Identifier
Values
[.,.] => [1] => 0
[.,[.,.]] => [2,1] => 0
[[.,.],.] => [1,2] => 0
[.,[.,[.,.]]] => [3,2,1] => 0
[.,[[.,.],.]] => [2,3,1] => 0
[[.,.],[.,.]] => [1,3,2] => 0
[[.,[.,.]],.] => [2,1,3] => 0
[[[.,.],.],.] => [1,2,3] => 0
[.,[.,[.,[.,.]]]] => [4,3,2,1] => 0
[.,[.,[[.,.],.]]] => [3,4,2,1] => 1
[.,[[.,.],[.,.]]] => [2,4,3,1] => 0
[.,[[.,[.,.]],.]] => [3,2,4,1] => 0
[.,[[[.,.],.],.]] => [2,3,4,1] => 0
[[.,.],[.,[.,.]]] => [1,4,3,2] => 0
[[.,.],[[.,.],.]] => [1,3,4,2] => 0
[[.,[.,.]],[.,.]] => [2,1,4,3] => 0
[[[.,.],.],[.,.]] => [1,2,4,3] => 0
[[.,[.,[.,.]]],.] => [3,2,1,4] => 0
[[.,[[.,.],.]],.] => [2,3,1,4] => 0
[[[.,.],[.,.]],.] => [1,3,2,4] => 0
[[[.,[.,.]],.],.] => [2,1,3,4] => 0
[[[[.,.],.],.],.] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => 0
[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => 1
[.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => 1
[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => 1
[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => 0
[.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => 0
[.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => 1
[.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => 0
[.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => 0
[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => 0
[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => 1
[.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => 0
[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => 0
[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => 0
[[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => 0
[[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => 1
[[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => 0
[[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => 0
[[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => 0
[[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => 0
[[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => 0
[[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => 0
[[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => 0
[[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => 0
[[.,[[.,.],.]],[.,.]] => [2,3,1,5,4] => 0
[[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => 0
[[[.,[.,.]],.],[.,.]] => [2,1,3,5,4] => 0
[[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => 0
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => 0
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => 1
[[.,[[.,.],[.,.]]],.] => [2,4,3,1,5] => 0
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => 0
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => 0
[[[.,.],[.,[.,.]]],.] => [1,4,3,2,5] => 0
[[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => 0
[[[.,[.,.]],[.,.]],.] => [2,1,4,3,5] => 0
[[[[.,.],.],[.,.]],.] => [1,2,4,3,5] => 0
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => 0
[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => 0
[[[[.,.],[.,.]],.],.] => [1,3,2,4,5] => 0
[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => 0
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => 0
[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => 0
[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => 1
[.,[.,[.,[[.,.],[.,.]]]]] => [4,6,5,3,2,1] => 2
[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => 2
[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => 1
