Identifier
Values
[.,.] => [1] => 1
[.,[.,.]] => [2,1] => 1
[[.,.],.] => [1,2] => 1
[.,[.,[.,.]]] => [3,2,1] => 1
[.,[[.,.],.]] => [2,3,1] => 2
[[.,.],[.,.]] => [1,3,2] => 2
[[.,[.,.]],.] => [2,1,3] => 1
[[[.,.],.],.] => [1,2,3] => 1
[.,[.,[.,[.,.]]]] => [4,3,2,1] => 1
[.,[.,[[.,.],.]]] => [3,4,2,1] => 2
[.,[[.,.],[.,.]]] => [2,4,3,1] => 2
[.,[[.,[.,.]],.]] => [3,2,4,1] => 2
[.,[[[.,.],.],.]] => [2,3,4,1] => 2
[[.,.],[.,[.,.]]] => [1,4,3,2] => 2
[[.,.],[[.,.],.]] => [1,3,4,2] => 2
[[.,[.,.]],[.,.]] => [2,1,4,3] => 2
[[[.,.],.],[.,.]] => [1,2,4,3] => 2
[[.,[.,[.,.]]],.] => [3,2,1,4] => 1
[[.,[[.,.],.]],.] => [2,3,1,4] => 2
[[[.,.],[.,.]],.] => [1,3,2,4] => 2
[[[.,[.,.]],.],.] => [2,1,3,4] => 1
[[[[.,.],.],.],.] => [1,2,3,4] => 1
[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => 1
[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => 2
[.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => 2
[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => 2
[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => 2
[.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => 2
[.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => 2
[.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => 2
[.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => 2
[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => 2
[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => 3
[.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => 3
[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => 2
[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => 2
[[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => 2
[[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => 2
[[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => 2
[[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => 3
[[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => 2
[[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => 2
[[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => 2
[[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => 2
[[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => 2
[[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => 2
[[.,[[.,.],.]],[.,.]] => [2,3,1,5,4] => 3
[[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => 3
[[[.,[.,.]],.],[.,.]] => [2,1,3,5,4] => 2
[[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => 2
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => 1
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => 2
[[.,[[.,.],[.,.]]],.] => [2,4,3,1,5] => 2
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => 2
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => 2
[[[.,.],[.,[.,.]]],.] => [1,4,3,2,5] => 2
[[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => 2
[[[.,[.,.]],[.,.]],.] => [2,1,4,3,5] => 2
[[[[.,.],.],[.,.]],.] => [1,2,4,3,5] => 2
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => 1
[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => 2
[[[[.,.],[.,.]],.],.] => [1,3,2,4,5] => 2
[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => 1
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => 1
[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => 1
[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => 2
[.,[.,[.,[[.,.],[.,.]]]]] => [4,6,5,3,2,1] => 2
[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => 2
