- FindStat

Identifier

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000007: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 445 entries. <<<

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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 saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern

$([1], {(1,1)})$ , i.e., the upper right quadrant is shaded, see [1].

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.