Identifier
-
Mp00017:
Binary trees
—to 312-avoiding permutation⟶
Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St001928: Permutations ⟶ ℤ
Values
[.,.] => [1] => [1] => [1] => 0
[.,[.,.]] => [2,1] => [2,1] => [2,1] => 1
[[.,.],.] => [1,2] => [1,2] => [1,2] => 0
[.,[.,[.,.]]] => [3,2,1] => [3,1,2] => [3,1,2] => 1
[.,[[.,.],.]] => [2,3,1] => [3,2,1] => [2,3,1] => 1
[[.,.],[.,.]] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[[.,[.,.]],.] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[[[.,.],.],.] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]] => [4,3,2,1] => [4,1,2,3] => [4,1,2,3] => 1
[.,[.,[[.,.],.]]] => [3,4,2,1] => [4,1,3,2] => [3,1,4,2] => 2
[.,[[.,.],[.,.]]] => [2,4,3,1] => [4,2,1,3] => [2,4,1,3] => 1
[.,[[.,[.,.]],.]] => [3,2,4,1] => [4,3,2,1] => [3,4,1,2] => 1
[.,[[[.,.],.],.]] => [2,3,4,1] => [4,2,3,1] => [2,3,4,1] => 1
[[.,.],[.,[.,.]]] => [1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 1
[[.,.],[[.,.],.]] => [1,3,4,2] => [1,4,3,2] => [1,3,4,2] => 1
[[.,[.,.]],[.,.]] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[[[.,.],.],[.,.]] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[[.,[.,[.,.]]],.] => [3,2,1,4] => [3,1,2,4] => [3,1,2,4] => 1
[[.,[[.,.],.]],.] => [2,3,1,4] => [3,2,1,4] => [2,3,1,4] => 1
[[[.,.],[.,.]],.] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[[[.,[.,.]],.],.] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[[[[.,.],.],.],.] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [5,1,2,3,4] => [5,1,2,3,4] => 1
[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => [5,1,2,4,3] => [4,1,2,5,3] => 2
[.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => [5,1,3,2,4] => [3,1,5,2,4] => 2
[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => [5,1,4,3,2] => [4,1,5,2,3] => 2
[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => [5,1,3,4,2] => [3,1,4,5,2] => 2
[.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => [5,2,1,3,4] => [2,5,1,3,4] => 1
[.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => [5,2,1,4,3] => [2,4,1,5,3] => 2
[.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => [5,3,2,1,4] => [3,5,1,2,4] => 1
[.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => [5,2,3,1,4] => [2,3,5,1,4] => 1
[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [5,4,2,3,1] => [4,5,1,2,3] => 1
[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => [5,4,3,2,1] => [3,4,5,1,2] => 1
[.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => [5,2,4,3,1] => [2,4,5,1,3] => 1
[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => [5,3,2,4,1] => [3,4,1,5,2] => 2
[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => [5,2,3,4,1] => [2,3,4,5,1] => 1
[[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => [1,5,2,3,4] => [1,5,2,3,4] => 1
[[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => [1,5,2,4,3] => [1,4,2,5,3] => 2
[[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => [1,5,3,2,4] => [1,3,5,2,4] => 1
[[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => [1,5,4,3,2] => [1,4,5,2,3] => 1
[[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,5,2] => 1
[[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => [2,1,5,3,4] => [2,1,5,3,4] => 2
[[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,5,3] => 2
[[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,5,3] => 1
[[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => 2
[[.,[[.,.],.]],[.,.]] => [2,3,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 2
[[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[[[.,[.,.]],.],[.,.]] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [4,1,2,3,5] => [4,1,2,3,5] => 1
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => [4,1,3,2,5] => [3,1,4,2,5] => 2
[[.,[[.,.],[.,.]]],.] => [2,4,3,1,5] => [4,2,1,3,5] => [2,4,1,3,5] => 1
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [4,3,2,1,5] => [3,4,1,2,5] => 1
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => [4,2,3,1,5] => [2,3,4,1,5] => 1
[[[.,.],[.,[.,.]]],.] => [1,4,3,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => 1
[[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => 1
[[[.,[.,.]],[.,.]],.] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2
