Identifier
-
Mp00014:
Binary trees
—to 132-avoiding permutation⟶
Permutations
St000155: Permutations ⟶ ℤ
Values
=>
Cc0010;cc-rep-0
[.,.]=>[1]=>0
[.,[.,.]]=>[2,1]=>1
[[.,.],.]=>[1,2]=>0
[.,[.,[.,.]]]=>[3,2,1]=>1
[.,[[.,.],.]]=>[2,3,1]=>2
[[.,.],[.,.]]=>[3,1,2]=>1
[[.,[.,.]],.]=>[2,1,3]=>1
[[[.,.],.],.]=>[1,2,3]=>0
[.,[.,[.,[.,.]]]]=>[4,3,2,1]=>2
[.,[.,[[.,.],.]]]=>[3,4,2,1]=>2
[.,[[.,.],[.,.]]]=>[4,2,3,1]=>1
[.,[[.,[.,.]],.]]=>[3,2,4,1]=>2
[.,[[[.,.],.],.]]=>[2,3,4,1]=>3
[[.,.],[.,[.,.]]]=>[4,3,1,2]=>2
[[.,.],[[.,.],.]]=>[3,4,1,2]=>2
[[.,[.,.]],[.,.]]=>[4,2,1,3]=>1
[[[.,.],.],[.,.]]=>[4,1,2,3]=>1
[[.,[.,[.,.]]],.]=>[3,2,1,4]=>1
[[.,[[.,.],.]],.]=>[2,3,1,4]=>2
[[[.,.],[.,.]],.]=>[3,1,2,4]=>1
[[[.,[.,.]],.],.]=>[2,1,3,4]=>1
[[[[.,.],.],.],.]=>[1,2,3,4]=>0
[.,[.,[.,[.,[.,.]]]]]=>[5,4,3,2,1]=>2
[.,[.,[.,[[.,.],.]]]]=>[4,5,3,2,1]=>2
[.,[.,[[.,.],[.,.]]]]=>[5,3,4,2,1]=>3
[.,[.,[[.,[.,.]],.]]]=>[4,3,5,2,1]=>3
[.,[.,[[[.,.],.],.]]]=>[3,4,5,2,1]=>3
[.,[[.,.],[.,[.,.]]]]=>[5,4,2,3,1]=>2
[.,[[.,.],[[.,.],.]]]=>[4,5,2,3,1]=>2
[.,[[.,[.,.]],[.,.]]]=>[5,3,2,4,1]=>2
[.,[[[.,.],.],[.,.]]]=>[5,2,3,4,1]=>1
[.,[[.,[.,[.,.]]],.]]=>[4,3,2,5,1]=>3
[.,[[.,[[.,.],.]],.]]=>[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]=>2
[[.,.],[.,[[.,.],.]]]=>[4,5,3,1,2]=>2
[[.,.],[[.,.],[.,.]]]=>[5,3,4,1,2]=>3
[[.,.],[[.,[.,.]],.]]=>[4,3,5,1,2]=>3
[[.,.],[[[.,.],.],.]]=>[3,4,5,1,2]=>3
[[.,[.,.]],[.,[.,.]]]=>[5,4,2,1,3]=>2
[[.,[.,.]],[[.,.],.]]=>[4,5,2,1,3]=>2
[[[.,.],.],[.,[.,.]]]=>[5,4,1,2,3]=>2
[[[.,.],.],[[.,.],.]]=>[4,5,1,2,3]=>2
[[.,[.,[.,.]]],[.,.]]=>[5,3,2,1,4]=>2
[[.,[[.,.],.]],[.,.]]=>[5,2,3,1,4]=>1
[[[.,.],[.,.]],[.,.]]=>[5,3,1,2,4]=>2
[[[.,[.,.]],.],[.,.]]=>[5,2,1,3,4]=>1
[[[[.,.],.],.],[.,.]]=>[5,1,2,3,4]=>1
[[.,[.,[.,[.,.]]]],.]=>[4,3,2,1,5]=>2
[[.,[.,[[.,.],.]]],.]=>[3,4,2,1,5]=>2
[[.,[[.,.],[.,.]]],.]=>[4,2,3,1,5]=>1
[[.,[[.,[.,.]],.]],.]=>[3,2,4,1,5]=>2
[[.,[[[.,.],.],.]],.]=>[2,3,4,1,5]=>3
[[[.,.],[.,[.,.]]],.]=>[4,3,1,2,5]=>2
[[[.,.],[[.,.],.]],.]=>[3,4,1,2,5]=>2
[[[.,[.,.]],[.,.]],.]=>[4,2,1,3,5]=>1
