Identifier
-
Mp00029:
Dyck paths
—to binary tree: left tree, up step, right tree, down step⟶
Binary trees
St000082: Binary trees ⟶ ℤ
Values
[1,0,1,0] => [[.,.],.] => 1
[1,1,0,0] => [.,[.,.]] => 2
[1,0,1,0,1,0] => [[[.,.],.],.] => 1
[1,0,1,1,0,0] => [[.,.],[.,.]] => 2
[1,1,0,0,1,0] => [[.,[.,.]],.] => 2
[1,1,0,1,0,0] => [.,[[.,.],.]] => 3
[1,1,1,0,0,0] => [.,[.,[.,.]]] => 5
[1,0,1,0,1,0,1,0] => [[[[.,.],.],.],.] => 1
[1,0,1,0,1,1,0,0] => [[[.,.],.],[.,.]] => 2
[1,0,1,1,0,0,1,0] => [[[.,.],[.,.]],.] => 2
[1,0,1,1,0,1,0,0] => [[.,.],[[.,.],.]] => 3
[1,0,1,1,1,0,0,0] => [[.,.],[.,[.,.]]] => 5
[1,1,0,0,1,0,1,0] => [[[.,[.,.]],.],.] => 2
[1,1,0,0,1,1,0,0] => [[.,[.,.]],[.,.]] => 4
[1,1,0,1,0,0,1,0] => [[.,[[.,.],.]],.] => 3
[1,1,0,1,0,1,0,0] => [.,[[[.,.],.],.]] => 4
[1,1,0,1,1,0,0,0] => [.,[[.,.],[.,.]]] => 7
[1,1,1,0,0,0,1,0] => [[.,[.,[.,.]]],.] => 5
[1,1,1,0,0,1,0,0] => [.,[[.,[.,.]],.]] => 7
[1,1,1,0,1,0,0,0] => [.,[.,[[.,.],.]]] => 9
[1,1,1,1,0,0,0,0] => [.,[.,[.,[.,.]]]] => 14
[1,0,1,0,1,0,1,0,1,0] => [[[[[.,.],.],.],.],.] => 1
[1,0,1,0,1,0,1,1,0,0] => [[[[.,.],.],.],[.,.]] => 2
[1,0,1,0,1,1,0,0,1,0] => [[[[.,.],.],[.,.]],.] => 2
[1,0,1,0,1,1,0,1,0,0] => [[[.,.],.],[[.,.],.]] => 3
[1,0,1,0,1,1,1,0,0,0] => [[[.,.],.],[.,[.,.]]] => 5
[1,0,1,1,0,0,1,0,1,0] => [[[[.,.],[.,.]],.],.] => 2
[1,0,1,1,0,0,1,1,0,0] => [[[.,.],[.,.]],[.,.]] => 4
[1,0,1,1,0,1,0,0,1,0] => [[[.,.],[[.,.],.]],.] => 3
[1,0,1,1,0,1,0,1,0,0] => [[.,.],[[[.,.],.],.]] => 4
[1,0,1,1,0,1,1,0,0,0] => [[.,.],[[.,.],[.,.]]] => 7
[1,0,1,1,1,0,0,0,1,0] => [[[.,.],[.,[.,.]]],.] => 5
[1,0,1,1,1,0,0,1,0,0] => [[.,.],[[.,[.,.]],.]] => 7
[1,0,1,1,1,0,1,0,0,0] => [[.,.],[.,[[.,.],.]]] => 9
[1,0,1,1,1,1,0,0,0,0] => [[.,.],[.,[.,[.,.]]]] => 14
[1,1,0,0,1,0,1,0,1,0] => [[[[.,[.,.]],.],.],.] => 2
[1,1,0,0,1,0,1,1,0,0] => [[[.,[.,.]],.],[.,.]] => 4
[1,1,0,0,1,1,0,0,1,0] => [[[.,[.,.]],[.,.]],.] => 4
[1,1,0,0,1,1,0,1,0,0] => [[.,[.,.]],[[.,.],.]] => 6
