Identifier
-
Mp00014:
Binary trees
—to 132-avoiding permutation⟶
Permutations
Mp00329: Permutations —Tanimoto⟶ Permutations
St001086: Permutations ⟶ ℤ
Values
[.,.] => [1] => [1] => 0
[.,[.,.]] => [2,1] => [2,1] => 0
[[.,.],.] => [1,2] => [1,2] => 0
[.,[.,[.,.]]] => [3,2,1] => [3,2,1] => 0
[.,[[.,.],.]] => [2,3,1] => [3,1,2] => 0
[[.,.],[.,.]] => [3,1,2] => [2,3,1] => 0
[[.,[.,.]],.] => [2,1,3] => [1,3,2] => 1
[[[.,.],.],.] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]] => [4,3,2,1] => [4,3,2,1] => 0
[.,[.,[[.,.],.]]] => [3,4,2,1] => [4,1,3,2] => 1
[.,[[.,.],[.,.]]] => [4,2,3,1] => [3,4,2,1] => 0
[.,[[.,[.,.]],.]] => [3,2,4,1] => [4,3,1,2] => 0
[.,[[[.,.],.],.]] => [2,3,4,1] => [3,4,1,2] => 0
[[.,.],[.,[.,.]]] => [4,3,1,2] => [4,2,3,1] => 0
[[.,.],[[.,.],.]] => [3,4,1,2] => [4,1,2,3] => 0
[[.,[.,.]],[.,.]] => [4,2,1,3] => [3,2,4,1] => 0
[[[.,.],.],[.,.]] => [4,1,2,3] => [2,3,4,1] => 0
[[.,[.,[.,.]]],.] => [3,2,1,4] => [1,4,3,2] => 1
[[.,[[.,.],.]],.] => [2,3,1,4] => [1,3,4,2] => 0
[[[.,.],[.,.]],.] => [3,1,2,4] => [1,4,2,3] => 1
[[[.,[.,.]],.],.] => [2,1,3,4] => [1,3,2,4] => 1
[[[[.,.],.],.],.] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => [5,1,4,3,2] => 1
[.,[.,[[.,.],[.,.]]]] => [5,3,4,2,1] => [4,5,3,2,1] => 0
[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => [5,4,1,3,2] => 1
[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => [4,5,1,3,2] => 1
[.,[[.,.],[.,[.,.]]]] => [5,4,2,3,1] => [5,3,4,2,1] => 0
[.,[[.,.],[[.,.],.]]] => [4,5,2,3,1] => [5,1,3,4,2] => 0
[.,[[.,[.,.]],[.,.]]] => [5,3,2,4,1] => [4,3,5,2,1] => 0
[.,[[[.,.],.],[.,.]]] => [5,2,3,4,1] => [3,4,5,2,1] => 0
[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [5,4,3,1,2] => 0
[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => [4,5,3,1,2] => 0
[.,[[[.,.],[.,.]],.]] => [4,2,3,5,1] => [5,3,4,1,2] => 0
[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => [4,3,5,1,2] => 0
[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => [3,4,5,1,2] => 0
[[.,.],[.,[.,[.,.]]]] => [5,4,3,1,2] => [5,4,2,3,1] => 0
[[.,.],[.,[[.,.],.]]] => [4,5,3,1,2] => [5,1,4,2,3] => 1
[[.,.],[[.,.],[.,.]]] => [5,3,4,1,2] => [4,5,2,3,1] => 0
[[.,.],[[.,[.,.]],.]] => [4,3,5,1,2] => [5,4,1,2,3] => 0
[[.,.],[[[.,.],.],.]] => [3,4,5,1,2] => [4,5,1,2,3] => 0
[[.,[.,.]],[.,[.,.]]] => [5,4,2,1,3] => [5,3,2,4,1] => 0
[[.,[.,.]],[[.,.],.]] => [4,5,2,1,3] => [5,1,3,2,4] => 1
[[[.,.],.],[.,[.,.]]] => [5,4,1,2,3] => [5,2,3,4,1] => 0
[[[.,.],.],[[.,.],.]] => [4,5,1,2,3] => [5,1,2,3,4] => 0
[[.,[.,[.,.]]],[.,.]] => [5,3,2,1,4] => [4,3,2,5,1] => 0
[[.,[[.,.],.]],[.,.]] => [5,2,3,1,4] => [3,4,2,5,1] => 0
[[[.,.],[.,.]],[.,.]] => [5,3,1,2,4] => [4,2,3,5,1] => 0
[[[.,[.,.]],.],[.,.]] => [5,2,1,3,4] => [3,2,4,5,1] => 0
[[[[.,.],.],.],[.,.]] => [5,1,2,3,4] => [2,3,4,5,1] => 0
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [1,5,4,3,2] => 1
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => [1,4,5,3,2] => 0
[[.,[[.,.],[.,.]]],.] => [4,2,3,1,5] => [1,5,3,4,2] => 1
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [1,4,3,5,2] => 1
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => [1,3,4,5,2] => 0
[[[.,.],[.,[.,.]]],.] => [4,3,1,2,5] => [1,5,4,2,3] => 1
[[[.,.],[[.,.],.]],.] => [3,4,1,2,5] => [1,4,5,2,3] => 0
[[[.,[.,.]],[.,.]],.] => [4,2,1,3,5] => [1,5,3,2,4] => 1
[[[[.,.],.],[.,.]],.] => [4,1,2,3,5] => [1,5,2,3,4] => 1
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [1,4,3,2,5] => 1
[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => [1,3,4,2,5] => 0