[.,[.,[[.,.],[.,[.,.]]]]] => [3,6,5,4,2,1] => 1
[.,[.,[[.,.],[[.,.],.]]]] => [3,5,6,4,2,1] => 0
[.,[.,[[.,[.,.]],[.,.]]]] => [4,3,6,5,2,1] => 1
[.,[.,[[[.,.],.],[.,.]]]] => [3,4,6,5,2,1] => 0
[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => 1
[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => 0
[.,[.,[[[.,.],[.,.]],.]]] => [3,5,4,6,2,1] => 0
[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => 0
[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => 1
[.,[[.,.],[.,[.,[.,.]]]]] => [2,6,5,4,3,1] => 0
[.,[[.,.],[.,[[.,.],.]]]] => [2,5,6,4,3,1] => 1
[.,[[.,.],[[.,.],[.,.]]]] => [2,4,6,5,3,1] => 1
[.,[[.,.],[[.,[.,.]],.]]] => [2,5,4,6,3,1] => 1
[.,[[.,.],[[[.,.],.],.]]] => [2,4,5,6,3,1] => 0
[.,[[.,[.,.]],[.,[.,.]]]] => [3,2,6,5,4,1] => 0
[.,[[.,[.,.]],[[.,.],.]]] => [3,2,5,6,4,1] => 1
[.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => 0
[.,[[[.,.],.],[[.,.],.]]] => [2,3,5,6,4,1] => 1
[.,[[.,[.,[.,.]]],[.,.]]] => [4,3,2,6,5,1] => 0
[.,[[.,[[.,.],.]],[.,.]]] => [3,4,2,6,5,1] => 1
[.,[[[.,.],[.,.]],[.,.]]] => [2,4,3,6,5,1] => 0
[.,[[[.,[.,.]],.],[.,.]]] => [3,2,4,6,5,1] => 0
[.,[[[[.,.],.],.],[.,.]]] => [2,3,4,6,5,1] => 0
[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => 0
[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => 1
[.,[[.,[[.,.],[.,.]]],.]] => [3,5,4,2,6,1] => 1
[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => 1
[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => 0
[.,[[[.,.],[.,[.,.]]],.]] => [2,5,4,3,6,1] => 0
[.,[[[.,.],[[.,.],.]],.]] => [2,4,5,3,6,1] => 1
[.,[[[.,[.,.]],[.,.]],.]] => [3,2,5,4,6,1] => 0
[.,[[[[.,.],.],[.,.]],.]] => [2,3,5,4,6,1] => 0
>>> Load all 196 entries. <<<
[.,[[[.,[.,[.,.]]],.],.]] => [4,3,2,5,6,1] => 0
[.,[[[.,[[.,.],.]],.],.]] => [3,4,2,5,6,1] => 1
[.,[[[[.,.],[.,.]],.],.]] => [2,4,3,5,6,1] => 0
[.,[[[[.,[.,.]],.],.],.]] => [3,2,4,5,6,1] => 0
[.,[[[[[.,.],.],.],.],.]] => [2,3,4,5,6,1] => 0
[[.,.],[.,[.,[.,[.,.]]]]] => [1,6,5,4,3,2] => 0
[[.,.],[.,[.,[[.,.],.]]]] => [1,5,6,4,3,2] => 1
[[.,.],[.,[[.,.],[.,.]]]] => [1,4,6,5,3,2] => 1
[[.,.],[.,[[.,[.,.]],.]]] => [1,5,4,6,3,2] => 1
[[.,.],[.,[[[.,.],.],.]]] => [1,4,5,6,3,2] => 0
[[.,.],[[.,.],[.,[.,.]]]] => [1,3,6,5,4,2] => 0
[[.,.],[[.,.],[[.,.],.]]] => [1,3,5,6,4,2] => 1
[[.,.],[[.,[.,.]],[.,.]]] => [1,4,3,6,5,2] => 0
[[.,.],[[[.,.],.],[.,.]]] => [1,3,4,6,5,2] => 0
[[.,.],[[.,[.,[.,.]]],.]] => [1,5,4,3,6,2] => 0
[[.,.],[[.,[[.,.],.]],.]] => [1,4,5,3,6,2] => 1
[[.,.],[[[.,.],[.,.]],.]] => [1,3,5,4,6,2] => 0
[[.,.],[[[.,[.,.]],.],.]] => [1,4,3,5,6,2] => 0
[[.,.],[[[[.,.],.],.],.]] => [1,3,4,5,6,2] => 0
[[.,[.,.]],[.,[.,[.,.]]]] => [2,1,6,5,4,3] => 0
[[.,[.,.]],[.,[[.,.],.]]] => [2,1,5,6,4,3] => 1
[[.,[.,.]],[[.,.],[.,.]]] => [2,1,4,6,5,3] => 0
[[.,[.,.]],[[.,[.,.]],.]] => [2,1,5,4,6,3] => 0
[[.,[.,.]],[[[.,.],.],.]] => [2,1,4,5,6,3] => 0
[[[.,.],.],[.,[.,[.,.]]]] => [1,2,6,5,4,3] => 0
[[[.,.],.],[.,[[.,.],.]]] => [1,2,5,6,4,3] => 1
[[[.,.],.],[[.,.],[.,.]]] => [1,2,4,6,5,3] => 0
[[[.,.],.],[[.,[.,.]],.]] => [1,2,5,4,6,3] => 0
[[[.,.],.],[[[.,.],.],.]] => [1,2,4,5,6,3] => 0
[[.,[.,[.,.]]],[.,[.,.]]] => [3,2,1,6,5,4] => 0
[[.,[.,[.,.]]],[[.,.],.]] => [3,2,1,5,6,4] => 0
[[.,[[.,.],.]],[.,[.,.]]] => [2,3,1,6,5,4] => 0
[[.,[[.,.],.]],[[.,.],.]] => [2,3,1,5,6,4] => 0
[[[.,.],[.,.]],[.,[.,.]]] => [1,3,2,6,5,4] => 0
[[[.,.],[.,.]],[[.,.],.]] => [1,3,2,5,6,4] => 0
