Identifier
Values
[.,.] => [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
>>> Load all 196 entries. <<<
[.,[[[.,[.,[.,.]]],.],.]] => [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..
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.