- FindStat

Identifier

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000740: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{2} = 2\ q$

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

$F_{4} = 4\ q + 6\ q^{2} + 4\ q^{3}$

$F_{5} = 5\ q + 12\ q^{2} + 15\ q^{3} + 10\ q^{4}$

$F_{6} = 6\ q + 20\ q^{2} + 36\ q^{3} + 42\ q^{4} + 28\ q^{5}$

$F_{7} = 7\ q + 30\ q^{2} + 70\ q^{3} + 112\ q^{4} + 126\ q^{5} + 84\ q^{6}$

Description

The last entry of a permutation.
This statistic is undefined for the empty permutation.

Map

restriction

Description

The permutation obtained by removing the largest letter.
This map is undefined for the empty 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.

Map

inverse

Description

Sends a permutation to its inverse.