[[[.,[.,.]],.],[.,[.,.]]] => [2,1,3,6,5,4] => 0
[[[.,[.,.]],.],[[.,.],.]] => [2,1,3,5,6,4] => 0
[[[[.,.],.],.],[.,[.,.]]] => [1,2,3,6,5,4] => 0
[[[[.,.],.],.],[[.,.],.]] => [1,2,3,5,6,4] => 0
[[.,[.,[.,[.,.]]]],[.,.]] => [4,3,2,1,6,5] => 0
[[.,[.,[[.,.],.]]],[.,.]] => [3,4,2,1,6,5] => 1
[[.,[[.,.],[.,.]]],[.,.]] => [2,4,3,1,6,5] => 0
[[.,[[.,[.,.]],.]],[.,.]] => [3,2,4,1,6,5] => 0
[[.,[[[.,.],.],.]],[.,.]] => [2,3,4,1,6,5] => 0
[[[.,.],[.,[.,.]]],[.,.]] => [1,4,3,2,6,5] => 0
[[[.,.],[[.,.],.]],[.,.]] => [1,3,4,2,6,5] => 0
[[[.,[.,.]],[.,.]],[.,.]] => [2,1,4,3,6,5] => 0
[[[[.,.],.],[.,.]],[.,.]] => [1,2,4,3,6,5] => 0
[[[.,[.,[.,.]]],.],[.,.]] => [3,2,1,4,6,5] => 0
[[[.,[[.,.],.]],.],[.,.]] => [2,3,1,4,6,5] => 0
[[[[.,.],[.,.]],.],[.,.]] => [1,3,2,4,6,5] => 0
[[[[.,[.,.]],.],.],[.,.]] => [2,1,3,4,6,5] => 0
[[[[[.,.],.],.],.],[.,.]] => [1,2,3,4,6,5] => 0
[[.,[.,[.,[.,[.,.]]]]],.] => [5,4,3,2,1,6] => 0
[[.,[.,[.,[[.,.],.]]]],.] => [4,5,3,2,1,6] => 1
[[.,[.,[[.,.],[.,.]]]],.] => [3,5,4,2,1,6] => 1
[[.,[.,[[.,[.,.]],.]]],.] => [4,3,5,2,1,6] => 1
[[.,[.,[[[.,.],.],.]]],.] => [3,4,5,2,1,6] => 0
[[.,[[.,.],[.,[.,.]]]],.] => [2,5,4,3,1,6] => 0
[[.,[[.,.],[[.,.],.]]],.] => [2,4,5,3,1,6] => 1
[[.,[[.,[.,.]],[.,.]]],.] => [3,2,5,4,1,6] => 0
[[.,[[[.,.],.],[.,.]]],.] => [2,3,5,4,1,6] => 0
[[.,[[.,[.,[.,.]]],.]],.] => [4,3,2,5,1,6] => 0
[[.,[[.,[[.,.],.]],.]],.] => [3,4,2,5,1,6] => 1
[[.,[[[.,.],[.,.]],.]],.] => [2,4,3,5,1,6] => 0
[[.,[[[.,[.,.]],.],.]],.] => [3,2,4,5,1,6] => 0
[[.,[[[[.,.],.],.],.]],.] => [2,3,4,5,1,6] => 0
[[[.,.],[.,[.,[.,.]]]],.] => [1,5,4,3,2,6] => 0
[[[.,.],[.,[[.,.],.]]],.] => [1,4,5,3,2,6] => 1
[[[.,.],[[.,.],[.,.]]],.] => [1,3,5,4,2,6] => 0
[[[.,.],[[.,[.,.]],.]],.] => [1,4,3,5,2,6] => 0
[[[.,.],[[[.,.],.],.]],.] => [1,3,4,5,2,6] => 0
[[[.,[.,.]],[.,[.,.]]],.] => [2,1,5,4,3,6] => 0
[[[.,[.,.]],[[.,.],.]],.] => [2,1,4,5,3,6] => 0
[[[[.,.],.],[.,[.,.]]],.] => [1,2,5,4,3,6] => 0
[[[[.,.],.],[[.,.],.]],.] => [1,2,4,5,3,6] => 0
[[[.,[.,[.,.]]],[.,.]],.] => [3,2,1,5,4,6] => 0
[[[.,[[.,.],.]],[.,.]],.] => [2,3,1,5,4,6] => 0
[[[[.,.],[.,.]],[.,.]],.] => [1,3,2,5,4,6] => 0
[[[[.,[.,.]],.],[.,.]],.] => [2,1,3,5,4,6] => 0
[[[[[.,.],.],.],[.,.]],.] => [1,2,3,5,4,6] => 0
[[[.,[.,[.,[.,.]]]],.],.] => [4,3,2,1,5,6] => 0
[[[.,[.,[[.,.],.]]],.],.] => [3,4,2,1,5,6] => 1
[[[.,[[.,.],[.,.]]],.],.] => [2,4,3,1,5,6] => 0
[[[.,[[.,[.,.]],.]],.],.] => [3,2,4,1,5,6] => 0
[[[.,[[[.,.],.],.]],.],.] => [2,3,4,1,5,6] => 0
[[[[.,.],[.,[.,.]]],.],.] => [1,4,3,2,5,6] => 0
[[[[.,.],[[.,.],.]],.],.] => [1,3,4,2,5,6] => 0
[[[[.,[.,.]],[.,.]],.],.] => [2,1,4,3,5,6] => 0
[[[[[.,.],.],[.,.]],.],.] => [1,2,4,3,5,6] => 0
[[[[.,[.,[.,.]]],.],.],.] => [3,2,1,4,5,6] => 0
[[[[.,[[.,.],.]],.],.],.] => [2,3,1,4,5,6] => 0
[[[[[.,.],[.,.]],.],.],.] => [1,3,2,4,5,6] => 0
[[[[[.,[.,.]],.],.],.],.] => [2,1,3,4,5,6] => 0
[[[[[[.,.],.],.],.],.],.] => [1,2,3,4,5,6] => 0
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Description
The cycle descent number of a permutation.
Let $(i_1,\ldots,i_k)$ be a cycle of a permutation $\pi$ such that $i_1$ is its smallest element. A **cycle descent** of $(i_1,\ldots,i_k)$ is an $i_a$ for $1 \leq a < k$ such that $i_a > i_{a+1}$. The **cycle descent set** of $\pi$ is then the set of descents in all the cycles of $\pi$, and the **cycle descent number** is its cardinality.
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.