Identifier
Values
[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] => [[[.,.],.],[.,.]] => 3
[1,2,1] => [1,0,1,1,0,0,1,0] => [[[.,.],[.,.]],.] => 2
[1,3] => [1,0,1,1,1,0,0,0] => [[.,.],[.,[.,.]]] => 3
[2,1,1] => [1,1,0,0,1,0,1,0] => [[[.,[.,.]],.],.] => 1
[2,2] => [1,1,0,0,1,1,0,0] => [[.,[.,.]],[.,.]] => 3
[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] => [[[[.,.],.],.],[.,.]] => 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [[[[.,.],.],[.,.]],.] => 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [[[.,.],.],[.,[.,.]]] => 6
[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] => [[[.,.],[.,.]],[.,.]] => 8
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [[[.,.],[.,[.,.]]],.] => 3
[1,4] => [1,0,1,1,1,1,0,0,0,0] => [[.,.],[.,[.,[.,.]]]] => 4
[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] => [[[.,[.,.]],.],[.,.]] => 4
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [[[.,[.,.]],[.,.]],.] => 3
[2,3] => [1,1,0,0,1,1,1,0,0,0] => [[.,[.,.]],[.,[.,.]]] => 6
[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] => [[.,[.,[.,.]]],[.,.]] => 4
[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] => [[[[[.,.],.],.],.],[.,.]] => 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [[[[[.,.],.],.],[.,.]],.] => 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [[[[.,.],.],.],[.,[.,.]]] => 10
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [[[[[.,.],.],[.,.]],.],.] => 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [[[[.,.],.],[.,.]],[.,.]] => 15
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [[[[.,.],.],[.,[.,.]]],.] => 6
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [[[.,.],.],[.,[.,[.,.]]]] => 10
[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] => [[[[.,.],[.,.]],.],[.,.]] => 10
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [[[[.,.],[.,.]],[.,.]],.] => 8
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [[[.,.],[.,.]],[.,[.,.]]] => 20
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [[[[.,.],[.,[.,.]]],.],.] => 3
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [[[.,.],[.,[.,.]]],[.,.]] => 15
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [[[.,.],[.,[.,[.,.]]]],.] => 4
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [[.,.],[.,[.,[.,[.,.]]]]] => 5
[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] => [[[[.,[.,.]],.],.],[.,.]] => 5
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [[[[.,[.,.]],.],[.,.]],.] => 4
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [[[.,[.,.]],.],[.,[.,.]]] => 10
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [[[[.,[.,.]],[.,.]],.],.] => 3
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [[[.,[.,.]],[.,.]],[.,.]] => 15
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [[[.,[.,.]],[.,[.,.]]],.] => 6
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [[.,[.,.]],[.,[.,[.,.]]]] => 10
[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] => [[[.,[.,[.,.]]],.],[.,.]] => 5
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [[[.,[.,[.,.]]],[.,.]],.] => 4
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [[.,[.,[.,.]]],[.,[.,.]]] => 10
[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] => [[.,[.,[.,[.,.]]]],[.,.]] => 5
[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
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Description
The number of linear extensions of a binary tree.
Also, the number of increasing / decreasing binary trees labelled by $1, \dots, n$ of this shape.
Also, the size of the sylvester class corresponding to this tree when the Tamari order is seen as a quotient poset of the right weak order on permutations.
Also, the number of permutations which give this tree shape when inserted in a binary search tree.
Also, the number of permutations which increasing / decreasing tree is of this shape.
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.
Map
bounce path
Description
The bounce path determined by an integer composition.