- FindStat

Identifier

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000220: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => [1,6,2,5,3,4] => 8

[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => [1,5,2,6,3,4] => 7

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

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

[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => [1,4,3,6,2,5] => 7

[.,[.,[[.,.],[.,[.,.]]]]] => [3,6,5,4,2,1] => [1,3,5,2,6,4] => 7

[.,[.,[[.,.],[[.,.],.]]]] => [3,5,6,4,2,1] => [1,3,6,2,5,4] => 9

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

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

[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => [1,5,2,4,6,3] => 7

[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => [1,4,6,2,5,3] => 8

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

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

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

[.,[[.,.],[.,[.,[.,.]]]]] => [2,6,5,4,3,1] => [1,2,6,3,5,4] => 9

[.,[[.,.],[.,[[.,.],.]]]] => [2,5,6,4,3,1] => [1,2,5,3,6,4] => 7

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

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

[.,[[.,.],[[[.,.],.],.]]] => [2,4,5,6,3,1] => [1,2,4,6,3,5] => 7

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

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

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

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

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

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

[.,[[[.,.],[.,.]],[.,.]]] => [2,4,3,6,5,1] => [1,2,4,6,3,5] => 7

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

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

[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => [1,5,6,2,4,3] => 8

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

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

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

[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => [1,3,5,6,2,4] => 7

[.,[[[.,.],[.,[.,.]]],.]] => [2,5,4,3,6,1] => [1,2,5,6,3,4] => 8

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

[.,[[[.,[.,.]],[.,.]],.]] => [3,2,5,4,6,1] => [1,3,5,6,2,4] => 7

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

>>> Load all 695 entries. <<<

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 4 + q$

$F_{4} = 8 + 2\ q + 4\ q^{2}$

$F_{5} = 16 + 4\ q + 8\ q^{2} + 5\ q^{3} + 8\ q^{4} + q^{5}$

$F_{6} = 32 + 8\ q + 16\ q^{2} + 9\ q^{3} + 24\ q^{4} + 4\ q^{5} + 19\ q^{6} + 12\ q^{7} + 5\ q^{8} + 3\ q^{9}$

Description

The number of occurrences of the pattern 132 in a permutation.

Map

cycle-as-one-line notation

Description

Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.

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.