Identifier
-
Mp00017:
Binary trees
—to 312-avoiding permutation⟶
Permutations
Mp00329: Permutations —Tanimoto⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001086: Permutations ⟶ ℤ
Values
[.,.] => [1] => [1] => [1] => 0
[.,[.,.]] => [2,1] => [2,1] => [1,2] => 0
[[.,.],.] => [1,2] => [1,2] => [2,1] => 0
[.,[.,[.,.]]] => [3,2,1] => [3,2,1] => [1,2,3] => 0
[.,[[.,.],.]] => [2,3,1] => [3,1,2] => [1,3,2] => 1
[[.,.],[.,.]] => [1,3,2] => [2,1,3] => [2,3,1] => 0
[[.,[.,.]],.] => [2,1,3] => [1,3,2] => [3,1,2] => 0
[[[.,.],.],.] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[.,[.,[.,[.,.]]]] => [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[.,[.,[[.,.],.]]] => [3,4,2,1] => [4,1,3,2] => [1,4,2,3] => 1
[.,[[.,.],[.,.]]] => [2,4,3,1] => [3,1,4,2] => [2,4,1,3] => 0
[.,[[.,[.,.]],.]] => [3,2,4,1] => [4,3,1,2] => [1,2,4,3] => 1
[.,[[[.,.],.],.]] => [2,3,4,1] => [3,4,1,2] => [2,1,4,3] => 1
[[.,.],[.,[.,.]]] => [1,4,3,2] => [2,1,4,3] => [3,4,1,2] => 0
[[.,.],[[.,.],.]] => [1,3,4,2] => [2,4,1,3] => [3,1,4,2] => 1
[[.,[.,.]],[.,.]] => [2,1,4,3] => [3,2,1,4] => [2,3,4,1] => 0
[[[.,.],.],[.,.]] => [1,2,4,3] => [2,3,1,4] => [3,2,4,1] => 0
[[.,[.,[.,.]]],.] => [3,2,1,4] => [1,4,3,2] => [4,1,2,3] => 0
[[.,[[.,.],.]],.] => [2,3,1,4] => [1,3,4,2] => [4,2,1,3] => 0
[[[.,.],[.,.]],.] => [1,3,2,4] => [1,2,4,3] => [4,3,1,2] => 0
[[[.,[.,.]],.],.] => [2,1,3,4] => [1,3,2,4] => [4,2,3,1] => 0
[[[[.,.],.],.],.] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => [5,1,4,3,2] => [1,5,2,3,4] => 1
[.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => [4,1,5,3,2] => [2,5,1,3,4] => 0
[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => [5,4,1,3,2] => [1,2,5,3,4] => 1
[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => [4,5,1,3,2] => [2,1,5,3,4] => 1
[.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => [3,1,5,4,2] => [3,5,1,2,4] => 0
[.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => [3,5,1,4,2] => [3,1,5,2,4] => 1
[.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => [4,3,1,5,2] => [2,3,5,1,4] => 0
[.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => [3,4,1,5,2] => [3,2,5,1,4] => 0
[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [5,4,3,1,2] => [1,2,3,5,4] => 1
[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => [4,5,3,1,2] => [2,1,3,5,4] => 1
[.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => [3,5,4,1,2] => [3,1,2,5,4] => 1
[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => [4,3,5,1,2] => [2,3,1,5,4] => 1
[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => [3,4,5,1,2] => [3,2,1,5,4] => 1
[[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => [2,1,5,4,3] => [4,5,1,2,3] => 0
[[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => [2,5,1,4,3] => [4,1,5,2,3] => 1
[[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => [2,4,1,5,3] => [4,2,5,1,3] => 0
[[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => [2,5,4,1,3] => [4,1,2,5,3] => 1
[[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => [2,4,5,1,3] => [4,2,1,5,3] => 1
[[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => [3,2,1,5,4] => [3,4,5,1,2] => 0
[[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => [3,2,5,1,4] => [3,4,1,5,2] => 1
[[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => [2,3,1,5,4] => [4,3,5,1,2] => 0
[[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => [2,3,5,1,4] => [4,3,1,5,2] => 1
[[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => [4,3,2,1,5] => [2,3,4,5,1] => 0
[[.,[[.,.],.]],[.,.]] => [2,3,1,5,4] => [3,4,2,1,5] => [3,2,4,5,1] => 0
[[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => [2,4,3,1,5] => [4,2,3,5,1] => 0
[[[.,[.,.]],.],[.,.]] => [2,1,3,5,4] => [3,2,4,1,5] => [3,4,2,5,1] => 0
[[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => [2,3,4,1,5] => [4,3,2,5,1] => 0
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => [1,4,5,3,2] => [5,2,1,3,4] => 0
[[.,[[.,.],[.,.]]],.] => [2,4,3,1,5] => [1,3,5,4,2] => [5,3,1,2,4] => 0
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [1,4,3,5,2] => [5,2,3,1,4] => 0
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => [1,3,4,5,2] => [5,3,2,1,4] => 0