[[[[.,.],[.,.]],.],.] => [3,1,2,4,5] => [1,4,2,3,5] => 1
[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => [1,3,2,4,5] => 1
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 0
[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => [6,1,5,4,3,2] => 1
[.,[.,[.,[[.,.],[.,.]]]]] => [6,4,5,3,2,1] => [5,6,4,3,2,1] => 0
[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => [6,5,1,4,3,2] => 1
[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => [5,6,1,4,3,2] => 1
[.,[.,[[.,.],[.,[.,.]]]]] => [6,5,3,4,2,1] => [6,4,5,3,2,1] => 0
[.,[.,[[.,.],[[.,.],.]]]] => [5,6,3,4,2,1] => [6,1,4,5,3,2] => 0
[.,[.,[[.,[.,.]],[.,.]]]] => [6,4,3,5,2,1] => [5,4,6,3,2,1] => 0
[.,[.,[[[.,.],.],[.,.]]]] => [6,3,4,5,2,1] => [4,5,6,3,2,1] => 0
[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => [6,5,4,1,3,2] => 1
[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => [5,6,4,1,3,2] => 1
[.,[.,[[[.,.],[.,.]],.]]] => [5,3,4,6,2,1] => [6,4,5,1,3,2] => 1
[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => [5,4,6,1,3,2] => 1
[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => [4,5,6,1,3,2] => 1
[.,[[.,.],[.,[.,[.,.]]]]] => [6,5,4,2,3,1] => [6,5,3,4,2,1] => 0
[.,[[.,.],[.,[[.,.],.]]]] => [5,6,4,2,3,1] => [6,1,5,3,4,2] => 1
[.,[[.,.],[[.,.],[.,.]]]] => [6,4,5,2,3,1] => [5,6,3,4,2,1] => 0
[.,[[.,.],[[.,[.,.]],.]]] => [5,4,6,2,3,1] => [6,5,1,3,4,2] => 0
[.,[[.,.],[[[.,.],.],.]]] => [4,5,6,2,3,1] => [5,6,1,3,4,2] => 0
[.,[[.,[.,.]],[.,[.,.]]]] => [6,5,3,2,4,1] => [6,4,3,5,2,1] => 0
[.,[[.,[.,.]],[[.,.],.]]] => [5,6,3,2,4,1] => [6,1,4,3,5,2] => 1
[.,[[[.,.],.],[.,[.,.]]]] => [6,5,2,3,4,1] => [6,3,4,5,2,1] => 0
[.,[[[.,.],.],[[.,.],.]]] => [5,6,2,3,4,1] => [6,1,3,4,5,2] => 0
[.,[[.,[.,[.,.]]],[.,.]]] => [6,4,3,2,5,1] => [5,4,3,6,2,1] => 0
[.,[[.,[[.,.],.]],[.,.]]] => [6,3,4,2,5,1] => [4,5,3,6,2,1] => 0
[.,[[[.,.],[.,.]],[.,.]]] => [6,4,2,3,5,1] => [5,3,4,6,2,1] => 0
[.,[[[.,[.,.]],.],[.,.]]] => [6,3,2,4,5,1] => [4,3,5,6,2,1] => 0
[.,[[[[.,.],.],.],[.,.]]] => [6,2,3,4,5,1] => [3,4,5,6,2,1] => 0
[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => [6,5,4,3,1,2] => 0
[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => [5,6,4,3,1,2] => 0
[.,[[.,[[.,.],[.,.]]],.]] => [5,3,4,2,6,1] => [6,4,5,3,1,2] => 0
[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => [5,4,6,3,1,2] => 0
[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => [4,5,6,3,1,2] => 0
[.,[[[.,.],[.,[.,.]]],.]] => [5,4,2,3,6,1] => [6,5,3,4,1,2] => 0
[.,[[[.,.],[[.,.],.]],.]] => [4,5,2,3,6,1] => [5,6,3,4,1,2] => 0
[.,[[[.,[.,.]],[.,.]],.]] => [5,3,2,4,6,1] => [6,4,3,5,1,2] => 0
[.,[[[[.,.],.],[.,.]],.]] => [5,2,3,4,6,1] => [6,3,4,5,1,2] => 0
>>> Load all 261 entries. <<<
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Description
The number of occurrences of the consecutive pattern 132 in a permutation.
This is the number of occurrences of the pattern $132$, where the matched entries are all adjacent.
This is the number of occurrences of the pattern $132$, where the matched entries are all adjacent.
Map
to 132-avoiding permutation
Description
Return a 132-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 maximal element of the 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 maximal element of the Sylvester class.
Map
Tanimoto
Description
Add 1 to every entry of the permutation (n becomes 1 instead of n+1), except that when n appears at the front or the back of the permutation, instead remove it and place 1 at the other end of the permutation.
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