Identifier
-
Mp00014:
Binary trees
—to 132-avoiding permutation⟶
Permutations
St000314: Permutations ⟶ ℤ
Values
=>
Cc0010;cc-rep-0
[.,.]=>[1]=>1
[.,[.,.]]=>[2,1]=>1
[[.,.],.]=>[1,2]=>2
[.,[.,[.,.]]]=>[3,2,1]=>1
[.,[[.,.],.]]=>[2,3,1]=>2
[[.,.],[.,.]]=>[3,1,2]=>1
[[.,[.,.]],.]=>[2,1,3]=>2
[[[.,.],.],.]=>[1,2,3]=>3
[.,[.,[.,[.,.]]]]=>[4,3,2,1]=>1
[.,[.,[[.,.],.]]]=>[3,4,2,1]=>2
[.,[[.,.],[.,.]]]=>[4,2,3,1]=>1
[.,[[.,[.,.]],.]]=>[3,2,4,1]=>2
[.,[[[.,.],.],.]]=>[2,3,4,1]=>3
[[.,.],[.,[.,.]]]=>[4,3,1,2]=>1
[[.,.],[[.,.],.]]=>[3,4,1,2]=>2
[[.,[.,.]],[.,.]]=>[4,2,1,3]=>1
[[[.,.],.],[.,.]]=>[4,1,2,3]=>1
[[.,[.,[.,.]]],.]=>[3,2,1,4]=>2
[[.,[[.,.],.]],.]=>[2,3,1,4]=>3
[[[.,.],[.,.]],.]=>[3,1,2,4]=>2
[[[.,[.,.]],.],.]=>[2,1,3,4]=>3
[[[[.,.],.],.],.]=>[1,2,3,4]=>4
[.,[.,[.,[.,[.,.]]]]]=>[5,4,3,2,1]=>1
[.,[.,[.,[[.,.],.]]]]=>[4,5,3,2,1]=>2
[.,[.,[[.,.],[.,.]]]]=>[5,3,4,2,1]=>1
[.,[.,[[.,[.,.]],.]]]=>[4,3,5,2,1]=>2
[.,[.,[[[.,.],.],.]]]=>[3,4,5,2,1]=>3
[.,[[.,.],[.,[.,.]]]]=>[5,4,2,3,1]=>1
[.,[[.,.],[[.,.],.]]]=>[4,5,2,3,1]=>2
[.,[[.,[.,.]],[.,.]]]=>[5,3,2,4,1]=>1
[.,[[[.,.],.],[.,.]]]=>[5,2,3,4,1]=>1
[.,[[.,[.,[.,.]]],.]]=>[4,3,2,5,1]=>2
[.,[[.,[[.,.],.]],.]]=>[3,4,2,5,1]=>3
[.,[[[.,.],[.,.]],.]]=>[4,2,3,5,1]=>2
[.,[[[.,[.,.]],.],.]]=>[3,2,4,5,1]=>3
[.,[[[[.,.],.],.],.]]=>[2,3,4,5,1]=>4
[[.,.],[.,[.,[.,.]]]]=>[5,4,3,1,2]=>1
[[.,.],[.,[[.,.],.]]]=>[4,5,3,1,2]=>2
[[.,.],[[.,.],[.,.]]]=>[5,3,4,1,2]=>1
[[.,.],[[.,[.,.]],.]]=>[4,3,5,1,2]=>2
[[.,.],[[[.,.],.],.]]=>[3,4,5,1,2]=>3
[[.,[.,.]],[.,[.,.]]]=>[5,4,2,1,3]=>1
[[.,[.,.]],[[.,.],.]]=>[4,5,2,1,3]=>2
[[[.,.],.],[.,[.,.]]]=>[5,4,1,2,3]=>1
[[[.,.],.],[[.,.],.]]=>[4,5,1,2,3]=>2
[[.,[.,[.,.]]],[.,.]]=>[5,3,2,1,4]=>1
[[.,[[.,.],.]],[.,.]]=>[5,2,3,1,4]=>1
[[[.,.],[.,.]],[.,.]]=>[5,3,1,2,4]=>1
[[[.,[.,.]],.],[.,.]]=>[5,2,1,3,4]=>1
[[[[.,.],.],.],[.,.]]=>[5,1,2,3,4]=>1
[[.,[.,[.,[.,.]]]],.]=>[4,3,2,1,5]=>2
[[.,[.,[[.,.],.]]],.]=>[3,4,2,1,5]=>3
[[.,[[.,.],[.,.]]],.]=>[4,2,3,1,5]=>2
[[.,[[.,[.,.]],.]],.]=>[3,2,4,1,5]=>3
[[.,[[[.,.],.],.]],.]=>[2,3,4,1,5]=>4
[[[.,.],[.,[.,.]]],.]=>[4,3,1,2,5]=>2
[[[.,.],[[.,.],.]],.]=>[3,4,1,2,5]=>3
[[[.,[.,.]],[.,.]],.]=>[4,2,1,3,5]=>2
[[[[.,.],.],[.,.]],.]=>[4,1,2,3,5]=>2