[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => 2
[.,[.,[[.,.],[.,[.,.]]]]] => [3,6,5,4,2,1] => 2
[.,[.,[[.,.],[[.,.],.]]]] => [3,5,6,4,2,1] => 2
[.,[.,[[.,[.,.]],[.,.]]]] => [4,3,6,5,2,1] => 2
[.,[.,[[[.,.],.],[.,.]]]] => [3,4,6,5,2,1] => 2
[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => 2
[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => 3
[.,[.,[[[.,.],[.,.]],.]]] => [3,5,4,6,2,1] => 3
[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => 2
[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => 2
[.,[[.,.],[.,[.,[.,.]]]]] => [2,6,5,4,3,1] => 2
[.,[[.,.],[.,[[.,.],.]]]] => [2,5,6,4,3,1] => 2
[.,[[.,.],[[.,.],[.,.]]]] => [2,4,6,5,3,1] => 2
[.,[[.,.],[[.,[.,.]],.]]] => [2,5,4,6,3,1] => 3
[.,[[.,.],[[[.,.],.],.]]] => [2,4,5,6,3,1] => 2
[.,[[.,[.,.]],[.,[.,.]]]] => [3,2,6,5,4,1] => 2
[.,[[.,[.,.]],[[.,.],.]]] => [3,2,5,6,4,1] => 2
[.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => 2
[.,[[[.,.],.],[[.,.],.]]] => [2,3,5,6,4,1] => 2
[.,[[.,[.,[.,.]]],[.,.]]] => [4,3,2,6,5,1] => 2
[.,[[.,[[.,.],.]],[.,.]]] => [3,4,2,6,5,1] => 3
[.,[[[.,.],[.,.]],[.,.]]] => [2,4,3,6,5,1] => 3
[.,[[[.,[.,.]],.],[.,.]]] => [3,2,4,6,5,1] => 2
[.,[[[[.,.],.],.],[.,.]]] => [2,3,4,6,5,1] => 2
[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => 2
[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => 3
[.,[[.,[[.,.],[.,.]]],.]] => [3,5,4,2,6,1] => 3
[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => 3
[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => 3
[.,[[[.,.],[.,[.,.]]],.]] => [2,5,4,3,6,1] => 3
[.,[[[.,.],[[.,.],.]],.]] => [2,4,5,3,6,1] => 3
[.,[[[.,[.,.]],[.,.]],.]] => [3,2,5,4,6,1] => 3
[.,[[[[.,.],.],[.,.]],.]] => [2,3,5,4,6,1] => 3
>>> Load all 196 entries. <<<
[.,[[[.,[.,[.,.]]],.],.]] => [4,3,2,5,6,1] => 2
[.,[[[.,[[.,.],.]],.],.]] => [3,4,2,5,6,1] => 3
[.,[[[[.,.],[.,.]],.],.]] => [2,4,3,5,6,1] => 3
[.,[[[[.,[.,.]],.],.],.]] => [3,2,4,5,6,1] => 2
[.,[[[[[.,.],.],.],.],.]] => [2,3,4,5,6,1] => 2
[[.,.],[.,[.,[.,[.,.]]]]] => [1,6,5,4,3,2] => 2
[[.,.],[.,[.,[[.,.],.]]]] => [1,5,6,4,3,2] => 2
[[.,.],[.,[[.,.],[.,.]]]] => [1,4,6,5,3,2] => 2
[[.,.],[.,[[.,[.,.]],.]]] => [1,5,4,6,3,2] => 3
[[.,.],[.,[[[.,.],.],.]]] => [1,4,5,6,3,2] => 2
[[.,.],[[.,.],[.,[.,.]]]] => [1,3,6,5,4,2] => 2
[[.,.],[[.,.],[[.,.],.]]] => [1,3,5,6,4,2] => 2
[[.,.],[[.,[.,.]],[.,.]]] => [1,4,3,6,5,2] => 3
[[.,.],[[[.,.],.],[.,.]]] => [1,3,4,6,5,2] => 2
[[.,.],[[.,[.,[.,.]]],.]] => [1,5,4,3,6,2] => 3
[[.,.],[[.,[[.,.],.]],.]] => [1,4,5,3,6,2] => 3
[[.,.],[[[.,.],[.,.]],.]] => [1,3,5,4,6,2] => 3
[[.,.],[[[.,[.,.]],.],.]] => [1,4,3,5,6,2] => 3
[[.,.],[[[[.,.],.],.],.]] => [1,3,4,5,6,2] => 2
[[.,[.,.]],[.,[.,[.,.]]]] => [2,1,6,5,4,3] => 2
[[.,[.,.]],[.,[[.,.],.]]] => [2,1,5,6,4,3] => 2
[[.,[.,.]],[[.,.],[.,.]]] => [2,1,4,6,5,3] => 2
[[.,[.,.]],[[.,[.,.]],.]] => [2,1,5,4,6,3] => 3
[[.,[.,.]],[[[.,.],.],.]] => [2,1,4,5,6,3] => 2
[[[.,.],.],[.,[.,[.,.]]]] => [1,2,6,5,4,3] => 2
[[[.,.],.],[.,[[.,.],.]]] => [1,2,5,6,4,3] => 2
[[[.,.],.],[[.,.],[.,.]]] => [1,2,4,6,5,3] => 2
[[[.,.],.],[[.,[.,.]],.]] => [1,2,5,4,6,3] => 3
[[[.,.],.],[[[.,.],.],.]] => [1,2,4,5,6,3] => 2
[[.,[.,[.,.]]],[.,[.,.]]] => [3,2,1,6,5,4] => 2
[[.,[.,[.,.]]],[[.,.],.]] => [3,2,1,5,6,4] => 2
[[.,[[.,.],.]],[.,[.,.]]] => [2,3,1,6,5,4] => 3
[[.,[[.,.],.]],[[.,.],.]] => [2,3,1,5,6,4] => 3
[[[.,.],[.,.]],[.,[.,.]]] => [1,3,2,6,5,4] => 3
