- FindStat

Identifier

Mp00297: Parking functions —ordered tree⟶ Ordered trees
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000929: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 16668 entries. <<<

search for individual values

searching the database for the individual values of this statistic

Description

The constant term of the character polynomial of an integer partition.
The definition of the character polynomial can be found in [1]. Indeed, this constant term is

$0$ for partitions

$\lambda \neq 1^n$ and

$1$ for

$\lambda = 1^n$ .

Map

ordered tree

Description

The ordered tree associated with the parking function by Pollak.
Let

$(p_1,\dots,p_n)$ be a parking function. Set

$c_i = p_{i+1}-p_i \mod (n+1)$ to obtain a Prüfer code, and return the underlying ordered tree rooted at

$0$ .

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

$(n-1,\ldots,2,1)$ .

Map

to Dyck path

Description

Return the Dyck path of the corresponding ordered tree induced by the recurrence of the Catalan numbers, see wikipedia:Catalan_number.
This sends the maximal height of the Dyck path to the depth of the tree.