

$\sigma_i$ in the one-line notation of a permutation

$\sigma$ is a left-to-right-maximum if there does not exist a

$j < i$ such that

$\sigma_j > \sigma_i$ .
This is also the number of weak exceedences of a permutation that are not mid-points of a decreasing subsequence of length 3, see [1] for more on the later description.

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.