[[[.,.],[.,.]],[[.,.],.]] => [1,3,2,5,6,4] => 3
[[[.,[.,.]],.],[.,[.,.]]] => [2,1,3,6,5,4] => 2
[[[.,[.,.]],.],[[.,.],.]] => [2,1,3,5,6,4] => 2
[[[[.,.],.],.],[.,[.,.]]] => [1,2,3,6,5,4] => 2
[[[[.,.],.],.],[[.,.],.]] => [1,2,3,5,6,4] => 2
[[.,[.,[.,[.,.]]]],[.,.]] => [4,3,2,1,6,5] => 2
[[.,[.,[[.,.],.]]],[.,.]] => [3,4,2,1,6,5] => 3
[[.,[[.,.],[.,.]]],[.,.]] => [2,4,3,1,6,5] => 3
[[.,[[.,[.,.]],.]],[.,.]] => [3,2,4,1,6,5] => 3
[[.,[[[.,.],.],.]],[.,.]] => [2,3,4,1,6,5] => 3
[[[.,.],[.,[.,.]]],[.,.]] => [1,4,3,2,6,5] => 3
[[[.,.],[[.,.],.]],[.,.]] => [1,3,4,2,6,5] => 3
[[[.,[.,.]],[.,.]],[.,.]] => [2,1,4,3,6,5] => 3
[[[[.,.],.],[.,.]],[.,.]] => [1,2,4,3,6,5] => 3
[[[.,[.,[.,.]]],.],[.,.]] => [3,2,1,4,6,5] => 2
[[[.,[[.,.],.]],.],[.,.]] => [2,3,1,4,6,5] => 3
[[[[.,.],[.,.]],.],[.,.]] => [1,3,2,4,6,5] => 3
[[[[.,[.,.]],.],.],[.,.]] => [2,1,3,4,6,5] => 2
[[[[[.,.],.],.],.],[.,.]] => [1,2,3,4,6,5] => 2
[[.,[.,[.,[.,[.,.]]]]],.] => [5,4,3,2,1,6] => 1
[[.,[.,[.,[[.,.],.]]]],.] => [4,5,3,2,1,6] => 2
[[.,[.,[[.,.],[.,.]]]],.] => [3,5,4,2,1,6] => 2
[[.,[.,[[.,[.,.]],.]]],.] => [4,3,5,2,1,6] => 2
[[.,[.,[[[.,.],.],.]]],.] => [3,4,5,2,1,6] => 2
[[.,[[.,.],[.,[.,.]]]],.] => [2,5,4,3,1,6] => 2
[[.,[[.,.],[[.,.],.]]],.] => [2,4,5,3,1,6] => 2
[[.,[[.,[.,.]],[.,.]]],.] => [3,2,5,4,1,6] => 2
[[.,[[[.,.],.],[.,.]]],.] => [2,3,5,4,1,6] => 2
[[.,[[.,[.,[.,.]]],.]],.] => [4,3,2,5,1,6] => 2
[[.,[[.,[[.,.],.]],.]],.] => [3,4,2,5,1,6] => 3
[[.,[[[.,.],[.,.]],.]],.] => [2,4,3,5,1,6] => 3
[[.,[[[.,[.,.]],.],.]],.] => [3,2,4,5,1,6] => 2
[[.,[[[[.,.],.],.],.]],.] => [2,3,4,5,1,6] => 2
[[[.,.],[.,[.,[.,.]]]],.] => [1,5,4,3,2,6] => 2
[[[.,.],[.,[[.,.],.]]],.] => [1,4,5,3,2,6] => 2
[[[.,.],[[.,.],[.,.]]],.] => [1,3,5,4,2,6] => 2
[[[.,.],[[.,[.,.]],.]],.] => [1,4,3,5,2,6] => 3
[[[.,.],[[[.,.],.],.]],.] => [1,3,4,5,2,6] => 2
[[[.,[.,.]],[.,[.,.]]],.] => [2,1,5,4,3,6] => 2
[[[.,[.,.]],[[.,.],.]],.] => [2,1,4,5,3,6] => 2
[[[[.,.],.],[.,[.,.]]],.] => [1,2,5,4,3,6] => 2
[[[[.,.],.],[[.,.],.]],.] => [1,2,4,5,3,6] => 2
[[[.,[.,[.,.]]],[.,.]],.] => [3,2,1,5,4,6] => 2
[[[.,[[.,.],.]],[.,.]],.] => [2,3,1,5,4,6] => 3
[[[[.,.],[.,.]],[.,.]],.] => [1,3,2,5,4,6] => 3
[[[[.,[.,.]],.],[.,.]],.] => [2,1,3,5,4,6] => 2
[[[[[.,.],.],.],[.,.]],.] => [1,2,3,5,4,6] => 2
[[[.,[.,[.,[.,.]]]],.],.] => [4,3,2,1,5,6] => 1
[[[.,[.,[[.,.],.]]],.],.] => [3,4,2,1,5,6] => 2
[[[.,[[.,.],[.,.]]],.],.] => [2,4,3,1,5,6] => 2
[[[.,[[.,[.,.]],.]],.],.] => [3,2,4,1,5,6] => 2
[[[.,[[[.,.],.],.]],.],.] => [2,3,4,1,5,6] => 2
[[[[.,.],[.,[.,.]]],.],.] => [1,4,3,2,5,6] => 2
[[[[.,.],[[.,.],.]],.],.] => [1,3,4,2,5,6] => 2
[[[[.,[.,.]],[.,.]],.],.] => [2,1,4,3,5,6] => 2
[[[[[.,.],.],[.,.]],.],.] => [1,2,4,3,5,6] => 2
[[[[.,[.,[.,.]]],.],.],.] => [3,2,1,4,5,6] => 1
[[[[.,[[.,.],.]],.],.],.] => [2,3,1,4,5,6] => 2
[[[[[.,.],[.,.]],.],.],.] => [1,3,2,4,5,6] => 2
[[[[[.,[.,.]],.],.],.],.] => [2,1,3,4,5,6] => 1
[[[[[[.,.],.],.],.],.],.] => [1,2,3,4,5,6] => 1
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Description
The number of valleys of a permutation, including the boundary.
The number of valleys excluding the boundary is St000353The number of inner valleys of a permutation..
Map
to 312-avoiding permutation
Description
Return a 312-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 minimal element of this Sylvester class.