Identifier
-
Mp00014:
Binary trees
—to 132-avoiding permutation⟶
Permutations
St000541: Permutations ⟶ ℤ (values match St000542The number of left-to-right-minima of a permutation.)
Values
=>
Cc0010;cc-rep-0
[.,[.,.]]=>[2,1]=>1
[[.,.],.]=>[1,2]=>0
[.,[.,[.,.]]]=>[3,2,1]=>2
[.,[[.,.],.]]=>[2,3,1]=>1
[[.,.],[.,.]]=>[3,1,2]=>1
[[.,[.,.]],.]=>[2,1,3]=>1
[[[.,.],.],.]=>[1,2,3]=>0
[.,[.,[.,[.,.]]]]=>[4,3,2,1]=>3
[.,[.,[[.,.],.]]]=>[3,4,2,1]=>2
[.,[[.,.],[.,.]]]=>[4,2,3,1]=>2
[.,[[.,[.,.]],.]]=>[3,2,4,1]=>2
[.,[[[.,.],.],.]]=>[2,3,4,1]=>1
[[.,.],[.,[.,.]]]=>[4,3,1,2]=>2
[[.,.],[[.,.],.]]=>[3,4,1,2]=>1
[[.,[.,.]],[.,.]]=>[4,2,1,3]=>2
[[[.,.],.],[.,.]]=>[4,1,2,3]=>1
[[.,[.,[.,.]]],.]=>[3,2,1,4]=>2
[[.,[[.,.],.]],.]=>[2,3,1,4]=>1
[[[.,.],[.,.]],.]=>[3,1,2,4]=>1
[[[.,[.,.]],.],.]=>[2,1,3,4]=>1
[[[[.,.],.],.],.]=>[1,2,3,4]=>0
[.,[.,[.,[.,[.,.]]]]]=>[5,4,3,2,1]=>4
[.,[.,[.,[[.,.],.]]]]=>[4,5,3,2,1]=>3
[.,[.,[[.,.],[.,.]]]]=>[5,3,4,2,1]=>3
[.,[.,[[.,[.,.]],.]]]=>[4,3,5,2,1]=>3
[.,[.,[[[.,.],.],.]]]=>[3,4,5,2,1]=>2
[.,[[.,.],[.,[.,.]]]]=>[5,4,2,3,1]=>3
[.,[[.,.],[[.,.],.]]]=>[4,5,2,3,1]=>2
[.,[[.,[.,.]],[.,.]]]=>[5,3,2,4,1]=>3
[.,[[[.,.],.],[.,.]]]=>[5,2,3,4,1]=>2
[.,[[.,[.,[.,.]]],.]]=>[4,3,2,5,1]=>3
[.,[[.,[[.,.],.]],.]]=>[3,4,2,5,1]=>2
[.,[[[.,.],[.,.]],.]]=>[4,2,3,5,1]=>2
[.,[[[.,[.,.]],.],.]]=>[3,2,4,5,1]=>2
[.,[[[[.,.],.],.],.]]=>[2,3,4,5,1]=>1
[[.,.],[.,[.,[.,.]]]]=>[5,4,3,1,2]=>3
[[.,.],[.,[[.,.],.]]]=>[4,5,3,1,2]=>2
[[.,.],[[.,.],[.,.]]]=>[5,3,4,1,2]=>2
[[.,.],[[.,[.,.]],.]]=>[4,3,5,1,2]=>2
[[.,.],[[[.,.],.],.]]=>[3,4,5,1,2]=>1
[[.,[.,.]],[.,[.,.]]]=>[5,4,2,1,3]=>3
[[.,[.,.]],[[.,.],.]]=>[4,5,2,1,3]=>2
[[[.,.],.],[.,[.,.]]]=>[5,4,1,2,3]=>2
[[[.,.],.],[[.,.],.]]=>[4,5,1,2,3]=>1
[[.,[.,[.,.]]],[.,.]]=>[5,3,2,1,4]=>3
[[.,[[.,.],.]],[.,.]]=>[5,2,3,1,4]=>2
[[[.,.],[.,.]],[.,.]]=>[5,3,1,2,4]=>2
[[[.,[.,.]],.],[.,.]]=>[5,2,1,3,4]=>2
[[[[.,.],.],.],[.,.]]=>[5,1,2,3,4]=>1
[[.,[.,[.,[.,.]]]],.]=>[4,3,2,1,5]=>3
[[.,[.,[[.,.],.]]],.]=>[3,4,2,1,5]=>2
[[.,[[.,.],[.,.]]],.]=>[4,2,3,1,5]=>2
[[.,[[.,[.,.]],.]],.]=>[3,2,4,1,5]=>2
[[.,[[[.,.],.],.]],.]=>[2,3,4,1,5]=>1
[[[.,.],[.,[.,.]]],.]=>[4,3,1,2,5]=>2
[[[.,.],[[.,.],.]],.]=>[3,4,1,2,5]=>1