[[[[.,.],.],[.,.]],.]=>[4,1,2,3,5]=>1
[[[.,[.,[.,.]]],.],.]=>[3,2,1,4,5]=>1
[[[.,[[.,.],.]],.],.]=>[2,3,1,4,5]=>2
[[[[.,.],[.,.]],.],.]=>[3,1,2,4,5]=>1
[[[[.,[.,.]],.],.],.]=>[2,1,3,4,5]=>1
[[[[[.,.],.],.],.],.]=>[1,2,3,4,5]=>0
[.,[.,[.,[.,[.,[.,.]]]]]]=>[6,5,4,3,2,1]=>3
[.,[.,[.,[.,[[.,.],.]]]]]=>[5,6,4,3,2,1]=>3
[.,[.,[.,[[.,.],[.,.]]]]]=>[6,4,5,3,2,1]=>3
[.,[.,[.,[[.,[.,.]],.]]]]=>[5,4,6,3,2,1]=>3
[.,[.,[.,[[[.,.],.],.]]]]=>[4,5,6,3,2,1]=>3
[.,[.,[[.,.],[.,[.,.]]]]]=>[6,5,3,4,2,1]=>2
[.,[.,[[.,.],[[.,.],.]]]]=>[5,6,3,4,2,1]=>2
[.,[.,[[.,[.,.]],[.,.]]]]=>[6,4,3,5,2,1]=>3
[.,[.,[[[.,.],.],[.,.]]]]=>[6,3,4,5,2,1]=>4
[.,[.,[[.,[.,[.,.]]],.]]]=>[5,4,3,6,2,1]=>3
[.,[.,[[.,[[.,.],.]],.]]]=>[4,5,3,6,2,1]=>3
[.,[.,[[[.,.],[.,.]],.]]]=>[5,3,4,6,2,1]=>4
[.,[.,[[[.,[.,.]],.],.]]]=>[4,3,5,6,2,1]=>4
[.,[.,[[[[.,.],.],.],.]]]=>[3,4,5,6,2,1]=>4
[.,[[.,.],[.,[.,[.,.]]]]]=>[6,5,4,2,3,1]=>3
[.,[[.,.],[.,[[.,.],.]]]]=>[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]=>3
[.,[[.,[.,.]],[.,[.,.]]]]=>[6,5,3,2,4,1]=>2
[.,[[.,[.,.]],[[.,.],.]]]=>[5,6,3,2,4,1]=>2
[.,[[[.,.],.],[.,[.,.]]]]=>[6,5,2,3,4,1]=>2
[.,[[[.,.],.],[[.,.],.]]]=>[5,6,2,3,4,1]=>2
[.,[[.,[.,[.,.]]],[.,.]]]=>[6,4,3,2,5,1]=>2
[.,[[.,[[.,.],.]],[.,.]]]=>[6,3,4,2,5,1]=>3
[.,[[[.,.],[.,.]],[.,.]]]=>[6,4,2,3,5,1]=>2
[.,[[[.,[.,.]],.],[.,.]]]=>[6,3,2,4,5,1]=>2
[.,[[[[.,.],.],.],[.,.]]]=>[6,2,3,4,5,1]=>1
[.,[[.,[.,[.,[.,.]]]],.]]=>[5,4,3,2,6,1]=>3
[.,[[.,[.,[[.,.],.]]],.]]=>[4,5,3,2,6,1]=>3
[.,[[.,[[.,.],[.,.]]],.]]=>[5,3,4,2,6,1]=>4
[.,[[.,[[.,[.,.]],.]],.]]=>[4,3,5,2,6,1]=>4
[.,[[.,[[[.,.],.],.]],.]]=>[3,4,5,2,6,1]=>4
[.,[[[.,.],[.,[.,.]]],.]]=>[5,4,2,3,6,1]=>3
[.,[[[.,.],[[.,.],.]],.]]=>[4,5,2,3,6,1]=>3
[.,[[[.,[.,.]],[.,.]],.]]=>[5,3,2,4,6,1]=>3
[.,[[[[.,.],.],[.,.]],.]]=>[5,2,3,4,6,1]=>2
[.,[[[.,[.,[.,.]]],.],.]]=>[4,3,2,5,6,1]=>4
[.,[[[.,[[.,.],.]],.],.]]=>[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]=>3
[[.,.],[.,[.,[[.,.],.]]]]=>[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]=>3
[[.,.],[[.,.],[.,[.,.]]]]=>[6,5,3,4,1,2]=>2