[[[.,[.,[.,.]]],.],.]=>[3,2,1,4,5]=>3
[[[.,[[.,.],.]],.],.]=>[2,3,1,4,5]=>4
[[[[.,.],[.,.]],.],.]=>[3,1,2,4,5]=>3
[[[[.,[.,.]],.],.],.]=>[2,1,3,4,5]=>4
[[[[[.,.],.],.],.],.]=>[1,2,3,4,5]=>5
[.,[.,[.,[.,[.,[.,.]]]]]]=>[6,5,4,3,2,1]=>1
[.,[.,[.,[.,[[.,.],.]]]]]=>[5,6,4,3,2,1]=>2
[.,[.,[.,[[.,.],[.,.]]]]]=>[6,4,5,3,2,1]=>1
[.,[.,[.,[[.,[.,.]],.]]]]=>[5,4,6,3,2,1]=>2
[.,[.,[.,[[[.,.],.],.]]]]=>[4,5,6,3,2,1]=>3
[.,[.,[[.,.],[.,[.,.]]]]]=>[6,5,3,4,2,1]=>1
[.,[.,[[.,.],[[.,.],.]]]]=>[5,6,3,4,2,1]=>2
[.,[.,[[.,[.,.]],[.,.]]]]=>[6,4,3,5,2,1]=>1
[.,[.,[[[.,.],.],[.,.]]]]=>[6,3,4,5,2,1]=>1
[.,[.,[[.,[.,[.,.]]],.]]]=>[5,4,3,6,2,1]=>2
[.,[.,[[.,[[.,.],.]],.]]]=>[4,5,3,6,2,1]=>3
[.,[.,[[[.,.],[.,.]],.]]]=>[5,3,4,6,2,1]=>2
[.,[.,[[[.,[.,.]],.],.]]]=>[4,3,5,6,2,1]=>3
[.,[.,[[[[.,.],.],.],.]]]=>[3,4,5,6,2,1]=>4
[.,[[.,.],[.,[.,[.,.]]]]]=>[6,5,4,2,3,1]=>1
[.,[[.,.],[.,[[.,.],.]]]]=>[5,6,4,2,3,1]=>2
[.,[[.,.],[[.,.],[.,.]]]]=>[6,4,5,2,3,1]=>1
[.,[[.,.],[[.,[.,.]],.]]]=>[5,4,6,2,3,1]=>2
[.,[[.,.],[[[.,.],.],.]]]=>[4,5,6,2,3,1]=>3
[.,[[.,[.,.]],[.,[.,.]]]]=>[6,5,3,2,4,1]=>1
[.,[[.,[.,.]],[[.,.],.]]]=>[5,6,3,2,4,1]=>2
[.,[[[.,.],.],[.,[.,.]]]]=>[6,5,2,3,4,1]=>1
[.,[[[.,.],.],[[.,.],.]]]=>[5,6,2,3,4,1]=>2
[.,[[.,[.,[.,.]]],[.,.]]]=>[6,4,3,2,5,1]=>1
[.,[[.,[[.,.],.]],[.,.]]]=>[6,3,4,2,5,1]=>1
[.,[[[.,.],[.,.]],[.,.]]]=>[6,4,2,3,5,1]=>1
[.,[[[.,[.,.]],.],[.,.]]]=>[6,3,2,4,5,1]=>1
[.,[[[[.,.],.],.],[.,.]]]=>[6,2,3,4,5,1]=>1
[.,[[.,[.,[.,[.,.]]]],.]]=>[5,4,3,2,6,1]=>2
[.,[[.,[.,[[.,.],.]]],.]]=>[4,5,3,2,6,1]=>3
[.,[[.,[[.,.],[.,.]]],.]]=>[5,3,4,2,6,1]=>2
[.,[[.,[[.,[.,.]],.]],.]]=>[4,3,5,2,6,1]=>3
[.,[[.,[[[.,.],.],.]],.]]=>[3,4,5,2,6,1]=>4
[.,[[[.,.],[.,[.,.]]],.]]=>[5,4,2,3,6,1]=>2
[.,[[[.,.],[[.,.],.]],.]]=>[4,5,2,3,6,1]=>3
[.,[[[.,[.,.]],[.,.]],.]]=>[5,3,2,4,6,1]=>2
[.,[[[[.,.],.],[.,.]],.]]=>[5,2,3,4,6,1]=>2
[.,[[[.,[.,[.,.]]],.],.]]=>[4,3,2,5,6,1]=>3
[.,[[[.,[[.,.],.]],.],.]]=>[3,4,2,5,6,1]=>4
[.,[[[[.,.],[.,.]],.],.]]=>[4,2,3,5,6,1]=>3
[.,[[[[.,[.,.]],.],.],.]]=>[3,2,4,5,6,1]=>4
[.,[[[[[.,.],.],.],.],.]]=>[2,3,4,5,6,1]=>5
[[.,.],[.,[.,[.,[.,.]]]]]=>[6,5,4,3,1,2]=>1
[[.,.],[.,[.,[[.,.],.]]]]=>[5,6,4,3,1,2]=>2
[[.,.],[.,[[.,.],[.,.]]]]=>[6,4,5,3,1,2]=>1
[[.,.],[.,[[.,[.,.]],.]]]=>[5,4,6,3,1,2]=>2