[1,1,0,0,1,1,1,0,0,0] => [[.,[.,.]],[.,[.,.]]] => 10
[1,1,0,1,0,0,1,0,1,0] => [[[.,[[.,.],.]],.],.] => 3
[1,1,0,1,0,0,1,1,0,0] => [[.,[[.,.],.]],[.,.]] => 6
[1,1,0,1,0,1,0,0,1,0] => [[.,[[[.,.],.],.]],.] => 4
[1,1,0,1,0,1,0,1,0,0] => [.,[[[[.,.],.],.],.]] => 5
[1,1,0,1,0,1,1,0,0,0] => [.,[[[.,.],.],[.,.]]] => 9
[1,1,0,1,1,0,0,0,1,0] => [[.,[[.,.],[.,.]]],.] => 7
[1,1,0,1,1,0,0,1,0,0] => [.,[[[.,.],[.,.]],.]] => 9
[1,1,0,1,1,0,1,0,0,0] => [.,[[.,.],[[.,.],.]]] => 12
[1,1,0,1,1,1,0,0,0,0] => [.,[[.,.],[.,[.,.]]]] => 19
[1,1,1,0,0,0,1,0,1,0] => [[[.,[.,[.,.]]],.],.] => 5
[1,1,1,0,0,0,1,1,0,0] => [[.,[.,[.,.]]],[.,.]] => 10
[1,1,1,0,0,1,0,0,1,0] => [[.,[[.,[.,.]],.]],.] => 7
[1,1,1,0,0,1,0,1,0,0] => [.,[[[.,[.,.]],.],.]] => 9
[1,1,1,0,0,1,1,0,0,0] => [.,[[.,[.,.]],[.,.]]] => 16
[1,1,1,0,1,0,0,0,1,0] => [[.,[.,[[.,.],.]]],.] => 9
[1,1,1,0,1,0,0,1,0,0] => [.,[[.,[[.,.],.]],.]] => 12
[1,1,1,0,1,0,1,0,0,0] => [.,[.,[[[.,.],.],.]]] => 14
[1,1,1,0,1,1,0,0,0,0] => [.,[.,[[.,.],[.,.]]]] => 23
[1,1,1,1,0,0,0,0,1,0] => [[.,[.,[.,[.,.]]]],.] => 14
[1,1,1,1,0,0,0,1,0,0] => [.,[[.,[.,[.,.]]],.]] => 19
[1,1,1,1,0,0,1,0,0,0] => [.,[.,[[.,[.,.]],.]]] => 23
[1,1,1,1,0,1,0,0,0,0] => [.,[.,[.,[[.,.],.]]]] => 28
[1,1,1,1,1,0,0,0,0,0] => [.,[.,[.,[.,[.,.]]]]] => 42
[1,0,1,0,1,0,1,0,1,0,1,0] => [[[[[[.,.],.],.],.],.],.] => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [[[[[.,.],.],.],.],[.,.]] => 2
[1,0,1,0,1,0,1,1,0,0,1,0] => [[[[[.,.],.],.],[.,.]],.] => 2
[1,0,1,0,1,0,1,1,0,1,0,0] => [[[[.,.],.],.],[[.,.],.]] => 3
[1,0,1,0,1,0,1,1,1,0,0,0] => [[[[.,.],.],.],[.,[.,.]]] => 5
[1,0,1,0,1,1,0,0,1,0,1,0] => [[[[[.,.],.],[.,.]],.],.] => 2
[1,0,1,0,1,1,0,0,1,1,0,0] => [[[[.,.],.],[.,.]],[.,.]] => 4
[1,0,1,0,1,1,0,1,0,0,1,0] => [[[[.,.],.],[[.,.],.]],.] => 3
[1,0,1,0,1,1,0,1,0,1,0,0] => [[[.,.],.],[[[.,.],.],.]] => 4
[1,0,1,0,1,1,0,1,1,0,0,0] => [[[.,.],.],[[.,.],[.,.]]] => 7