[[.,.],[[.,.],[[.,.],.]]]=>[5,6,3,4,1,2]=>2
[[.,.],[[.,[.,.]],[.,.]]]=>[6,4,3,5,1,2]=>3
[[.,.],[[[.,.],.],[.,.]]]=>[6,3,4,5,1,2]=>4
[[.,.],[[.,[.,[.,.]]],.]]=>[5,4,3,6,1,2]=>3
[[.,.],[[.,[[.,.],.]],.]]=>[4,5,3,6,1,2]=>3
[[.,.],[[[.,.],[.,.]],.]]=>[5,3,4,6,1,2]=>4
[[.,.],[[[.,[.,.]],.],.]]=>[4,3,5,6,1,2]=>4
[[.,.],[[[[.,.],.],.],.]]=>[3,4,5,6,1,2]=>4
[[.,[.,.]],[.,[.,[.,.]]]]=>[6,5,4,2,1,3]=>3
[[.,[.,.]],[.,[[.,.],.]]]=>[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]=>3
[[[.,.],.],[.,[.,[.,.]]]]=>[6,5,4,1,2,3]=>3
[[[.,.],.],[.,[[.,.],.]]]=>[5,6,4,1,2,3]=>3
[[[.,.],.],[[.,.],[.,.]]]=>[6,4,5,1,2,3]=>3
[[[.,.],.],[[.,[.,.]],.]]=>[5,4,6,1,2,3]=>3
[[[.,.],.],[[[.,.],.],.]]=>[4,5,6,1,2,3]=>3
[[.,[.,[.,.]]],[.,[.,.]]]=>[6,5,3,2,1,4]=>2
[[.,[.,[.,.]]],[[.,.],.]]=>[5,6,3,2,1,4]=>2
[[.,[[.,.],.]],[.,[.,.]]]=>[6,5,2,3,1,4]=>2
[[.,[[.,.],.]],[[.,.],.]]=>[5,6,2,3,1,4]=>2
[[[.,.],[.,.]],[.,[.,.]]]=>[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]=>2
[[[[.,.],.],.],[[.,.],.]]=>[5,6,1,2,3,4]=>2
[[.,[.,[.,[.,.]]]],[.,.]]=>[6,4,3,2,1,5]=>2
[[.,[.,[[.,.],.]]],[.,.]]=>[6,3,4,2,1,5]=>3
[[.,[[.,.],[.,.]]],[.,.]]=>[6,4,2,3,1,5]=>2
[[.,[[.,[.,.]],.]],[.,.]]=>[6,3,2,4,1,5]=>2
[[.,[[[.,.],.],.]],[.,.]]=>[6,2,3,4,1,5]=>1
[[[.,.],[.,[.,.]]],[.,.]]=>[6,4,3,1,2,5]=>2
[[[.,.],[[.,.],.]],[.,.]]=>[6,3,4,1,2,5]=>3
[[[.,[.,.]],[.,.]],[.,.]]=>[6,4,2,1,3,5]=>2
[[[[.,.],.],[.,.]],[.,.]]=>[6,4,1,2,3,5]=>2
[[[.,[.,[.,.]]],.],[.,.]]=>[6,3,2,1,4,5]=>2
[[[.,[[.,.],.]],.],[.,.]]=>[6,2,3,1,4,5]=>1
[[[[.,.],[.,.]],.],[.,.]]=>[6,3,1,2,4,5]=>2
[[[[.,[.,.]],.],.],[.,.]]=>[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]=>2
[[.,[.,[[.,.],[.,.]]]],.]=>[5,3,4,2,1,6]=>3
[[.,[.,[[.,[.,.]],.]]],.]=>[4,3,5,2,1,6]=>3
[[.,[.,[[[.,.],.],.]]],.]=>[3,4,5,2,1,6]=>3
[[.,[[.,.],[.,[.,.]]]],.]=>[5,4,2,3,1,6]=>2
[[.,[[.,.],[[.,.],.]]],.]=>[4,5,2,3,1,6]=>2
[[.,[[.,[.,.]],[.,.]]],.]=>[5,3,2,4,1,6]=>2
[[.,[[[.,.],.],[.,.]]],.]=>[5,2,3,4,1,6]=>1
[[.,[[.,[.,[.,.]]],.]],.]=>[4,3,2,5,1,6]=>3
[[.,[[.,[[.,.],.]],.]],.]=>[3,4,2,5,1,6]=>3
