- FindStat

Identifier

Mp00057: Parking functions —to touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001099: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

[1,2,2] => [1,2] => [1,1] => [1,1] => 1

[2,1,2] => [1,2] => [1,1] => [1,1] => 1

[2,2,1] => [1,2] => [1,1] => [1,1] => 1

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

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

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

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

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

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

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

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

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

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

[1,1,2,4] => [3,1] => [1,1] => [1,1] => 1

[1,1,4,2] => [3,1] => [1,1] => [1,1] => 1

[1,2,1,4] => [3,1] => [1,1] => [1,1] => 1

[1,2,4,1] => [3,1] => [1,1] => [1,1] => 1

[1,4,1,2] => [3,1] => [1,1] => [1,1] => 1

[1,4,2,1] => [3,1] => [1,1] => [1,1] => 1

[2,1,1,4] => [3,1] => [1,1] => [1,1] => 1

[2,1,4,1] => [3,1] => [1,1] => [1,1] => 1

[2,4,1,1] => [3,1] => [1,1] => [1,1] => 1

[4,1,1,2] => [3,1] => [1,1] => [1,1] => 1

[4,1,2,1] => [3,1] => [1,1] => [1,1] => 1

[4,2,1,1] => [3,1] => [1,1] => [1,1] => 1

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

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

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

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

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

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

[1,1,3,4] => [2,1,1] => [1,2] => [2,1] => 2

[1,1,4,3] => [2,1,1] => [1,2] => [2,1] => 2

[1,3,1,4] => [2,1,1] => [1,2] => [2,1] => 2

[1,3,4,1] => [2,1,1] => [1,2] => [2,1] => 2

[1,4,1,3] => [2,1,1] => [1,2] => [2,1] => 2

[1,4,3,1] => [2,1,1] => [1,2] => [2,1] => 2

[3,1,1,4] => [2,1,1] => [1,2] => [2,1] => 2

[3,1,4,1] => [2,1,1] => [1,2] => [2,1] => 2

[3,4,1,1] => [2,1,1] => [1,2] => [2,1] => 2

[4,1,1,3] => [2,1,1] => [1,2] => [2,1] => 2

[4,1,3,1] => [2,1,1] => [1,2] => [2,1] => 2

[4,3,1,1] => [2,1,1] => [1,2] => [2,1] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,2,2,4] => [1,2,1] => [1,1,1] => [1,1,1] => 3

[1,2,4,2] => [1,2,1] => [1,1,1] => [1,1,1] => 3

[1,4,2,2] => [1,2,1] => [1,1,1] => [1,1,1] => 3

[2,1,2,4] => [1,2,1] => [1,1,1] => [1,1,1] => 3

[2,1,4,2] => [1,2,1] => [1,1,1] => [1,1,1] => 3

[2,2,1,4] => [1,2,1] => [1,1,1] => [1,1,1] => 3

[2,2,4,1] => [1,2,1] => [1,1,1] => [1,1,1] => 3

[2,4,1,2] => [1,2,1] => [1,1,1] => [1,1,1] => 3

[2,4,2,1] => [1,2,1] => [1,1,1] => [1,1,1] => 3

[4,1,2,2] => [1,2,1] => [1,1,1] => [1,1,1] => 3

[4,2,1,2] => [1,2,1] => [1,1,1] => [1,1,1] => 3

[4,2,2,1] => [1,2,1] => [1,1,1] => [1,1,1] => 3

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

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

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

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

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

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

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

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

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

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

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

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

[1,2,3,4] => [1,1,1,1] => [4] => [4] => 0

[1,2,4,3] => [1,1,1,1] => [4] => [4] => 0

[1,3,2,4] => [1,1,1,1] => [4] => [4] => 0

[1,3,4,2] => [1,1,1,1] => [4] => [4] => 0

[1,4,2,3] => [1,1,1,1] => [4] => [4] => 0

[1,4,3,2] => [1,1,1,1] => [4] => [4] => 0

[2,1,3,4] => [1,1,1,1] => [4] => [4] => 0

[2,1,4,3] => [1,1,1,1] => [4] => [4] => 0

[2,3,1,4] => [1,1,1,1] => [4] => [4] => 0

[2,3,4,1] => [1,1,1,1] => [4] => [4] => 0

[2,4,1,3] => [1,1,1,1] => [4] => [4] => 0

[2,4,3,1] => [1,1,1,1] => [4] => [4] => 0

[3,1,2,4] => [1,1,1,1] => [4] => [4] => 0

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search for individual values

searching the database for the individual values of this statistic

Description

The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees.
For a generating function

$f$ the associated formal group law is the symmetric function

$f(f^{(-1)}(x_1) + f^{(-1)}(x_2), \dots)$ , see [1].
This statistic records the coefficient of the monomial symmetric function

$m_\lambda$ times the product of the factorials of the parts of

$\lambda$ in the formal group law for leaf labelled binary trees, with generating function

$f(x) = 1-\sqrt{1-2x}$ , see [1, sec. 3.2]
Fix a set of distinguishable vertices and a coloring of the vertices so that

$\lambda_i$ are colored

$i$ . This statistic gives the number of rooted binary trees with leaves labeled with this set of vertices and internal vertices unlabeled so that no pair of 'twin' leaves have the same color.

Map

delta morphism

Description

Apply the delta morphism to an integer composition.
The delta morphism of a composition

$C$ is the compositions composed of the lengths of consecutive runs of the same integer in

$C$ .

Map

to touch composition

Description

Sends a parking function to the composition of its labelled Dyck path.

Map

to partition

Description

Sends a composition to the partition obtained by sorting the entries.