- FindStat

Identifier

Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000161: Binary trees ⟶ ℤ

Values

[1] => [1,0] => [.,.] => 0

[1,1] => [1,1,0,0] => [.,[.,.]] => 1

[1,2] => [1,0,1,0] => [[.,.],.] => 0

[2,1] => [1,0,1,0] => [[.,.],.] => 0

[1,1,1] => [1,1,1,0,0,0] => [.,[.,[.,.]]] => 3

[1,1,2] => [1,1,0,1,0,0] => [.,[[.,.],.]] => 2

[1,2,1] => [1,1,0,1,0,0] => [.,[[.,.],.]] => 2

[2,1,1] => [1,1,0,1,0,0] => [.,[[.,.],.]] => 2

[1,1,3] => [1,1,0,0,1,0] => [[.,[.,.]],.] => 1

[1,3,1] => [1,1,0,0,1,0] => [[.,[.,.]],.] => 1

[3,1,1] => [1,1,0,0,1,0] => [[.,[.,.]],.] => 1

[1,2,2] => [1,0,1,1,0,0] => [[.,.],[.,.]] => 1

[2,1,2] => [1,0,1,1,0,0] => [[.,.],[.,.]] => 1

[2,2,1] => [1,0,1,1,0,0] => [[.,.],[.,.]] => 1

[1,2,3] => [1,0,1,0,1,0] => [[[.,.],.],.] => 0

[1,3,2] => [1,0,1,0,1,0] => [[[.,.],.],.] => 0

[2,1,3] => [1,0,1,0,1,0] => [[[.,.],.],.] => 0

[2,3,1] => [1,0,1,0,1,0] => [[[.,.],.],.] => 0

[3,1,2] => [1,0,1,0,1,0] => [[[.,.],.],.] => 0

[3,2,1] => [1,0,1,0,1,0] => [[[.,.],.],.] => 0

[1,1,1,1] => [1,1,1,1,0,0,0,0] => [.,[.,[.,[.,.]]]] => 6

[1,1,1,2] => [1,1,1,0,1,0,0,0] => [.,[.,[[.,.],.]]] => 5

[1,1,2,1] => [1,1,1,0,1,0,0,0] => [.,[.,[[.,.],.]]] => 5

[1,2,1,1] => [1,1,1,0,1,0,0,0] => [.,[.,[[.,.],.]]] => 5

[2,1,1,1] => [1,1,1,0,1,0,0,0] => [.,[.,[[.,.],.]]] => 5

[1,1,1,3] => [1,1,1,0,0,1,0,0] => [.,[[.,[.,.]],.]] => 4

[1,1,3,1] => [1,1,1,0,0,1,0,0] => [.,[[.,[.,.]],.]] => 4

[1,3,1,1] => [1,1,1,0,0,1,0,0] => [.,[[.,[.,.]],.]] => 4

[3,1,1,1] => [1,1,1,0,0,1,0,0] => [.,[[.,[.,.]],.]] => 4

[1,1,1,4] => [1,1,1,0,0,0,1,0] => [[.,[.,[.,.]]],.] => 3

[1,1,4,1] => [1,1,1,0,0,0,1,0] => [[.,[.,[.,.]]],.] => 3

[1,4,1,1] => [1,1,1,0,0,0,1,0] => [[.,[.,[.,.]]],.] => 3

[4,1,1,1] => [1,1,1,0,0,0,1,0] => [[.,[.,[.,.]]],.] => 3

[1,1,2,2] => [1,1,0,1,1,0,0,0] => [.,[[.,.],[.,.]]] => 4

[1,2,1,2] => [1,1,0,1,1,0,0,0] => [.,[[.,.],[.,.]]] => 4

[1,2,2,1] => [1,1,0,1,1,0,0,0] => [.,[[.,.],[.,.]]] => 4

[2,1,1,2] => [1,1,0,1,1,0,0,0] => [.,[[.,.],[.,.]]] => 4

[2,1,2,1] => [1,1,0,1,1,0,0,0] => [.,[[.,.],[.,.]]] => 4

[2,2,1,1] => [1,1,0,1,1,0,0,0] => [.,[[.,.],[.,.]]] => 4

[1,1,2,3] => [1,1,0,1,0,1,0,0] => [.,[[[.,.],.],.]] => 3

[1,1,3,2] => [1,1,0,1,0,1,0,0] => [.,[[[.,.],.],.]] => 3

[1,2,1,3] => [1,1,0,1,0,1,0,0] => [.,[[[.,.],.],.]] => 3

