Identifier
Values
[1,0] => [.,.] => [1] => [1] => 0
[1,0,1,0] => [.,[.,.]] => [2,1] => [2,1] => 0
[1,1,0,0] => [[.,.],.] => [1,2] => [1,2] => 0
[1,0,1,0,1,0] => [.,[.,[.,.]]] => [3,2,1] => [3,2,1] => 0
[1,0,1,1,0,0] => [.,[[.,.],.]] => [2,3,1] => [2,3,1] => 1
[1,1,0,0,1,0] => [[.,.],[.,.]] => [1,3,2] => [1,3,2] => 0
[1,1,0,1,0,0] => [[.,[.,.]],.] => [2,1,3] => [2,1,3] => 0
[1,1,1,0,0,0] => [[[.,.],.],.] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0] => [.,[.,[.,[.,.]]]] => [4,3,2,1] => [4,3,2,1] => 0
[1,0,1,0,1,1,0,0] => [.,[.,[[.,.],.]]] => [3,4,2,1] => [3,4,2,1] => 1
[1,0,1,1,0,0,1,0] => [.,[[.,.],[.,.]]] => [2,4,3,1] => [2,4,3,1] => 1
[1,0,1,1,0,1,0,0] => [.,[[.,[.,.]],.]] => [3,2,4,1] => [3,2,4,1] => 1
[1,0,1,1,1,0,0,0] => [.,[[[.,.],.],.]] => [2,3,4,1] => [2,3,4,1] => 2
[1,1,0,0,1,0,1,0] => [[.,.],[.,[.,.]]] => [1,4,3,2] => [1,4,3,2] => 0
[1,1,0,0,1,1,0,0] => [[.,.],[[.,.],.]] => [1,3,4,2] => [1,3,4,2] => 1
[1,1,0,1,0,0,1,0] => [[.,[.,.]],[.,.]] => [2,1,4,3] => [2,1,4,3] => 0
[1,1,0,1,0,1,0,0] => [[.,[.,[.,.]]],.] => [3,2,1,4] => [3,2,1,4] => 0
[1,1,0,1,1,0,0,0] => [[.,[[.,.],.]],.] => [2,3,1,4] => [2,3,1,4] => 1
[1,1,1,0,0,0,1,0] => [[[.,.],.],[.,.]] => [1,2,4,3] => [1,2,4,3] => 0
[1,1,1,0,0,1,0,0] => [[[.,.],[.,.]],.] => [1,3,2,4] => [1,3,2,4] => 0
[1,1,1,0,1,0,0,0] => [[[.,[.,.]],.],.] => [2,1,3,4] => [2,1,3,4] => 0
[1,1,1,1,0,0,0,0] => [[[[.,.],.],.],.] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0] => [[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[1,1,0,0,1,0,1,1,0,0] => [[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => [1,4,5,3,2] => 1
[1,1,0,0,1,1,0,0,1,0] => [[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => [1,3,5,4,2] => 1
[1,1,0,0,1,1,0,1,0,0] => [[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => [1,4,3,5,2] => 1
[1,1,0,0,1,1,1,0,0,0] => [[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => [1,3,4,5,2] => 2
[1,1,1,0,0,0,1,0,1,0] => [[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[1,1,1,0,0,0,1,1,0,0] => [[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => [1,2,4,5,3] => 1
[1,1,1,0,0,1,0,0,1,0] => [[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,1,1,0,0,1,0,1,0,0] => [[[.,.],[.,[.,.]]],.] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[1,1,1,0,0,1,1,0,0,0] => [[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => [1,3,4,2,5] => 1
[1,1,1,1,0,0,0,0,1,0] => [[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,1,1,1,0,0,0,1,0,0] => [[[[.,.],.],[.,.]],.] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,1,1,1,0,0,1,0,0,0] => [[[[.,.],[.,.]],.],.] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,1,1,1,1,0,0,0,0,0] => [[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [1,2,3,4,5] => 0
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Description
The number of crossings of a signed permutation.
A crossing of a signed permutation $\pi$ is a pair $(i, j)$ of indices such that
  • $i < j \leq \pi(i) < \pi(j)$, or
  • $-i < j \leq -\pi(i) < \pi(j)$, or
  • $i > j > \pi(i) > \pi(j)$.
Map
to signed permutation
Description
The signed permutation with all signs positive.
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.
Map
to binary tree: up step, left tree, down step, right tree
Description
Return the binary tree corresponding to the Dyck path under the transformation up step - left tree - down step - right tree.
A Dyck path $D$ of semilength $n$ with $ n > 1$ may be uniquely decomposed into $1L0R$ for Dyck paths L,R of respective semilengths $n_1, n_2$ with $n_1 + n_2 = n-1$.
This map sends $D$ to the binary tree $T$ consisting of a root node with a left child according to $L$ and a right child according to $R$ and then recursively proceeds.
The base case of the unique Dyck path of semilength $1$ is sent to a single node.