[[.,.],[.,[[[.,.],.],.]]]=>[4,5,6,3,1,2]=>3
[[.,.],[[.,.],[.,[.,.]]]]=>[6,5,3,4,1,2]=>1
[[.,.],[[.,.],[[.,.],.]]]=>[5,6,3,4,1,2]=>2
[[.,.],[[.,[.,.]],[.,.]]]=>[6,4,3,5,1,2]=>1
[[.,.],[[[.,.],.],[.,.]]]=>[6,3,4,5,1,2]=>1
[[.,.],[[.,[.,[.,.]]],.]]=>[5,4,3,6,1,2]=>2
[[.,.],[[.,[[.,.],.]],.]]=>[4,5,3,6,1,2]=>3
[[.,.],[[[.,.],[.,.]],.]]=>[5,3,4,6,1,2]=>2
[[.,.],[[[.,[.,.]],.],.]]=>[4,3,5,6,1,2]=>3
[[.,.],[[[[.,.],.],.],.]]=>[3,4,5,6,1,2]=>4
[[.,[.,.]],[.,[.,[.,.]]]]=>[6,5,4,2,1,3]=>1
[[.,[.,.]],[.,[[.,.],.]]]=>[5,6,4,2,1,3]=>2
[[.,[.,.]],[[.,.],[.,.]]]=>[6,4,5,2,1,3]=>1
[[.,[.,.]],[[.,[.,.]],.]]=>[5,4,6,2,1,3]=>2
[[.,[.,.]],[[[.,.],.],.]]=>[4,5,6,2,1,3]=>3
[[[.,.],.],[.,[.,[.,.]]]]=>[6,5,4,1,2,3]=>1
[[[.,.],.],[.,[[.,.],.]]]=>[5,6,4,1,2,3]=>2
[[[.,.],.],[[.,.],[.,.]]]=>[6,4,5,1,2,3]=>1
[[[.,.],.],[[.,[.,.]],.]]=>[5,4,6,1,2,3]=>2
[[[.,.],.],[[[.,.],.],.]]=>[4,5,6,1,2,3]=>3
[[.,[.,[.,.]]],[.,[.,.]]]=>[6,5,3,2,1,4]=>1
[[.,[.,[.,.]]],[[.,.],.]]=>[5,6,3,2,1,4]=>2
[[.,[[.,.],.]],[.,[.,.]]]=>[6,5,2,3,1,4]=>1
[[.,[[.,.],.]],[[.,.],.]]=>[5,6,2,3,1,4]=>2
[[[.,.],[.,.]],[.,[.,.]]]=>[6,5,3,1,2,4]=>1
[[[.,.],[.,.]],[[.,.],.]]=>[5,6,3,1,2,4]=>2
[[[.,[.,.]],.],[.,[.,.]]]=>[6,5,2,1,3,4]=>1
[[[.,[.,.]],.],[[.,.],.]]=>[5,6,2,1,3,4]=>2
[[[[.,.],.],.],[.,[.,.]]]=>[6,5,1,2,3,4]=>1
[[[[.,.],.],.],[[.,.],.]]=>[5,6,1,2,3,4]=>2
[[.,[.,[.,[.,.]]]],[.,.]]=>[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]=>1
[[[.,.],[[.,.],.]],[.,.]]=>[6,3,4,1,2,5]=>1
[[[.,[.,.]],[.,.]],[.,.]]=>[6,4,2,1,3,5]=>1
[[[[.,.],.],[.,.]],[.,.]]=>[6,4,1,2,3,5]=>1
[[[.,[.,[.,.]]],.],[.,.]]=>[6,3,2,1,4,5]=>1
[[[.,[[.,.],.]],.],[.,.]]=>[6,2,3,1,4,5]=>1
[[[[.,.],[.,.]],.],[.,.]]=>[6,3,1,2,4,5]=>1
[[[[.,[.,.]],.],.],[.,.]]=>[6,2,1,3,4,5]=>1
[[[[[.,.],.],.],.],[.,.]]=>[6,1,2,3,4,5]=>1
[[.,[.,[.,[.,[.,.]]]]],.]=>[5,4,3,2,1,6]=>2
[[.,[.,[.,[[.,.],.]]]],.]=>[4,5,3,2,1,6]=>3
[[.,[.,[[.,.],[.,.]]]],.]=>[5,3,4,2,1,6]=>2
[[.,[.,[[.,[.,.]],.]]],.]=>[4,3,5,2,1,6]=>3
[[.,[.,[[[.,.],.],.]]],.]=>[3,4,5,2,1,6]=>4
[[.,[[.,.],[.,[.,.]]]],.]=>[5,4,2,3,1,6]=>2
[[.,[[.,.],[[.,.],.]]],.]=>[4,5,2,3,1,6]=>3
[[.,[[.,[.,.]],[.,.]]],.]=>[5,3,2,4,1,6]=>2
[[.,[[[.,.],.],[.,.]]],.]=>[5,2,3,4,1,6]=>2