[[.,[[[.,.],[.,.]],.]],.]=>[4,2,3,5,1,6]=>2
[[.,[[[.,[.,.]],.],.]],.]=>[3,2,4,5,1,6]=>3
[[.,[[[[.,.],.],.],.]],.]=>[2,3,4,5,1,6]=>4
[[[.,.],[.,[.,[.,.]]]],.]=>[5,4,3,1,2,6]=>2
[[[.,.],[.,[[.,.],.]]],.]=>[4,5,3,1,2,6]=>2
[[[.,.],[[.,.],[.,.]]],.]=>[5,3,4,1,2,6]=>3
[[[.,.],[[.,[.,.]],.]],.]=>[4,3,5,1,2,6]=>3
[[[.,.],[[[.,.],.],.]],.]=>[3,4,5,1,2,6]=>3
[[[.,[.,.]],[.,[.,.]]],.]=>[5,4,2,1,3,6]=>2
[[[.,[.,.]],[[.,.],.]],.]=>[4,5,2,1,3,6]=>2
[[[[.,.],.],[.,[.,.]]],.]=>[5,4,1,2,3,6]=>2
[[[[.,.],.],[[.,.],.]],.]=>[4,5,1,2,3,6]=>2
[[[.,[.,[.,.]]],[.,.]],.]=>[5,3,2,1,4,6]=>2
[[[.,[[.,.],.]],[.,.]],.]=>[5,2,3,1,4,6]=>1
[[[[.,.],[.,.]],[.,.]],.]=>[5,3,1,2,4,6]=>2
[[[[.,[.,.]],.],[.,.]],.]=>[5,2,1,3,4,6]=>1
[[[[[.,.],.],.],[.,.]],.]=>[5,1,2,3,4,6]=>1
[[[.,[.,[.,[.,.]]]],.],.]=>[4,3,2,1,5,6]=>2
[[[.,[.,[[.,.],.]]],.],.]=>[3,4,2,1,5,6]=>2
[[[.,[[.,.],[.,.]]],.],.]=>[4,2,3,1,5,6]=>1
[[[.,[[.,[.,.]],.]],.],.]=>[3,2,4,1,5,6]=>2
[[[.,[[[.,.],.],.]],.],.]=>[2,3,4,1,5,6]=>3
[[[[.,.],[.,[.,.]]],.],.]=>[4,3,1,2,5,6]=>2
[[[[.,.],[[.,.],.]],.],.]=>[3,4,1,2,5,6]=>2
[[[[.,[.,.]],[.,.]],.],.]=>[4,2,1,3,5,6]=>1
[[[[[.,.],.],[.,.]],.],.]=>[4,1,2,3,5,6]=>1
[[[[.,[.,[.,.]]],.],.],.]=>[3,2,1,4,5,6]=>1
[[[[.,[[.,.],.]],.],.],.]=>[2,3,1,4,5,6]=>2
[[[[[.,.],[.,.]],.],.],.]=>[3,1,2,4,5,6]=>1
[[[[[.,[.,.]],.],.],.],.]=>[2,1,3,4,5,6]=>1
[[[[[[.,.],.],.],.],.],.]=>[1,2,3,4,5,6]=>0
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Description
The number of exceedances (also excedences) of a permutation.
This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$.
It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is Euler-Mahonian. Here, $den$ is the Denert index of a permutation, see St000156The Denert index of a permutation..
This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$.
It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is Euler-Mahonian. Here, $den$ is the Denert index of a permutation, see St000156The Denert index 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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