- FindStat

Identifier

Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000133: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 196 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

search for individual values

searching the database for the individual values of this statistic

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searching the database for statistics with the same generating function

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$F_{1} = 1$

$F_{2} = 1 + q$

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

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

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

$F_{6} = 42 + 14\ q + 10\ q^{2} + 10\ q^{3} + 14\ q^{4} + 42\ q^{5}$

Description

The "bounce" of a permutation.

Map

left border symmetry

Description

Return the tree where a symmetry has been applied recursively on all left borders. If a tree is made of three trees

$T_1, T_2, T_3$ on its left border, it becomes

$T_3, T_2, T_1$ where same symmetry has been applied to

$T_1, T_2, T_3$ .

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

Simion-Schmidt map

Description

The Simion-Schmidt map sends any permutation to a

$123$ -avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between

$132$ -avoiding permutations and

$123$ -avoiding permutations, see [1, Proposition 19].