[[[.,[.,.]],[.,.]],.]=>[4,2,1,3,5]=>2
[[[[.,.],.],[.,.]],.]=>[4,1,2,3,5]=>1
[[[.,[.,[.,.]]],.],.]=>[3,2,1,4,5]=>2
[[[.,[[.,.],.]],.],.]=>[2,3,1,4,5]=>1
[[[[.,.],[.,.]],.],.]=>[3,1,2,4,5]=>1
[[[[.,[.,.]],.],.],.]=>[2,1,3,4,5]=>1
[[[[[.,.],.],.],.],.]=>[1,2,3,4,5]=>0
[.,[.,[.,[.,[.,[.,.]]]]]]=>[6,5,4,3,2,1]=>5
[.,[.,[.,[.,[[.,.],.]]]]]=>[5,6,4,3,2,1]=>4
[.,[.,[.,[[.,.],[.,.]]]]]=>[6,4,5,3,2,1]=>4
[.,[.,[.,[[.,[.,.]],.]]]]=>[5,4,6,3,2,1]=>4
[.,[.,[.,[[[.,.],.],.]]]]=>[4,5,6,3,2,1]=>3
[.,[.,[[.,.],[.,[.,.]]]]]=>[6,5,3,4,2,1]=>4
[.,[.,[[.,.],[[.,.],.]]]]=>[5,6,3,4,2,1]=>3
[.,[.,[[.,[.,.]],[.,.]]]]=>[6,4,3,5,2,1]=>4
[.,[.,[[[.,.],.],[.,.]]]]=>[6,3,4,5,2,1]=>3
[.,[.,[[.,[.,[.,.]]],.]]]=>[5,4,3,6,2,1]=>4
[.,[.,[[.,[[.,.],.]],.]]]=>[4,5,3,6,2,1]=>3
[.,[.,[[[.,.],[.,.]],.]]]=>[5,3,4,6,2,1]=>3
[.,[.,[[[.,[.,.]],.],.]]]=>[4,3,5,6,2,1]=>3
[.,[.,[[[[.,.],.],.],.]]]=>[3,4,5,6,2,1]=>2
[.,[[.,.],[.,[.,[.,.]]]]]=>[6,5,4,2,3,1]=>4
[.,[[.,.],[.,[[.,.],.]]]]=>[5,6,4,2,3,1]=>3
[.,[[.,.],[[.,.],[.,.]]]]=>[6,4,5,2,3,1]=>3
[.,[[.,.],[[.,[.,.]],.]]]=>[5,4,6,2,3,1]=>3
[.,[[.,.],[[[.,.],.],.]]]=>[4,5,6,2,3,1]=>2
[.,[[.,[.,.]],[.,[.,.]]]]=>[6,5,3,2,4,1]=>4
[.,[[.,[.,.]],[[.,.],.]]]=>[5,6,3,2,4,1]=>3
[.,[[[.,.],.],[.,[.,.]]]]=>[6,5,2,3,4,1]=>3
[.,[[[.,.],.],[[.,.],.]]]=>[5,6,2,3,4,1]=>2
[.,[[.,[.,[.,.]]],[.,.]]]=>[6,4,3,2,5,1]=>4
[.,[[.,[[.,.],.]],[.,.]]]=>[6,3,4,2,5,1]=>3
[.,[[[.,.],[.,.]],[.,.]]]=>[6,4,2,3,5,1]=>3
[.,[[[.,[.,.]],.],[.,.]]]=>[6,3,2,4,5,1]=>3
[.,[[[[.,.],.],.],[.,.]]]=>[6,2,3,4,5,1]=>2
[.,[[.,[.,[.,[.,.]]]],.]]=>[5,4,3,2,6,1]=>4
[.,[[.,[.,[[.,.],.]]],.]]=>[4,5,3,2,6,1]=>3
[.,[[.,[[.,.],[.,.]]],.]]=>[5,3,4,2,6,1]=>3
[.,[[.,[[.,[.,.]],.]],.]]=>[4,3,5,2,6,1]=>3
[.,[[.,[[[.,.],.],.]],.]]=>[3,4,5,2,6,1]=>2
[.,[[[.,.],[.,[.,.]]],.]]=>[5,4,2,3,6,1]=>3
[.,[[[.,.],[[.,.],.]],.]]=>[4,5,2,3,6,1]=>2
[.,[[[.,[.,.]],[.,.]],.]]=>[5,3,2,4,6,1]=>3
[.,[[[[.,.],.],[.,.]],.]]=>[5,2,3,4,6,1]=>2
[.,[[[.,[.,[.,.]]],.],.]]=>[4,3,2,5,6,1]=>3
[.,[[[.,[[.,.],.]],.],.]]=>[3,4,2,5,6,1]=>2
[.,[[[[.,.],[.,.]],.],.]]=>[4,2,3,5,6,1]=>2
[.,[[[[.,[.,.]],.],.],.]]=>[3,2,4,5,6,1]=>2
[.,[[[[[.,.],.],.],.],.]]=>[2,3,4,5,6,1]=>1