[[[[.,.],.],[.,.]],.] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => 1
[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => [3,2,1,4,5] => [2,3,1,4,5] => 1
[[[[.,.],[.,.]],.],.] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => [6,1,2,3,4,5] => [6,1,2,3,4,5] => 1
[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => [6,1,2,3,5,4] => [5,1,2,3,6,4] => 2
[.,[.,[.,[[.,.],[.,.]]]]] => [4,6,5,3,2,1] => [6,1,2,4,3,5] => [4,1,2,6,3,5] => 2
[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => [6,1,2,5,4,3] => [5,1,2,6,3,4] => 2
[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => [6,1,2,4,5,3] => [4,1,2,5,6,3] => 2
[.,[.,[[.,.],[.,[.,.]]]]] => [3,6,5,4,2,1] => [6,1,3,2,4,5] => [3,1,6,2,4,5] => 2
[.,[.,[[.,.],[[.,.],.]]]] => [3,5,6,4,2,1] => [6,1,3,2,5,4] => [3,1,5,2,6,4] => 3
[.,[.,[[.,[.,.]],[.,.]]]] => [4,3,6,5,2,1] => [6,1,4,3,2,5] => [4,1,6,2,3,5] => 2
[.,[.,[[[.,.],.],[.,.]]]] => [3,4,6,5,2,1] => [6,1,3,4,2,5] => [3,1,4,6,2,5] => 2
[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => [6,1,5,3,4,2] => [5,1,6,2,3,4] => 2
[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => [6,1,5,4,3,2] => [4,1,5,6,2,3] => 2
[.,[.,[[[.,.],[.,.]],.]]] => [3,5,4,6,2,1] => [6,1,3,5,4,2] => [3,1,5,6,2,4] => 2
[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => [6,1,4,3,5,2] => [4,1,5,2,6,3] => 3
[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => [6,1,3,4,5,2] => [3,1,4,5,6,2] => 2
[.,[[.,.],[.,[.,[.,.]]]]] => [2,6,5,4,3,1] => [6,2,1,3,4,5] => [2,6,1,3,4,5] => 1
[.,[[.,.],[.,[[.,.],.]]]] => [2,5,6,4,3,1] => [6,2,1,3,5,4] => [2,5,1,3,6,4] => 2
[.,[[.,.],[[.,.],[.,.]]]] => [2,4,6,5,3,1] => [6,2,1,4,3,5] => [2,4,1,6,3,5] => 2
[.,[[.,.],[[.,[.,.]],.]]] => [2,5,4,6,3,1] => [6,2,1,5,4,3] => [2,5,1,6,3,4] => 2
[.,[[.,.],[[[.,.],.],.]]] => [2,4,5,6,3,1] => [6,2,1,4,5,3] => [2,4,1,5,6,3] => 2
[.,[[.,[.,.]],[.,[.,.]]]] => [3,2,6,5,4,1] => [6,3,2,1,4,5] => [3,6,1,2,4,5] => 1
[.,[[.,[.,.]],[[.,.],.]]] => [3,2,5,6,4,1] => [6,3,2,1,5,4] => [3,5,1,2,6,4] => 2
[.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => [6,2,3,1,4,5] => [2,3,6,1,4,5] => 1
[.,[[[.,.],.],[[.,.],.]]] => [2,3,5,6,4,1] => [6,2,3,1,5,4] => [2,3,5,1,6,4] => 2
[.,[[.,[.,[.,.]]],[.,.]]] => [4,3,2,6,5,1] => [6,4,2,3,1,5] => [4,6,1,2,3,5] => 1
[.,[[.,[[.,.],.]],[.,.]]] => [3,4,2,6,5,1] => [6,4,3,2,1,5] => [3,4,6,1,2,5] => 1
[.,[[[.,.],[.,.]],[.,.]]] => [2,4,3,6,5,1] => [6,2,4,3,1,5] => [2,4,6,1,3,5] => 1
[.,[[[.,[.,.]],.],[.,.]]] => [3,2,4,6,5,1] => [6,3,2,4,1,5] => [3,4,1,6,2,5] => 2
[.,[[[[.,.],.],.],[.,.]]] => [2,3,4,6,5,1] => [6,2,3,4,1,5] => [2,3,4,6,1,5] => 1
[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => [6,5,2,3,4,1] => [5,6,1,2,3,4] => 1
[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => [6,5,2,4,3,1] => [4,5,1,6,2,3] => 2
[.,[[.,[[.,.],[.,.]]],.]] => [3,5,4,2,6,1] => [6,5,3,2,4,1] => [3,5,6,1,2,4] => 1
[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => [6,5,4,3,2,1] => [4,5,6,1,2,3] => 1
[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => [6,5,3,4,2,1] => [3,4,5,6,1,2] => 1
[.,[[[.,.],[.,[.,.]]],.]] => [2,5,4,3,6,1] => [6,2,5,3,4,1] => [2,5,6,1,3,4] => 1
[.,[[[.,.],[[.,.],.]],.]] => [2,4,5,3,6,1] => [6,2,5,4,3,1] => [2,4,5,6,1,3] => 1
[.,[[[.,[.,.]],[.,.]],.]] => [3,2,5,4,6,1] => [6,3,2,5,4,1] => [3,5,1,6,2,4] => 2
[.,[[[[.,.],.],[.,.]],.]] => [2,3,5,4,6,1] => [6,2,3,5,4,1] => [2,3,5,6,1,4] => 1
>>> Load all 306 entries. <<<
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Description
The number of non-overlapping descents in a permutation.
In other words, any maximal descending subsequence $\pi_i,\pi_{i+1},\dots,\pi_k$ contributes $\lfloor\frac{k-i+1}{2}\rfloor$ to the total count.
In other words, any maximal descending subsequence $\pi_i,\pi_{i+1},\dots,\pi_k$ contributes $\lfloor\frac{k-i+1}{2}\rfloor$ to the total count.
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
Corteel
Description
Corteel's map interchanging the number of crossings and the number of nestings of a permutation.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.
Map
first fundamental transformation
Description
Return the permutation whose cycles are the subsequences between successive left to right maxima.
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