Identifier
Values
[.,[.,.]] => [2,1] => 0
[[.,.],.] => [1,2] => 1
[.,[.,[.,.]]] => [3,2,1] => 0
[.,[[.,.],.]] => [2,3,1] => 0
[[.,.],[.,.]] => [3,1,2] => 1
[[.,[.,.]],.] => [2,1,3] => 1
[[[.,.],.],.] => [1,2,3] => 2
[.,[.,[.,[.,.]]]] => [4,3,2,1] => 0
[.,[.,[[.,.],.]]] => [3,4,2,1] => 0
[.,[[.,.],[.,.]]] => [4,2,3,1] => 0
[.,[[.,[.,.]],.]] => [3,2,4,1] => 0
[.,[[[.,.],.],.]] => [2,3,4,1] => 0
[[.,.],[.,[.,.]]] => [4,3,1,2] => 1
[[.,.],[[.,.],.]] => [3,4,1,2] => 1
[[.,[.,.]],[.,.]] => [4,2,1,3] => 1
[[[.,.],.],[.,.]] => [4,1,2,3] => 2
[[.,[.,[.,.]]],.] => [3,2,1,4] => 1
[[.,[[.,.],.]],.] => [2,3,1,4] => 1
[[[.,.],[.,.]],.] => [3,1,2,4] => 2
[[[.,[.,.]],.],.] => [2,1,3,4] => 2
[[[[.,.],.],.],.] => [1,2,3,4] => 3
[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => 0
[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => 0
[.,[.,[[.,.],[.,.]]]] => [5,3,4,2,1] => 0
[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => 0
[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => 0
[.,[[.,.],[.,[.,.]]]] => [5,4,2,3,1] => 0
[.,[[.,.],[[.,.],.]]] => [4,5,2,3,1] => 0
[.,[[.,[.,.]],[.,.]]] => [5,3,2,4,1] => 0
[.,[[[.,.],.],[.,.]]] => [5,2,3,4,1] => 0
[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => 0
[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => 0
[.,[[[.,.],[.,.]],.]] => [4,2,3,5,1] => 0
[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => 0
[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => 0
[[.,.],[.,[.,[.,.]]]] => [5,4,3,1,2] => 1
[[.,.],[.,[[.,.],.]]] => [4,5,3,1,2] => 1
[[.,.],[[.,.],[.,.]]] => [5,3,4,1,2] => 1
[[.,.],[[.,[.,.]],.]] => [4,3,5,1,2] => 1
[[.,.],[[[.,.],.],.]] => [3,4,5,1,2] => 1
[[.,[.,.]],[.,[.,.]]] => [5,4,2,1,3] => 1
[[.,[.,.]],[[.,.],.]] => [4,5,2,1,3] => 1
[[[.,.],.],[.,[.,.]]] => [5,4,1,2,3] => 2
[[[.,.],.],[[.,.],.]] => [4,5,1,2,3] => 2
[[.,[.,[.,.]]],[.,.]] => [5,3,2,1,4] => 1
[[.,[[.,.],.]],[.,.]] => [5,2,3,1,4] => 1
[[[.,.],[.,.]],[.,.]] => [5,3,1,2,4] => 2
[[[.,[.,.]],.],[.,.]] => [5,2,1,3,4] => 2
[[[[.,.],.],.],[.,.]] => [5,1,2,3,4] => 3
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => 1
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => 1
[[.,[[.,.],[.,.]]],.] => [4,2,3,1,5] => 1
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => 1
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => 1
[[[.,.],[.,[.,.]]],.] => [4,3,1,2,5] => 2
[[[.,.],[[.,.],.]],.] => [3,4,1,2,5] => 2
[[[.,[.,.]],[.,.]],.] => [4,2,1,3,5] => 2