[[.,.],[.,[.,[.,[.,.]]]]]=>[6,5,4,3,1,2]=>4
[[.,.],[.,[.,[[.,.],.]]]]=>[5,6,4,3,1,2]=>3
[[.,.],[.,[[.,.],[.,.]]]]=>[6,4,5,3,1,2]=>3
[[.,.],[.,[[.,[.,.]],.]]]=>[5,4,6,3,1,2]=>3
[[.,.],[.,[[[.,.],.],.]]]=>[4,5,6,3,1,2]=>2
[[.,.],[[.,.],[.,[.,.]]]]=>[6,5,3,4,1,2]=>3
[[.,.],[[.,.],[[.,.],.]]]=>[5,6,3,4,1,2]=>2
[[.,.],[[.,[.,.]],[.,.]]]=>[6,4,3,5,1,2]=>3
[[.,.],[[[.,.],.],[.,.]]]=>[6,3,4,5,1,2]=>2
[[.,.],[[.,[.,[.,.]]],.]]=>[5,4,3,6,1,2]=>3
[[.,.],[[.,[[.,.],.]],.]]=>[4,5,3,6,1,2]=>2
[[.,.],[[[.,.],[.,.]],.]]=>[5,3,4,6,1,2]=>2
[[.,.],[[[.,[.,.]],.],.]]=>[4,3,5,6,1,2]=>2
[[.,.],[[[[.,.],.],.],.]]=>[3,4,5,6,1,2]=>1
[[.,[.,.]],[.,[.,[.,.]]]]=>[6,5,4,2,1,3]=>4
[[.,[.,.]],[.,[[.,.],.]]]=>[5,6,4,2,1,3]=>3
[[.,[.,.]],[[.,.],[.,.]]]=>[6,4,5,2,1,3]=>3
[[.,[.,.]],[[.,[.,.]],.]]=>[5,4,6,2,1,3]=>3
[[.,[.,.]],[[[.,.],.],.]]=>[4,5,6,2,1,3]=>2
[[[.,.],.],[.,[.,[.,.]]]]=>[6,5,4,1,2,3]=>3
[[[.,.],.],[.,[[.,.],.]]]=>[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]=>1
[[.,[.,[.,.]]],[.,[.,.]]]=>[6,5,3,2,1,4]=>4
[[.,[.,[.,.]]],[[.,.],.]]=>[5,6,3,2,1,4]=>3
[[.,[[.,.],.]],[.,[.,.]]]=>[6,5,2,3,1,4]=>3
[[.,[[.,.],.]],[[.,.],.]]=>[5,6,2,3,1,4]=>2
[[[.,.],[.,.]],[.,[.,.]]]=>[6,5,3,1,2,4]=>3
[[[.,.],[.,.]],[[.,.],.]]=>[5,6,3,1,2,4]=>2
[[[.,[.,.]],.],[.,[.,.]]]=>[6,5,2,1,3,4]=>3
[[[.,[.,.]],.],[[.,.],.]]=>[5,6,2,1,3,4]=>2
[[[[.,.],.],.],[.,[.,.]]]=>[6,5,1,2,3,4]=>2
[[[[.,.],.],.],[[.,.],.]]=>[5,6,1,2,3,4]=>1
[[.,[.,[.,[.,.]]]],[.,.]]=>[6,4,3,2,1,5]=>4
[[.,[.,[[.,.],.]]],[.,.]]=>[6,3,4,2,1,5]=>3
[[.,[[.,.],[.,.]]],[.,.]]=>[6,4,2,3,1,5]=>3
[[.,[[.,[.,.]],.]],[.,.]]=>[6,3,2,4,1,5]=>3
[[.,[[[.,.],.],.]],[.,.]]=>[6,2,3,4,1,5]=>2
[[[.,.],[.,[.,.]]],[.,.]]=>[6,4,3,1,2,5]=>3
[[[.,.],[[.,.],.]],[.,.]]=>[6,3,4,1,2,5]=>2
[[[.,[.,.]],[.,.]],[.,.]]=>[6,4,2,1,3,5]=>3
[[[[.,.],.],[.,.]],[.,.]]=>[6,4,1,2,3,5]=>2
[[[.,[.,[.,.]]],.],[.,.]]=>[6,3,2,1,4,5]=>3
[[[.,[[.,.],.]],.],[.,.]]=>[6,2,3,1,4,5]=>2
[[[[.,.],[.,.]],.],[.,.]]=>[6,3,1,2,4,5]=>2
[[[[.,[.,.]],.],.],[.,.]]=>[6,2,1,3,4,5]=>2
[[[[[.,.],.],.],.],[.,.]]=>[6,1,2,3,4,5]=>1
[[.,[.,[.,[.,[.,.]]]]],.]=>[5,4,3,2,1,6]=>4
[[.,[.,[.,[[.,.],.]]]],.]=>[4,5,3,2,1,6]=>3
[[.,[.,[[.,.],[.,.]]]],.]=>[5,3,4,2,1,6]=>3
