- FindStat

Identifier

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000654: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Description

The first descent of a permutation.
For a permutation

$\pi$ of

$\{1,\ldots,n\}$ , this is the smallest index

$0 < i \leq n$ such that

$\pi(i) > \pi(i+1)$ where one considers

$\pi(n+1)=0$ .

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.