[[[[.,.],.],[.,.]],.] => [4,1,2,3,5] => 3
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => 2
[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => 2
[[[[.,.],[.,.]],.],.] => [3,1,2,4,5] => 3
[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => 3
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => 4
[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => 0
[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => 0
[.,[.,[.,[[.,.],[.,.]]]]] => [6,4,5,3,2,1] => 0
[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => 0
[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => 0
[.,[.,[[.,.],[.,[.,.]]]]] => [6,5,3,4,2,1] => 0
[.,[.,[[.,.],[[.,.],.]]]] => [5,6,3,4,2,1] => 0
[.,[.,[[.,[.,.]],[.,.]]]] => [6,4,3,5,2,1] => 0
[.,[.,[[[.,.],.],[.,.]]]] => [6,3,4,5,2,1] => 0
[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => 0
[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => 0
[.,[.,[[[.,.],[.,.]],.]]] => [5,3,4,6,2,1] => 0
[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => 0
[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => 0
[.,[[.,.],[.,[.,[.,.]]]]] => [6,5,4,2,3,1] => 0
[.,[[.,.],[.,[[.,.],.]]]] => [5,6,4,2,3,1] => 0
[.,[[.,.],[[.,.],[.,.]]]] => [6,4,5,2,3,1] => 0
[.,[[.,.],[[.,[.,.]],.]]] => [5,4,6,2,3,1] => 0
[.,[[.,.],[[[.,.],.],.]]] => [4,5,6,2,3,1] => 0
[.,[[.,[.,.]],[.,[.,.]]]] => [6,5,3,2,4,1] => 0
[.,[[.,[.,.]],[[.,.],.]]] => [5,6,3,2,4,1] => 0
[.,[[[.,.],.],[.,[.,.]]]] => [6,5,2,3,4,1] => 0
[.,[[[.,.],.],[[.,.],.]]] => [5,6,2,3,4,1] => 0
[.,[[.,[.,[.,.]]],[.,.]]] => [6,4,3,2,5,1] => 0
[.,[[.,[[.,.],.]],[.,.]]] => [6,3,4,2,5,1] => 0
[.,[[[.,.],[.,.]],[.,.]]] => [6,4,2,3,5,1] => 0
[.,[[[.,[.,.]],.],[.,.]]] => [6,3,2,4,5,1] => 0
[.,[[[[.,.],.],.],[.,.]]] => [6,2,3,4,5,1] => 0
[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => 0
[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => 0
[.,[[.,[[.,.],[.,.]]],.]] => [5,3,4,2,6,1] => 0
[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => 0
[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => 0
[.,[[[.,.],[.,[.,.]]],.]] => [5,4,2,3,6,1] => 0
[.,[[[.,.],[[.,.],.]],.]] => [4,5,2,3,6,1] => 0
[.,[[[.,[.,.]],[.,.]],.]] => [5,3,2,4,6,1] => 0
[.,[[[[.,.],.],[.,.]],.]] => [5,2,3,4,6,1] => 0
[.,[[[.,[.,[.,.]]],.],.]] => [4,3,2,5,6,1] => 0
>>> Load all 196 entries. <<<
[.,[[[.,[[.,.],.]],.],.]] => [3,4,2,5,6,1] => 0
[.,[[[[.,.],[.,.]],.],.]] => [4,2,3,5,6,1] => 0
[.,[[[[.,[.,.]],.],.],.]] => [3,2,4,5,6,1] => 0
