- FindStat

Identifier

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000542: Permutations ⟶ ℤ (values match St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 305 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 left-to-right-minima of a permutation.
An integer

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

$\sigma$ is a left-to-right-minimum if there does not exist a j < i such that

$\sigma_j < \sigma_i$ .

Map

to 312-avoiding permutation

Description

Return a 312-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 minimal element of this Sylvester class.