Identifier
-
Mp00027:
Dyck paths
—to partition⟶
Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000045: Binary trees ⟶ ℤ
Values
[1,0,1,0,1,0] => [2,1] => [1,0,1,1,0,0] => [.,[[.,.],.]] => 1
[1,0,1,0,1,0,1,0] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [.,[[[.,.],[.,.]],.]] => 2
[1,0,1,0,1,1,0,0] => [2,2,1] => [1,1,1,0,0,1,0,0] => [[[.,.],[.,.]],.] => 2
[1,0,1,1,0,0,1,0] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [.,[.,[[.,[.,.]],.]]] => 1
[1,0,1,1,0,1,0,0] => [2,1,1] => [1,0,1,1,0,1,0,0] => [.,[[.,[.,.]],.]] => 1
[1,0,1,1,1,0,0,0] => [1,1,1] => [1,1,0,1,0,0] => [[.,[.,.]],.] => 1
[1,1,0,0,1,0,1,0] => [3,2] => [1,0,1,1,1,0,0,0] => [.,[[[.,.],.],.]] => 1
[1,1,0,0,1,1,0,0] => [2,2] => [1,1,1,0,0,0] => [[[.,.],.],.] => 1
[1,1,0,1,0,0,1,0] => [3,1] => [1,0,1,0,1,1,0,0] => [.,[.,[[.,.],.]]] => 1
[1,1,0,1,0,1,0,0] => [2,1] => [1,0,1,1,0,0] => [.,[[.,.],.]] => 1
[1,1,1,0,0,0,1,0] => [3] => [1,0,1,0,1,0] => [.,[.,[.,.]]] => 1
[1,0,1,0,1,0,1,1,0,0] => [3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => [[[.,[[.,.],.]],[.,.]],.] => 4
[1,0,1,0,1,1,0,1,0,0] => [3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [.,[[[[.,.],.],[.,.]],.]] => 3
[1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [[[[.,.],.],[.,.]],.] => 3
[1,0,1,1,0,0,1,1,0,0] => [3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => [[[.,[.,.]],[.,[.,.]]],.] => 6
[1,0,1,1,0,1,0,1,0,0] => [3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [.,[[[.,.],[.,[.,.]]],.]] => 3
[1,0,1,1,0,1,1,0,0,0] => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [[[.,.],[.,[.,.]]],.] => 3
[1,0,1,1,1,0,0,1,0,0] => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [.,[.,[[.,[.,[.,.]]],.]]] => 1
[1,0,1,1,1,0,1,0,0,0] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [.,[[.,[.,[.,.]]],.]] => 1
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [[.,[.,[.,.]]],.] => 1
[1,1,0,0,1,0,1,0,1,0] => [4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => [.,[[[.,[[.,.],.]],.],.]] => 1
[1,1,0,0,1,0,1,1,0,0] => [3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [[[.,[[.,.],.]],.],.] => 1
[1,1,0,0,1,1,0,0,1,0] => [4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [.,[.,[[[[.,.],.],.],.]]] => 1
[1,1,0,0,1,1,0,1,0,0] => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [.,[[[[.,.],.],.],.]] => 1
[1,1,0,0,1,1,1,0,0,0] => [2,2,2] => [1,1,1,1,0,0,0,0] => [[[[.,.],.],.],.] => 1
[1,1,0,1,0,0,1,0,1,0] => [4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [.,[[[.,[.,.]],[.,.]],.]] => 3
[1,1,0,1,0,0,1,1,0,0] => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [[[.,[.,.]],[.,.]],.] => 3
[1,1,0,1,0,1,0,0,1,0] => [4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [.,[.,[[[.,.],[.,.]],.]]] => 2
[1,1,0,1,0,1,0,1,0,0] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [.,[[[.,.],[.,.]],.]] => 2
[1,1,0,1,0,1,1,0,0,0] => [2,2,1] => [1,1,1,0,0,1,0,0] => [[[.,.],[.,.]],.] => 2
[1,1,0,1,1,0,0,0,1,0] => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [.,[.,[.,[[.,[.,.]],.]]]] => 1
[1,1,0,1,1,0,0,1,0,0] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [.,[.,[[.,[.,.]],.]]] => 1
[1,1,0,1,1,0,1,0,0,0] => [2,1,1] => [1,0,1,1,0,1,0,0] => [.,[[.,[.,.]],.]] => 1
[1,1,0,1,1,1,0,0,0,0] => [1,1,1] => [1,1,0,1,0,0] => [[.,[.,.]],.] => 1
[1,1,1,0,0,0,1,0,1,0] => [4,3] => [1,0,1,1,1,0,1,0,0,0] => [.,[[[.,[.,.]],.],.]] => 1
[1,1,1,0,0,0,1,1,0,0] => [3,3] => [1,1,1,0,1,0,0,0] => [[[.,[.,.]],.],.] => 1
[1,1,1,0,0,1,0,0,1,0] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => [.,[.,[[[.,.],.],.]]] => 1
[1,1,1,0,0,1,0,1,0,0] => [3,2] => [1,0,1,1,1,0,0,0] => [.,[[[.,.],.],.]] => 1
[1,1,1,0,0,1,1,0,0,0] => [2,2] => [1,1,1,0,0,0] => [[[.,.],.],.] => 1
[1,1,1,0,1,0,0,0,1,0] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [.,[.,[.,[[.,.],.]]]] => 1