[.,[[[[[.,.],.],.],.],.]] => [2,3,4,5,6,1] => 0
[[.,.],[.,[.,[.,[.,.]]]]] => [6,5,4,3,1,2] => 1
[[.,.],[.,[.,[[.,.],.]]]] => [5,6,4,3,1,2] => 1
[[.,.],[.,[[.,.],[.,.]]]] => [6,4,5,3,1,2] => 1
[[.,.],[.,[[.,[.,.]],.]]] => [5,4,6,3,1,2] => 1
[[.,.],[.,[[[.,.],.],.]]] => [4,5,6,3,1,2] => 1
[[.,.],[[.,.],[.,[.,.]]]] => [6,5,3,4,1,2] => 1
[[.,.],[[.,.],[[.,.],.]]] => [5,6,3,4,1,2] => 1
[[.,.],[[.,[.,.]],[.,.]]] => [6,4,3,5,1,2] => 1
[[.,.],[[[.,.],.],[.,.]]] => [6,3,4,5,1,2] => 1
[[.,.],[[.,[.,[.,.]]],.]] => [5,4,3,6,1,2] => 1
[[.,.],[[.,[[.,.],.]],.]] => [4,5,3,6,1,2] => 1
[[.,.],[[[.,.],[.,.]],.]] => [5,3,4,6,1,2] => 1
[[.,.],[[[.,[.,.]],.],.]] => [4,3,5,6,1,2] => 1
[[.,.],[[[[.,.],.],.],.]] => [3,4,5,6,1,2] => 1
[[.,[.,.]],[.,[.,[.,.]]]] => [6,5,4,2,1,3] => 1
[[.,[.,.]],[.,[[.,.],.]]] => [5,6,4,2,1,3] => 1
[[.,[.,.]],[[.,.],[.,.]]] => [6,4,5,2,1,3] => 1
[[.,[.,.]],[[.,[.,.]],.]] => [5,4,6,2,1,3] => 1
[[.,[.,.]],[[[.,.],.],.]] => [4,5,6,2,1,3] => 1
[[[.,.],.],[.,[.,[.,.]]]] => [6,5,4,1,2,3] => 2
[[[.,.],.],[.,[[.,.],.]]] => [5,6,4,1,2,3] => 2
[[[.,.],.],[[.,.],[.,.]]] => [6,4,5,1,2,3] => 2
[[[.,.],.],[[.,[.,.]],.]] => [5,4,6,1,2,3] => 2
[[[.,.],.],[[[.,.],.],.]] => [4,5,6,1,2,3] => 2
[[.,[.,[.,.]]],[.,[.,.]]] => [6,5,3,2,1,4] => 1
[[.,[.,[.,.]]],[[.,.],.]] => [5,6,3,2,1,4] => 1
[[.,[[.,.],.]],[.,[.,.]]] => [6,5,2,3,1,4] => 1
[[.,[[.,.],.]],[[.,.],.]] => [5,6,2,3,1,4] => 1
[[[.,.],[.,.]],[.,[.,.]]] => [6,5,3,1,2,4] => 2
[[[.,.],[.,.]],[[.,.],.]] => [5,6,3,1,2,4] => 2
[[[.,[.,.]],.],[.,[.,.]]] => [6,5,2,1,3,4] => 2
[[[.,[.,.]],.],[[.,.],.]] => [5,6,2,1,3,4] => 2
[[[[.,.],.],.],[.,[.,.]]] => [6,5,1,2,3,4] => 3
[[[[.,.],.],.],[[.,.],.]] => [5,6,1,2,3,4] => 3
[[.,[.,[.,[.,.]]]],[.,.]] => [6,4,3,2,1,5] => 1
[[.,[.,[[.,.],.]]],[.,.]] => [6,3,4,2,1,5] => 1
[[.,[[.,.],[.,.]]],[.,.]] => [6,4,2,3,1,5] => 1
[[.,[[.,[.,.]],.]],[.,.]] => [6,3,2,4,1,5] => 1
[[.,[[[.,.],.],.]],[.,.]] => [6,2,3,4,1,5] => 1
[[[.,.],[.,[.,.]]],[.,.]] => [6,4,3,1,2,5] => 2
[[[.,.],[[.,.],.]],[.,.]] => [6,3,4,1,2,5] => 2
[[[.,[.,.]],[.,.]],[.,.]] => [6,4,2,1,3,5] => 2
[[[[.,.],.],[.,.]],[.,.]] => [6,4,1,2,3,5] => 3
[[[.,[.,[.,.]]],.],[.,.]] => [6,3,2,1,4,5] => 2
[[[.,[[.,.],.]],.],[.,.]] => [6,2,3,1,4,5] => 2
[[[[.,.],[.,.]],.],[.,.]] => [6,3,1,2,4,5] => 3
[[[[.,[.,.]],.],.],[.,.]] => [6,2,1,3,4,5] => 3
[[[[[.,.],.],.],.],[.,.]] => [6,1,2,3,4,5] => 4
[[.,[.,[.,[.,[.,.]]]]],.] => [5,4,3,2,1,6] => 1
