Identifier
-
Mp00014:
Binary trees
—to 132-avoiding permutation⟶
Permutations
St000989: Permutations ⟶ ℤ
Values
=>
Cc0010;cc-rep-0
[.,[.,.]]=>[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
[.,[[[.,[[.,.],.]],.],.]]=>[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..
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.
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.
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