- FindStat

Identifier

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St001394: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => 0

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

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

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

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

[.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => 0

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

[.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => 0

[.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => 0

[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => 0

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

[.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => 0

[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => 0

[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => 0

[[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => 0

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

[[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => 0

[[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => 0

[[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => 0

[[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => 0

[[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => 0

[[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => 0

[[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => 0

[[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => 0

[[.,[[.,.],.]],[.,.]] => [2,3,1,5,4] => 0

[[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => 0

[[[.,[.,.]],.],[.,.]] => [2,1,3,5,4] => 0

[[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => 0

[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => 0

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

[[.,[[.,.],[.,.]]],.] => [2,4,3,1,5] => 0

[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => 0

[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => 0

[[[.,.],[.,[.,.]]],.] => [1,4,3,2,5] => 0

[[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => 0

[[[.,[.,.]],[.,.]],.] => [2,1,4,3,5] => 0

[[[[.,.],.],[.,.]],.] => [1,2,4,3,5] => 0

[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => 0

[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => 0

[[[[.,.],[.,.]],.],.] => [1,3,2,4,5] => 0

[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => 0

[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => 0

[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

[.,[[.,.],[.,[.,[.,.]]]]] => [2,6,5,4,3,1] => 0

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

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

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

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

[.,[[.,[.,.]],[.,[.,.]]]] => [3,2,6,5,4,1] => 0

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

[.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => 0

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

[.,[[.,[.,[.,.]]],[.,.]]] => [4,3,2,6,5,1] => 0

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

[.,[[[.,.],[.,.]],[.,.]]] => [2,4,3,6,5,1] => 0

[.,[[[.,[.,.]],.],[.,.]]] => [3,2,4,6,5,1] => 0

[.,[[[[.,.],.],.],[.,.]]] => [2,3,4,6,5,1] => 0

[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => 0

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

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

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

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

[.,[[[.,.],[.,[.,.]]],.]] => [2,5,4,3,6,1] => 0

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

[.,[[[.,[.,.]],[.,.]],.]] => [3,2,5,4,6,1] => 0

[.,[[[[.,.],.],[.,.]],.]] => [2,3,5,4,6,1] => 0

>>> Load all 492 entries. <<<

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 5$

$F_{4} = 13 + q$

$F_{5} = 34 + 8\ q$

$F_{6} = 89 + 40\ q + 3\ q^{2}$

Description

The genus of a permutation.
The genus

$g(\pi)$ of a permutation

$\pi\in\mathfrak S_n$ is defined via the relation

$n+1-2g(\pi) = z(\pi) + z(\pi^{-1} \zeta ),$
where

$\zeta = (1,2,\dots,n)$ is the long cycle and

$z(\cdot)$ is the number of cycles in the permutation.

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.