[[.,[.,[.,[[.,.],.]]]],.] => [4,5,3,2,1,6] => 1
[[.,[.,[[.,.],[.,.]]]],.] => [5,3,4,2,1,6] => 1
[[.,[.,[[.,[.,.]],.]]],.] => [4,3,5,2,1,6] => 1
[[.,[.,[[[.,.],.],.]]],.] => [3,4,5,2,1,6] => 1
[[.,[[.,.],[.,[.,.]]]],.] => [5,4,2,3,1,6] => 1
[[.,[[.,.],[[.,.],.]]],.] => [4,5,2,3,1,6] => 1
[[.,[[.,[.,.]],[.,.]]],.] => [5,3,2,4,1,6] => 1
[[.,[[[.,.],.],[.,.]]],.] => [5,2,3,4,1,6] => 1
[[.,[[.,[.,[.,.]]],.]],.] => [4,3,2,5,1,6] => 1
[[.,[[.,[[.,.],.]],.]],.] => [3,4,2,5,1,6] => 1
[[.,[[[.,.],[.,.]],.]],.] => [4,2,3,5,1,6] => 1
[[.,[[[.,[.,.]],.],.]],.] => [3,2,4,5,1,6] => 1
[[.,[[[[.,.],.],.],.]],.] => [2,3,4,5,1,6] => 1
[[[.,.],[.,[.,[.,.]]]],.] => [5,4,3,1,2,6] => 2
[[[.,.],[.,[[.,.],.]]],.] => [4,5,3,1,2,6] => 2
[[[.,.],[[.,.],[.,.]]],.] => [5,3,4,1,2,6] => 2
[[[.,.],[[.,[.,.]],.]],.] => [4,3,5,1,2,6] => 2
[[[.,.],[[[.,.],.],.]],.] => [3,4,5,1,2,6] => 2
[[[.,[.,.]],[.,[.,.]]],.] => [5,4,2,1,3,6] => 2
[[[.,[.,.]],[[.,.],.]],.] => [4,5,2,1,3,6] => 2
[[[[.,.],.],[.,[.,.]]],.] => [5,4,1,2,3,6] => 3
[[[[.,.],.],[[.,.],.]],.] => [4,5,1,2,3,6] => 3
[[[.,[.,[.,.]]],[.,.]],.] => [5,3,2,1,4,6] => 2
[[[.,[[.,.],.]],[.,.]],.] => [5,2,3,1,4,6] => 2
[[[[.,.],[.,.]],[.,.]],.] => [5,3,1,2,4,6] => 3
[[[[.,[.,.]],.],[.,.]],.] => [5,2,1,3,4,6] => 3
[[[[[.,.],.],.],[.,.]],.] => [5,1,2,3,4,6] => 4
[[[.,[.,[.,[.,.]]]],.],.] => [4,3,2,1,5,6] => 2
[[[.,[.,[[.,.],.]]],.],.] => [3,4,2,1,5,6] => 2
[[[.,[[.,.],[.,.]]],.],.] => [4,2,3,1,5,6] => 2
[[[.,[[.,[.,.]],.]],.],.] => [3,2,4,1,5,6] => 2
[[[.,[[[.,.],.],.]],.],.] => [2,3,4,1,5,6] => 2
[[[[.,.],[.,[.,.]]],.],.] => [4,3,1,2,5,6] => 3
[[[[.,.],[[.,.],.]],.],.] => [3,4,1,2,5,6] => 3
[[[[.,[.,.]],[.,.]],.],.] => [4,2,1,3,5,6] => 3
[[[[[.,.],.],[.,.]],.],.] => [4,1,2,3,5,6] => 4
[[[[.,[.,[.,.]]],.],.],.] => [3,2,1,4,5,6] => 3
[[[[.,[[.,.],.]],.],.],.] => [2,3,1,4,5,6] => 3
[[[[[.,.],[.,.]],.],.],.] => [3,1,2,4,5,6] => 4
[[[[[.,[.,.]],.],.],.],.] => [2,1,3,4,5,6] => 4
[[[[[[.,.],.],.],.],.],.] => [1,2,3,4,5,6] => 5
[[[[[[[.,.],.],.],.],.],.],.] => [1,2,3,4,5,6,7] => 6
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Description
The number of final rises of a permutation.
For a permutation $\pi$ of length $n$, this is the maximal $k$ such that
$$\pi(n-k) \leq \pi(n-k+1) \leq \cdots \leq \pi(n-1) \leq \pi(n).$$
Equivalently, this is $n-1$ minus the position of the last descent St000653The last descent of a permutation..
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.