[1,0,1,0,1,1,1,0,0,0,1,0] => [[[[.,.],.],[.,[.,.]]],.] => 5
[1,0,1,0,1,1,1,0,0,1,0,0] => [[[.,.],.],[[.,[.,.]],.]] => 7
[1,0,1,0,1,1,1,0,1,0,0,0] => [[[.,.],.],[.,[[.,.],.]]] => 9
[1,0,1,0,1,1,1,1,0,0,0,0] => [[[.,.],.],[.,[.,[.,.]]]] => 14
[1,0,1,1,0,0,1,0,1,0,1,0] => [[[[[.,.],[.,.]],.],.],.] => 2
[1,0,1,1,0,0,1,0,1,1,0,0] => [[[[.,.],[.,.]],.],[.,.]] => 4
[1,0,1,1,0,0,1,1,0,0,1,0] => [[[[.,.],[.,.]],[.,.]],.] => 4
[1,0,1,1,0,0,1,1,0,1,0,0] => [[[.,.],[.,.]],[[.,.],.]] => 6
[1,0,1,1,0,0,1,1,1,0,0,0] => [[[.,.],[.,.]],[.,[.,.]]] => 10
[1,0,1,1,0,1,0,0,1,0,1,0] => [[[[.,.],[[.,.],.]],.],.] => 3
[1,0,1,1,0,1,0,0,1,1,0,0] => [[[.,.],[[.,.],.]],[.,.]] => 6
[1,0,1,1,0,1,0,1,0,0,1,0] => [[[.,.],[[[.,.],.],.]],.] => 4
[1,0,1,1,0,1,0,1,0,1,0,0] => [[.,.],[[[[.,.],.],.],.]] => 5
[1,0,1,1,0,1,0,1,1,0,0,0] => [[.,.],[[[.,.],.],[.,.]]] => 9
[1,0,1,1,0,1,1,0,0,0,1,0] => [[[.,.],[[.,.],[.,.]]],.] => 7
[1,0,1,1,0,1,1,0,0,1,0,0] => [[.,.],[[[.,.],[.,.]],.]] => 9
[1,0,1,1,0,1,1,0,1,0,0,0] => [[.,.],[[.,.],[[.,.],.]]] => 12
[1,0,1,1,0,1,1,1,0,0,0,0] => [[.,.],[[.,.],[.,[.,.]]]] => 19
[1,0,1,1,1,0,0,0,1,0,1,0] => [[[[.,.],[.,[.,.]]],.],.] => 5
[1,0,1,1,1,0,0,0,1,1,0,0] => [[[.,.],[.,[.,.]]],[.,.]] => 10
[1,0,1,1,1,0,0,1,0,0,1,0] => [[[.,.],[[.,[.,.]],.]],.] => 7
[1,0,1,1,1,0,0,1,0,1,0,0] => [[.,.],[[[.,[.,.]],.],.]] => 9
[1,0,1,1,1,0,0,1,1,0,0,0] => [[.,.],[[.,[.,.]],[.,.]]] => 16
[1,0,1,1,1,0,1,0,0,0,1,0] => [[[.,.],[.,[[.,.],.]]],.] => 9
[1,0,1,1,1,0,1,0,0,1,0,0] => [[.,.],[[.,[[.,.],.]],.]] => 12
[1,0,1,1,1,0,1,0,1,0,0,0] => [[.,.],[.,[[[.,.],.],.]]] => 14
[1,0,1,1,1,0,1,1,0,0,0,0] => [[.,.],[.,[[.,.],[.,.]]]] => 23
[1,0,1,1,1,1,0,0,0,0,1,0] => [[[.,.],[.,[.,[.,.]]]],.] => 14
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Description
The number of elements smaller than a binary tree in Tamari order.
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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