[1,1,1,0,1,0,0,1,0,0] => [3,1] => [1,0,1,0,1,1,0,0] => [.,[.,[[.,.],.]]] => 1
[1,1,1,0,1,0,1,0,0,0] => [2,1] => [1,0,1,1,0,0] => [.,[[.,.],.]] => 1
[1,1,1,1,0,0,0,0,1,0] => [4] => [1,0,1,0,1,0,1,0] => [.,[.,[.,[.,.]]]] => 1
[1,1,1,1,0,0,0,1,0,0] => [3] => [1,0,1,0,1,0] => [.,[.,[.,.]]] => 1
[1,0,1,0,1,1,1,1,0,0,0,0] => [2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => [[[[.,[.,.]],.],[.,.]],.] => 4
[1,0,1,1,0,1,1,1,0,0,0,0] => [2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => [[[[.,.],.],[.,[.,.]]],.] => 6
[1,0,1,1,1,0,1,1,0,0,0,0] => [2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [[[.,.],[.,[.,[.,.]]]],.] => 4
[1,0,1,1,1,1,0,1,0,0,0,0] => [2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [.,[[.,[.,[.,[.,.]]]],.]] => 1
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [[.,[.,[.,[.,.]]]],.] => 1
[1,1,0,0,1,0,1,1,1,0,0,0] => [3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => [[[[[.,.],[.,.]],.],.],.] => 2
[1,1,0,0,1,1,0,1,1,0,0,0] => [3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => [[[.,[[.,[.,.]],.]],.],.] => 1
[1,1,0,0,1,1,1,0,1,0,0,0] => [3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [.,[[[[.,[.,.]],.],.],.]] => 1
[1,1,0,0,1,1,1,1,0,0,0,0] => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [[[[.,[.,.]],.],.],.] => 1
[1,1,0,1,0,0,1,1,1,0,0,0] => [3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [[[[[.,.],.],.],[.,.]],.] => 4
[1,1,0,1,0,1,0,1,1,0,0,0] => [3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => [[[.,[[.,.],.]],[.,.]],.] => 4
[1,1,0,1,0,1,1,0,1,0,0,0] => [3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [.,[[[[.,.],.],[.,.]],.]] => 3
[1,1,0,1,0,1,1,1,0,0,0,0] => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [[[[.,.],.],[.,.]],.] => 3
[1,1,0,1,1,0,0,1,1,0,0,0] => [3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => [[[.,[.,.]],[.,[.,.]]],.] => 6
[1,1,0,1,1,0,1,0,1,0,0,0] => [3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [.,[[[.,.],[.,[.,.]]],.]] => 3
[1,1,0,1,1,0,1,1,0,0,0,0] => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [[[.,.],[.,[.,.]]],.] => 3
[1,1,0,1,1,1,0,0,1,0,0,0] => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [.,[.,[[.,[.,[.,.]]],.]]] => 1
[1,1,0,1,1,1,0,1,0,0,0,0] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [.,[[.,[.,[.,.]]],.]] => 1
[1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [[.,[.,[.,.]]],.] => 1
[1,1,1,0,0,0,1,0,1,1,0,0] => [4,4,3] => [1,1,1,0,1,1,1,0,0,0,0,0] => [[[.,[[[.,.],.],.]],.],.] => 1
[1,1,1,0,0,0,1,1,0,1,0,0] => [4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => [.,[[[[[.,.],.],.],.],.]] => 1
[1,1,1,0,0,0,1,1,1,0,0,0] => [3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [[[[[.,.],.],.],.],.] => 1
[1,1,1,0,0,1,0,0,1,1,0,0] => [4,4,2] => [1,1,1,0,1,0,1,1,0,0,0,0] => [[[.,[.,[[.,.],.]]],.],.] => 1
[1,1,1,0,0,1,0,1,0,1,0,0] => [4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => [.,[[[.,[[.,.],.]],.],.]] => 1
[1,1,1,0,0,1,0,1,1,0,0,0] => [3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [[[.,[[.,.],.]],.],.] => 1
[1,1,1,0,0,1,1,0,0,1,0,0] => [4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [.,[.,[[[[.,.],.],.],.]]] => 1
[1,1,1,0,0,1,1,0,1,0,0,0] => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [.,[[[[.,.],.],.],.]] => 1
[1,1,1,0,0,1,1,1,0,0,0,0] => [2,2,2] => [1,1,1,1,0,0,0,0] => [[[[.,.],.],.],.] => 1
[1,1,1,0,1,0,0,0,1,1,0,0] => [4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => [[[.,[.,[.,.]]],[.,.]],.] => 4
[1,1,1,0,1,0,0,1,0,1,0,0] => [4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [.,[[[.,[.,.]],[.,.]],.]] => 3
[1,1,1,0,1,0,0,1,1,0,0,0] => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [[[.,[.,.]],[.,.]],.] => 3
[1,1,1,0,1,0,1,0,0,1,0,0] => [4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [.,[.,[[[.,.],[.,.]],.]]] => 2
