Identifier
-
Mp00029:
Dyck paths
—to binary tree: left tree, up step, right tree, down step⟶
Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001864: Signed permutations ⟶ ℤ
Values
[1,0] => [.,.] => [1] => [1] => 0
[1,0,1,0] => [[.,.],.] => [1,2] => [1,2] => 0
[1,1,0,0] => [.,[.,.]] => [2,1] => [2,1] => 1
[1,0,1,0,1,0] => [[[.,.],.],.] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0] => [[.,.],[.,.]] => [3,1,2] => [3,1,2] => 1
[1,1,0,0,1,0] => [[.,[.,.]],.] => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0] => [.,[[.,.],.]] => [2,3,1] => [2,3,1] => 2
[1,1,1,0,0,0] => [.,[.,[.,.]]] => [3,2,1] => [3,2,1] => 1
[1,0,1,0,1,0,1,0] => [[[[.,.],.],.],.] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0] => [[[.,.],.],[.,.]] => [4,1,2,3] => [4,1,2,3] => 1
[1,0,1,1,0,0,1,0] => [[[.,.],[.,.]],.] => [3,1,2,4] => [3,1,2,4] => 1
[1,0,1,1,0,1,0,0] => [[.,.],[[.,.],.]] => [3,4,1,2] => [3,4,1,2] => 2
[1,0,1,1,1,0,0,0] => [[.,.],[.,[.,.]]] => [4,3,1,2] => [4,3,1,2] => 2
[1,1,0,0,1,0,1,0] => [[[.,[.,.]],.],.] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0] => [[.,[.,.]],[.,.]] => [4,2,1,3] => [4,2,1,3] => 1
[1,1,0,1,0,0,1,0] => [[.,[[.,.],.]],.] => [2,3,1,4] => [2,3,1,4] => 2
[1,1,0,1,0,1,0,0] => [.,[[[.,.],.],.]] => [2,3,4,1] => [2,3,4,1] => 3
[1,1,0,1,1,0,0,0] => [.,[[.,.],[.,.]]] => [4,2,3,1] => [4,2,3,1] => 1
[1,1,1,0,0,0,1,0] => [[.,[.,[.,.]]],.] => [3,2,1,4] => [3,2,1,4] => 1
[1,1,1,0,0,1,0,0] => [.,[[.,[.,.]],.]] => [3,2,4,1] => [3,2,4,1] => 2
[1,1,1,0,1,0,0,0] => [.,[.,[[.,.],.]]] => [3,4,2,1] => [3,4,2,1] => 2
[1,1,1,1,0,0,0,0] => [.,[.,[.,[.,.]]]] => [4,3,2,1] => [4,3,2,1] => 2
[1,0,1,0,1,0,1,0,1,0] => [[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [1,2,3,4,5] => 0
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Description
The number of excedances of a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, this is $\lvert\{i\in[n] \mid \pi(i) > i\}\rvert$.
For a signed permutation $\pi\in\mathfrak H_n$, this is $\lvert\{i\in[n] \mid \pi(i) > i\}\rvert$.
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.
Map
to signed permutation
Description
The signed permutation with all signs positive.
Map
to binary tree: left tree, up step, right tree, down step
Description
Return the binary tree corresponding to the Dyck path under the transformation left tree - up step - right tree - down step.
A Dyck path $D$ of semilength $n$ with $n > 1$ may be uniquely decomposed into $L 1 R 0$ 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.
This map may also be described as the unique map sending the Tamari orders on Dyck paths to the Tamari order on binary trees.
A Dyck path $D$ of semilength $n$ with $n > 1$ may be uniquely decomposed into $L 1 R 0$ 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.
This map may also be described as the unique map sending the Tamari orders on Dyck paths to the Tamari order on binary trees.
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