[[.,[.,[[.,[.,.]],.]]],.]=>[4,3,5,2,1,6]=>3
[[.,[.,[[[.,.],.],.]]],.]=>[3,4,5,2,1,6]=>2
[[.,[[.,.],[.,[.,.]]]],.]=>[5,4,2,3,1,6]=>3
[[.,[[.,.],[[.,.],.]]],.]=>[4,5,2,3,1,6]=>2
[[.,[[.,[.,.]],[.,.]]],.]=>[5,3,2,4,1,6]=>3
[[.,[[[.,.],.],[.,.]]],.]=>[5,2,3,4,1,6]=>2
[[.,[[.,[.,[.,.]]],.]],.]=>[4,3,2,5,1,6]=>3
[[.,[[.,[[.,.],.]],.]],.]=>[3,4,2,5,1,6]=>2
[[.,[[[.,.],[.,.]],.]],.]=>[4,2,3,5,1,6]=>2
[[.,[[[.,[.,.]],.],.]],.]=>[3,2,4,5,1,6]=>2
[[.,[[[[.,.],.],.],.]],.]=>[2,3,4,5,1,6]=>1
[[[.,.],[.,[.,[.,.]]]],.]=>[5,4,3,1,2,6]=>3
[[[.,.],[.,[[.,.],.]]],.]=>[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]=>1
[[[.,[.,.]],[.,[.,.]]],.]=>[5,4,2,1,3,6]=>3
[[[.,[.,.]],[[.,.],.]],.]=>[4,5,2,1,3,6]=>2
[[[[.,.],.],[.,[.,.]]],.]=>[5,4,1,2,3,6]=>2
[[[[.,.],.],[[.,.],.]],.]=>[4,5,1,2,3,6]=>1
[[[.,[.,[.,.]]],[.,.]],.]=>[5,3,2,1,4,6]=>3
[[[.,[[.,.],.]],[.,.]],.]=>[5,2,3,1,4,6]=>2
[[[[.,.],[.,.]],[.,.]],.]=>[5,3,1,2,4,6]=>2
[[[[.,[.,.]],.],[.,.]],.]=>[5,2,1,3,4,6]=>2
[[[[[.,.],.],.],[.,.]],.]=>[5,1,2,3,4,6]=>1
[[[.,[.,[.,[.,.]]]],.],.]=>[4,3,2,1,5,6]=>3
[[[.,[.,[[.,.],.]]],.],.]=>[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]=>1
[[[[.,.],[.,[.,.]]],.],.]=>[4,3,1,2,5,6]=>2
[[[[.,.],[[.,.],.]],.],.]=>[3,4,1,2,5,6]=>1
[[[[.,[.,.]],[.,.]],.],.]=>[4,2,1,3,5,6]=>2
[[[[[.,.],.],[.,.]],.],.]=>[4,1,2,3,5,6]=>1
[[[[.,[.,[.,.]]],.],.],.]=>[3,2,1,4,5,6]=>2
[[[[.,[[.,.],.]],.],.],.]=>[2,3,1,4,5,6]=>1
[[[[[.,.],[.,.]],.],.],.]=>[3,1,2,4,5,6]=>1
[[[[[.,[.,.]],.],.],.],.]=>[2,1,3,4,5,6]=>1
[[[[[[.,.],.],.],.],.],.]=>[1,2,3,4,5,6]=>0
[[[[[[[.,.],.],.],.],.],.],.]=>[1,2,3,4,5,6,7]=>0
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Description
The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right.
For a permutation $\pi$ of length $n$, this is the number of indices $2 \leq j \leq n$ such that for all $1 \leq i < j$, the pair $(i,j)$ is an inversion of $\pi$.
For a permutation $\pi$ of length $n$, this is the number of indices $2 \leq j \leq n$ such that for all $1 \leq i < j$, the pair $(i,j)$ is an inversion of $\pi$.
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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