Identifier
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Mp00231:
Integer compositions
—bounce path⟶
Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000396: Binary trees ⟶ ℤ
Values
[1] => [1,0] => [.,.] => 1
[1,1] => [1,0,1,0] => [[.,.],.] => 1
[2] => [1,1,0,0] => [.,[.,.]] => 1
[1,1,1] => [1,0,1,0,1,0] => [[[.,.],.],.] => 1
[1,2] => [1,0,1,1,0,0] => [[.,.],[.,.]] => 2
[2,1] => [1,1,0,0,1,0] => [[.,[.,.]],.] => 1
[3] => [1,1,1,0,0,0] => [.,[.,[.,.]]] => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0] => [[[[.,.],.],.],.] => 1
[1,1,2] => [1,0,1,0,1,1,0,0] => [[[.,.],.],[.,.]] => 2
[1,2,1] => [1,0,1,1,0,0,1,0] => [[[.,.],[.,.]],.] => 2
[1,3] => [1,0,1,1,1,0,0,0] => [[.,.],[.,[.,.]]] => 2
[2,1,1] => [1,1,0,0,1,0,1,0] => [[[.,[.,.]],.],.] => 1
[2,2] => [1,1,0,0,1,1,0,0] => [[.,[.,.]],[.,.]] => 2
[3,1] => [1,1,1,0,0,0,1,0] => [[.,[.,[.,.]]],.] => 1
[4] => [1,1,1,1,0,0,0,0] => [.,[.,[.,[.,.]]]] => 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [[[[[.,.],.],.],.],.] => 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [[[[.,.],.],.],[.,.]] => 2
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [[[[.,.],.],[.,.]],.] => 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [[[.,.],.],[.,[.,.]]] => 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [[[[.,.],[.,.]],.],.] => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [[[.,.],[.,.]],[.,.]] => 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [[[.,.],[.,[.,.]]],.] => 2
[1,4] => [1,0,1,1,1,1,0,0,0,0] => [[.,.],[.,[.,[.,.]]]] => 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [[[[.,[.,.]],.],.],.] => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [[[.,[.,.]],.],[.,.]] => 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [[[.,[.,.]],[.,.]],.] => 2
[2,3] => [1,1,0,0,1,1,1,0,0,0] => [[.,[.,.]],[.,[.,.]]] => 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [[[.,[.,[.,.]]],.],.] => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0] => [[.,[.,[.,.]]],[.,.]] => 2
[4,1] => [1,1,1,1,0,0,0,0,1,0] => [[.,[.,[.,[.,.]]]],.] => 1
[5] => [1,1,1,1,1,0,0,0,0,0] => [.,[.,[.,[.,[.,.]]]]] => 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[[[[[.,.],.],.],.],.],.] => 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [[[[[.,.],.],.],.],[.,.]] => 2
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [[[[[.,.],.],.],[.,.]],.] => 2
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [[[[.,.],.],.],[.,[.,.]]] => 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [[[[[.,.],.],[.,.]],.],.] => 2
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [[[[.,.],.],[.,.]],[.,.]] => 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [[[[.,.],.],[.,[.,.]]],.] => 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [[[.,.],.],[.,[.,[.,.]]]] => 2
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [[[[[.,.],[.,.]],.],.],.] => 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [[[[.,.],[.,.]],.],[.,.]] => 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [[[[.,.],[.,.]],[.,.]],.] => 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [[[.,.],[.,.]],[.,[.,.]]] => 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [[[[.,.],[.,[.,.]]],.],.] => 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [[[.,.],[.,[.,.]]],[.,.]] => 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [[[.,.],[.,[.,[.,.]]]],.] => 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [[.,.],[.,[.,[.,[.,.]]]]] => 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [[[[[.,[.,.]],.],.],.],.] => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [[[[.,[.,.]],.],.],[.,.]] => 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [[[[.,[.,.]],.],[.,.]],.] => 2
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [[[.,[.,.]],.],[.,[.,.]]] => 2
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [[[[.,[.,.]],[.,.]],.],.] => 2
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [[[.,[.,.]],[.,.]],[.,.]] => 2
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [[[.,[.,.]],[.,[.,.]]],.] => 2
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [[.,[.,.]],[.,[.,[.,.]]]] => 2
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [[[[.,[.,[.,.]]],.],.],.] => 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [[[.,[.,[.,.]]],.],[.,.]] => 2
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [[[.,[.,[.,.]]],[.,.]],.] => 2
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [[.,[.,[.,.]]],[.,[.,.]]] => 2
