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. <<<
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searching the database for the individual values of this statistic
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searching the database for statistics with the same generating function
Description
The "bounce" of a permutation.
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].
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].
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.
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.
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