[1,1,1,0,1,0,1,0,1,0,0,0] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [.,[[[.,.],[.,.]],.]] => 2
[1,1,1,0,1,0,1,1,0,0,0,0] => [2,2,1] => [1,1,1,0,0,1,0,0] => [[[.,.],[.,.]],.] => 2
[1,1,1,0,1,1,0,0,0,1,0,0] => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [.,[.,[.,[[.,[.,.]],.]]]] => 1
[1,1,1,0,1,1,0,0,1,0,0,0] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [.,[.,[[.,[.,.]],.]]] => 1
[1,1,1,0,1,1,0,1,0,0,0,0] => [2,1,1] => [1,0,1,1,0,1,0,0] => [.,[[.,[.,.]],.]] => 1
[1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,1] => [1,1,0,1,0,0] => [[.,[.,.]],.] => 1
[1,1,1,1,0,0,0,0,1,0,1,0] => [5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => [.,[[[.,[.,[.,.]]],.],.]] => 1
[1,1,1,1,0,0,0,0,1,1,0,0] => [4,4] => [1,1,1,0,1,0,1,0,0,0] => [[[.,[.,[.,.]]],.],.] => 1
[1,1,1,1,0,0,0,1,0,0,1,0] => [5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [.,[.,[[[.,[.,.]],.],.]]] => 1
[1,1,1,1,0,0,0,1,0,1,0,0] => [4,3] => [1,0,1,1,1,0,1,0,0,0] => [.,[[[.,[.,.]],.],.]] => 1
[1,1,1,1,0,0,0,1,1,0,0,0] => [3,3] => [1,1,1,0,1,0,0,0] => [[[.,[.,.]],.],.] => 1
[1,1,1,1,0,0,1,0,0,0,1,0] => [5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [.,[.,[.,[[[.,.],.],.]]]] => 1
[1,1,1,1,0,0,1,0,0,1,0,0] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => [.,[.,[[[.,.],.],.]]] => 1
[1,1,1,1,0,0,1,0,1,0,0,0] => [3,2] => [1,0,1,1,1,0,0,0] => [.,[[[.,.],.],.]] => 1
[1,1,1,1,0,0,1,1,0,0,0,0] => [2,2] => [1,1,1,0,0,0] => [[[.,.],.],.] => 1
[1,1,1,1,0,1,0,0,0,0,1,0] => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [.,[.,[.,[.,[[.,.],.]]]]] => 1
[1,1,1,1,0,1,0,0,0,1,0,0] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [.,[.,[.,[[.,.],.]]]] => 1
[1,1,1,1,0,1,0,0,1,0,0,0] => [3,1] => [1,0,1,0,1,1,0,0] => [.,[.,[[.,.],.]]] => 1
[1,1,1,1,0,1,0,1,0,0,0,0] => [2,1] => [1,0,1,1,0,0] => [.,[[.,.],.]] => 1
[1,1,1,1,1,0,0,0,0,0,1,0] => [5] => [1,0,1,0,1,0,1,0,1,0] => [.,[.,[.,[.,[.,.]]]]] => 1
[1,1,1,1,1,0,0,0,0,1,0,0] => [4] => [1,0,1,0,1,0,1,0] => [.,[.,[.,[.,.]]]] => 1
[1,1,1,1,1,0,0,0,1,0,0,0] => [3] => [1,0,1,0,1,0] => [.,[.,[.,.]]] => 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [[.,[.,[.,[.,[.,.]]]]],.] => 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [[[[.,[.,[.,.]]],.],.],.] => 1
[1,1,0,1,0,1,1,1,1,0,0,0,0,0] => [2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => [[[[.,[.,.]],.],[.,.]],.] => 4
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searching the database for the individual values of this statistic
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.
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: 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.
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.
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
to partition
Description
The cut-out partition of a Dyck path.
The partition $\lambda$ associated to a Dyck path is defined to be the complementary partition inside the staircase partition $(n-1,\ldots,2,1)$ when cutting out $D$ considered as a path from $(0,0)$ to $(n,n)$.
In other words, $\lambda_{i}$ is the number of down-steps before the $(n+1-i)$-th up-step of $D$.
This map is a bijection between Dyck paths of size $n$ and partitions inside the staircase partition $(n-1,\ldots,2,1)$.
The partition $\lambda$ associated to a Dyck path is defined to be the complementary partition inside the staircase partition $(n-1,\ldots,2,1)$ when cutting out $D$ considered as a path from $(0,0)$ to $(n,n)$.
In other words, $\lambda_{i}$ is the number of down-steps before the $(n+1-i)$-th up-step of $D$.
This map is a bijection between Dyck paths of size $n$ and partitions inside the staircase partition $(n-1,\ldots,2,1)$.
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