[1,2,3,1] => [1,1,0,1,0,1,0,0] => [.,[[[.,.],.],.]] => 3

[1,3,1,2] => [1,1,0,1,0,1,0,0] => [.,[[[.,.],.],.]] => 3

[1,3,2,1] => [1,1,0,1,0,1,0,0] => [.,[[[.,.],.],.]] => 3

[2,1,1,3] => [1,1,0,1,0,1,0,0] => [.,[[[.,.],.],.]] => 3

[2,1,3,1] => [1,1,0,1,0,1,0,0] => [.,[[[.,.],.],.]] => 3

[2,3,1,1] => [1,1,0,1,0,1,0,0] => [.,[[[.,.],.],.]] => 3

[3,1,1,2] => [1,1,0,1,0,1,0,0] => [.,[[[.,.],.],.]] => 3

[3,1,2,1] => [1,1,0,1,0,1,0,0] => [.,[[[.,.],.],.]] => 3

[3,2,1,1] => [1,1,0,1,0,1,0,0] => [.,[[[.,.],.],.]] => 3

[1,1,2,4] => [1,1,0,1,0,0,1,0] => [[.,[[.,.],.]],.] => 2

[1,1,4,2] => [1,1,0,1,0,0,1,0] => [[.,[[.,.],.]],.] => 2

[1,2,1,4] => [1,1,0,1,0,0,1,0] => [[.,[[.,.],.]],.] => 2

[1,2,4,1] => [1,1,0,1,0,0,1,0] => [[.,[[.,.],.]],.] => 2

[1,4,1,2] => [1,1,0,1,0,0,1,0] => [[.,[[.,.],.]],.] => 2

[1,4,2,1] => [1,1,0,1,0,0,1,0] => [[.,[[.,.],.]],.] => 2

[2,1,1,4] => [1,1,0,1,0,0,1,0] => [[.,[[.,.],.]],.] => 2

[2,1,4,1] => [1,1,0,1,0,0,1,0] => [[.,[[.,.],.]],.] => 2

[2,4,1,1] => [1,1,0,1,0,0,1,0] => [[.,[[.,.],.]],.] => 2

[4,1,1,2] => [1,1,0,1,0,0,1,0] => [[.,[[.,.],.]],.] => 2

[4,1,2,1] => [1,1,0,1,0,0,1,0] => [[.,[[.,.],.]],.] => 2

[4,2,1,1] => [1,1,0,1,0,0,1,0] => [[.,[[.,.],.]],.] => 2

[1,1,3,3] => [1,1,0,0,1,1,0,0] => [[.,[.,.]],[.,.]] => 2

[1,3,1,3] => [1,1,0,0,1,1,0,0] => [[.,[.,.]],[.,.]] => 2

[1,3,3,1] => [1,1,0,0,1,1,0,0] => [[.,[.,.]],[.,.]] => 2

[3,1,1,3] => [1,1,0,0,1,1,0,0] => [[.,[.,.]],[.,.]] => 2

[3,1,3,1] => [1,1,0,0,1,1,0,0] => [[.,[.,.]],[.,.]] => 2

[3,3,1,1] => [1,1,0,0,1,1,0,0] => [[.,[.,.]],[.,.]] => 2

[1,1,3,4] => [1,1,0,0,1,0,1,0] => [[[.,[.,.]],.],.] => 1

[1,1,4,3] => [1,1,0,0,1,0,1,0] => [[[.,[.,.]],.],.] => 1

[1,3,1,4] => [1,1,0,0,1,0,1,0] => [[[.,[.,.]],.],.] => 1

[1,3,4,1] => [1,1,0,0,1,0,1,0] => [[[.,[.,.]],.],.] => 1

[1,4,1,3] => [1,1,0,0,1,0,1,0] => [[[.,[.,.]],.],.] => 1

[1,4,3,1] => [1,1,0,0,1,0,1,0] => [[[.,[.,.]],.],.] => 1

[3,1,1,4] => [1,1,0,0,1,0,1,0] => [[[.,[.,.]],.],.] => 1

[3,1,4,1] => [1,1,0,0,1,0,1,0] => [[[.,[.,.]],.],.] => 1

[3,4,1,1] => [1,1,0,0,1,0,1,0] => [[[.,[.,.]],.],.] => 1

[4,1,1,3] => [1,1,0,0,1,0,1,0] => [[[.,[.,.]],.],.] => 1

[4,1,3,1] => [1,1,0,0,1,0,1,0] => [[[.,[.,.]],.],.] => 1

[4,3,1,1] => [1,1,0,0,1,0,1,0] => [[[.,[.,.]],.],.] => 1