[[[.,.],[.,[.,.]]],.] => [1,4,3,2,5] => [1,2,5,4,3] => [5,4,1,2,3] => 0
[[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => [1,2,4,5,3] => [5,4,2,1,3] => 0
[[[.,[.,.]],[.,.]],.] => [2,1,4,3,5] => [1,3,2,5,4] => [5,3,4,1,2] => 0
[[[[.,.],.],[.,.]],.] => [1,2,4,3,5] => [1,2,3,5,4] => [5,4,3,1,2] => 0
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [1,4,3,2,5] => [5,2,3,4,1] => 0
[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => [1,3,4,2,5] => [5,3,2,4,1] => 0
[[[[.,.],[.,.]],.],.] => [1,3,2,4,5] => [1,2,4,3,5] => [5,4,2,3,1] => 0
[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 0
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 0
[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => [6,1,5,4,3,2] => [1,6,2,3,4,5] => 1
[.,[.,[.,[[.,.],[.,.]]]]] => [4,6,5,3,2,1] => [5,1,6,4,3,2] => [2,6,1,3,4,5] => 0
[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => [6,5,1,4,3,2] => [1,2,6,3,4,5] => 1
[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => [5,6,1,4,3,2] => [2,1,6,3,4,5] => 1
[.,[.,[[.,.],[.,[.,.]]]]] => [3,6,5,4,2,1] => [4,1,6,5,3,2] => [3,6,1,2,4,5] => 0
[.,[.,[[.,.],[[.,.],.]]]] => [3,5,6,4,2,1] => [4,6,1,5,3,2] => [3,1,6,2,4,5] => 1
[.,[.,[[.,[.,.]],[.,.]]]] => [4,3,6,5,2,1] => [5,4,1,6,3,2] => [2,3,6,1,4,5] => 0
[.,[.,[[[.,.],.],[.,.]]]] => [3,4,6,5,2,1] => [4,5,1,6,3,2] => [3,2,6,1,4,5] => 0
[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => [6,5,4,1,3,2] => [1,2,3,6,4,5] => 1
[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => [5,6,4,1,3,2] => [2,1,3,6,4,5] => 1
[.,[.,[[[.,.],[.,.]],.]]] => [3,5,4,6,2,1] => [4,6,5,1,3,2] => [3,1,2,6,4,5] => 1
[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => [5,4,6,1,3,2] => [2,3,1,6,4,5] => 1
[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => [4,5,6,1,3,2] => [3,2,1,6,4,5] => 1
[.,[[.,.],[.,[.,[.,.]]]]] => [2,6,5,4,3,1] => [3,1,6,5,4,2] => [4,6,1,2,3,5] => 0
[.,[[.,.],[.,[[.,.],.]]]] => [2,5,6,4,3,1] => [3,6,1,5,4,2] => [4,1,6,2,3,5] => 1
[.,[[.,.],[[.,.],[.,.]]]] => [2,4,6,5,3,1] => [3,5,1,6,4,2] => [4,2,6,1,3,5] => 0
[.,[[.,.],[[.,[.,.]],.]]] => [2,5,4,6,3,1] => [3,6,5,1,4,2] => [4,1,2,6,3,5] => 1
[.,[[.,.],[[[.,.],.],.]]] => [2,4,5,6,3,1] => [3,5,6,1,4,2] => [4,2,1,6,3,5] => 1
[.,[[.,[.,.]],[.,[.,.]]]] => [3,2,6,5,4,1] => [4,3,1,6,5,2] => [3,4,6,1,2,5] => 0
[.,[[.,[.,.]],[[.,.],.]]] => [3,2,5,6,4,1] => [4,3,6,1,5,2] => [3,4,1,6,2,5] => 1
[.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => [3,4,1,6,5,2] => [4,3,6,1,2,5] => 0
[.,[[[.,.],.],[[.,.],.]]] => [2,3,5,6,4,1] => [3,4,6,1,5,2] => [4,3,1,6,2,5] => 1
[.,[[.,[.,[.,.]]],[.,.]]] => [4,3,2,6,5,1] => [5,4,3,1,6,2] => [2,3,4,6,1,5] => 0
[.,[[.,[[.,.],.]],[.,.]]] => [3,4,2,6,5,1] => [4,5,3,1,6,2] => [3,2,4,6,1,5] => 0
[.,[[[.,.],[.,.]],[.,.]]] => [2,4,3,6,5,1] => [3,5,4,1,6,2] => [4,2,3,6,1,5] => 0
[.,[[[.,[.,.]],.],[.,.]]] => [3,2,4,6,5,1] => [4,3,5,1,6,2] => [3,4,2,6,1,5] => 0
[.,[[[[.,.],.],.],[.,.]]] => [2,3,4,6,5,1] => [3,4,5,1,6,2] => [4,3,2,6,1,5] => 0
[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => [6,5,4,3,1,2] => [1,2,3,4,6,5] => 1
[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => [5,6,4,3,1,2] => [2,1,3,4,6,5] => 1
[.,[[.,[[.,.],[.,.]]],.]] => [3,5,4,2,6,1] => [4,6,5,3,1,2] => [3,1,2,4,6,5] => 1
[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => [5,4,6,3,1,2] => [2,3,1,4,6,5] => 1
[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => [4,5,6,3,1,2] => [3,2,1,4,6,5] => 1
[.,[[[.,.],[.,[.,.]]],.]] => [2,5,4,3,6,1] => [3,6,5,4,1,2] => [4,1,2,3,6,5] => 1
[.,[[[.,.],[[.,.],.]],.]] => [2,4,5,3,6,1] => [3,5,6,4,1,2] => [4,2,1,3,6,5] => 1
[.,[[[.,[.,.]],[.,.]],.]] => [3,2,5,4,6,1] => [4,3,6,5,1,2] => [3,4,1,2,6,5] => 1
[.,[[[[.,.],.],[.,.]],.]] => [2,3,5,4,6,1] => [3,4,6,5,1,2] => [4,3,1,2,6,5] => 1
>>> Load all 260 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 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
complement
Description
Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
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.
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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