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [[[.,[.,[.,[.,.]]]],.],.] => 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [[.,[.,[.,[.,.]]]],[.,.]] => 2
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [[.,[.,[.,[.,[.,.]]]]],.] => 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [.,[.,[.,[.,[.,[.,.]]]]]] => 1
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [[[[[[[.,.],.],.],.],.],.],.] => 1
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [[[[[[.,.],.],.],.],.],[.,.]] => 2
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [[[[[[.,.],.],.],.],[.,.]],.] => 2
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [[[[[.,.],.],.],.],[.,[.,.]]] => 2
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [[[[[[.,.],.],.],[.,.]],.],.] => 2
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [[[[[.,.],.],.],[.,.]],[.,.]] => 2
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0] => [[[[[.,.],.],.],[.,[.,.]]],.] => 2
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [[[[.,.],.],.],[.,[.,[.,.]]]] => 2
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [[[[[[.,.],.],[.,.]],.],.],.] => 2
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [[[[[.,.],.],[.,.]],.],[.,.]] => 2
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [[[[[.,.],.],[.,.]],[.,.]],.] => 2
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0] => [[[[.,.],.],[.,.]],[.,[.,.]]] => 2
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0] => [[[[[.,.],.],[.,[.,.]]],.],.] => 2
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => [[[[.,.],.],[.,[.,.]]],[.,.]] => 2
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [[[[.,.],.],[.,[.,[.,.]]]],.] => 2
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [[[.,.],.],[.,[.,[.,[.,.]]]]] => 2
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [[[[[[.,.],[.,.]],.],.],.],.] => 2
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [[[[[.,.],[.,.]],.],.],[.,.]] => 2
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [[[[[.,.],[.,.]],.],[.,.]],.] => 2
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0] => [[[[.,.],[.,.]],.],[.,[.,.]]] => 2
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [[[[[.,.],[.,.]],[.,.]],.],.] => 2
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [[[[.,.],[.,.]],[.,.]],[.,.]] => 2
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [[[[.,.],[.,.]],[.,[.,.]]],.] => 2
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => [[[.,.],[.,.]],[.,[.,[.,.]]]] => 2
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0] => [[[[[.,.],[.,[.,.]]],.],.],.] => 2
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => [[[[.,.],[.,[.,.]]],.],[.,.]] => 2
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [[[[.,.],[.,[.,.]]],[.,.]],.] => 2
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => [[[.,.],[.,[.,.]]],[.,[.,.]]] => 2
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0] => [[[[.,.],[.,[.,[.,.]]]],.],.] => 2
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => [[[.,.],[.,[.,[.,.]]]],[.,.]] => 2
[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [[[.,.],[.,[.,[.,[.,.]]]]],.] => 2
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [[.,.],[.,[.,[.,[.,[.,.]]]]]] => 2
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [[[[[[.,[.,.]],.],.],.],.],.] => 1
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [[[[[.,[.,.]],.],.],.],[.,.]] => 2
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [[[[[.,[.,.]],.],.],[.,.]],.] => 2
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [[[[.,[.,.]],.],.],[.,[.,.]]] => 2
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [[[[[.,[.,.]],.],[.,.]],.],.] => 2
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [[[[.,[.,.]],.],[.,.]],[.,.]] => 2
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Description
The register function (or Horton-Strahler number) of a binary tree.
This is different from the dimension of the associated poset for the tree $[[[.,.],[.,.]],[[.,.],[.,.]]]$: its register function is 3, whereas the dimension of the associated poset is 2.
This is different from the dimension of the associated poset for the tree $[[[.,.],[.,.]],[[.,.],[.,.]]]$: its register function is 3, whereas the dimension of the associated poset is 2.
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.
Map
bounce path
Description
The bounce path determined by an integer composition.
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