[[.,[[.,[.,[.,.]]],.]],.]=>[4,3,2,5,1,6]=>3
[[.,[[.,[[.,.],.]],.]],.]=>[3,4,2,5,1,6]=>4
[[.,[[[.,.],[.,.]],.]],.]=>[4,2,3,5,1,6]=>3
[[.,[[[.,[.,.]],.],.]],.]=>[3,2,4,5,1,6]=>4
[[.,[[[[.,.],.],.],.]],.]=>[2,3,4,5,1,6]=>5
[[[.,.],[.,[.,[.,.]]]],.]=>[5,4,3,1,2,6]=>2
[[[.,.],[.,[[.,.],.]]],.]=>[4,5,3,1,2,6]=>3
[[[.,.],[[.,.],[.,.]]],.]=>[5,3,4,1,2,6]=>2
[[[.,.],[[.,[.,.]],.]],.]=>[4,3,5,1,2,6]=>3
[[[.,.],[[[.,.],.],.]],.]=>[3,4,5,1,2,6]=>4
[[[.,[.,.]],[.,[.,.]]],.]=>[5,4,2,1,3,6]=>2
[[[.,[.,.]],[[.,.],.]],.]=>[4,5,2,1,3,6]=>3
[[[[.,.],.],[.,[.,.]]],.]=>[5,4,1,2,3,6]=>2
[[[[.,.],.],[[.,.],.]],.]=>[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]=>2
[[[[.,[.,.]],.],[.,.]],.]=>[5,2,1,3,4,6]=>2
[[[[[.,.],.],.],[.,.]],.]=>[5,1,2,3,4,6]=>2
[[[.,[.,[.,[.,.]]]],.],.]=>[4,3,2,1,5,6]=>3
[[[.,[.,[[.,.],.]]],.],.]=>[3,4,2,1,5,6]=>4
[[[.,[[.,.],[.,.]]],.],.]=>[4,2,3,1,5,6]=>3
[[[.,[[.,[.,.]],.]],.],.]=>[3,2,4,1,5,6]=>4
[[[.,[[[.,.],.],.]],.],.]=>[2,3,4,1,5,6]=>5
[[[[.,.],[.,[.,.]]],.],.]=>[4,3,1,2,5,6]=>3
[[[[.,.],[[.,.],.]],.],.]=>[3,4,1,2,5,6]=>4
[[[[.,[.,.]],[.,.]],.],.]=>[4,2,1,3,5,6]=>3
[[[[[.,.],.],[.,.]],.],.]=>[4,1,2,3,5,6]=>3
[[[[.,[.,[.,.]]],.],.],.]=>[3,2,1,4,5,6]=>4
[[[[.,[[.,.],.]],.],.],.]=>[2,3,1,4,5,6]=>5
[[[[[.,.],[.,.]],.],.],.]=>[3,1,2,4,5,6]=>4
[[[[[.,[.,.]],.],.],.],.]=>[2,1,3,4,5,6]=>5
[[[[[[.,.],.],.],.],.],.]=>[1,2,3,4,5,6]=>6
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Description
The number of left-to-right-maxima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a left-to-right-maximum if there does not exist a $j < i$ such that $\sigma_j > \sigma_i$.
This is also the number of weak exceedences of a permutation that are not mid-points of a decreasing subsequence of length 3, see [1] for more on the later description.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a left-to-right-maximum if there does not exist a $j < i$ such that $\sigma_j > \sigma_i$.
This is also the number of weak exceedences of a permutation that are not mid-points of a decreasing subsequence of length 3, see [1] for more on the later description.
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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