[1,2,2,2] => [1,0,1,1,1,0,0,0] => [[.,.],[.,[.,.]]] => 3

[2,1,2,2] => [1,0,1,1,1,0,0,0] => [[.,.],[.,[.,.]]] => 3

[2,2,1,2] => [1,0,1,1,1,0,0,0] => [[.,.],[.,[.,.]]] => 3

[2,2,2,1] => [1,0,1,1,1,0,0,0] => [[.,.],[.,[.,.]]] => 3

[1,2,2,3] => [1,0,1,1,0,1,0,0] => [[.,.],[[.,.],.]] => 2

[1,2,3,2] => [1,0,1,1,0,1,0,0] => [[.,.],[[.,.],.]] => 2

[1,3,2,2] => [1,0,1,1,0,1,0,0] => [[.,.],[[.,.],.]] => 2

[2,1,2,3] => [1,0,1,1,0,1,0,0] => [[.,.],[[.,.],.]] => 2

[2,1,3,2] => [1,0,1,1,0,1,0,0] => [[.,.],[[.,.],.]] => 2

[2,2,1,3] => [1,0,1,1,0,1,0,0] => [[.,.],[[.,.],.]] => 2

[2,2,3,1] => [1,0,1,1,0,1,0,0] => [[.,.],[[.,.],.]] => 2

[2,3,1,2] => [1,0,1,1,0,1,0,0] => [[.,.],[[.,.],.]] => 2

[2,3,2,1] => [1,0,1,1,0,1,0,0] => [[.,.],[[.,.],.]] => 2

[3,1,2,2] => [1,0,1,1,0,1,0,0] => [[.,.],[[.,.],.]] => 2

[3,2,1,2] => [1,0,1,1,0,1,0,0] => [[.,.],[[.,.],.]] => 2

[3,2,2,1] => [1,0,1,1,0,1,0,0] => [[.,.],[[.,.],.]] => 2

[1,2,2,4] => [1,0,1,1,0,0,1,0] => [[[.,.],[.,.]],.] => 1

[1,2,4,2] => [1,0,1,1,0,0,1,0] => [[[.,.],[.,.]],.] => 1

[1,4,2,2] => [1,0,1,1,0,0,1,0] => [[[.,.],[.,.]],.] => 1

[2,1,2,4] => [1,0,1,1,0,0,1,0] => [[[.,.],[.,.]],.] => 1

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$F_{1} = 1$

$F_{2} = 2 + q$

$F_{3} = 6 + 6\ q + 3\ q^{2} + q^{3}$

$F_{4} = 24 + 36\ q + 30\ q^{2} + 20\ q^{3} + 10\ q^{4} + 4\ q^{5} + q^{6}$

$F_{5} = 120 + 240\ q + 270\ q^{2} + 240\ q^{3} + 180\ q^{4} + 120\ q^{5} + 70\ q^{6} + 35\ q^{7} + 15\ q^{8} + 5\ q^{9} + q^{10}$

$F_{6} = 720 + 1800\ q + 2520\ q^{2} + 2730\ q^{3} + 2520\ q^{4} + 2100\ q^{5} + 1610\ q^{6} + 1140\ q^{7} + 750\ q^{8} + 455\ q^{9} + 252\ q^{10} + 126\ q^{11} + 56\ q^{12} + 21\ q^{13} + 6\ q^{14} + q^{15}$

Description

The sum of the sizes of the right subtrees of a binary tree.
This statistic corresponds to St000012The area of a Dyck path. under the Tamari Dyck path-binary tree bijection, and to St000018The number of inversions of a permutation. of the

$312$ -avoiding permutation corresponding to the binary tree.
It is also the sum of all heights

$j$ of the coordinates

$(i,j)$ of the Dyck path corresponding to the binary tree.

Map

to Dyck path

Description

Sends a parking function to its support Dyck word.

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.