- FindStat

Identifier

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤ

Values

[1,0,1,0] => [1] => [1,0,1,0] => 1010 => 0

[1,0,1,0,1,0] => [2,1] => [1,0,1,0,1,0] => 101010 => 0

[1,0,1,1,0,0] => [1,1] => [1,0,1,1,0,0] => 101100 => 0

[1,1,0,0,1,0] => [2] => [1,1,0,0,1,0] => 110010 => 0

[1,1,0,1,0,0] => [1] => [1,0,1,0] => 1010 => 0

[1,0,1,0,1,0,1,0] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0

[1,0,1,0,1,1,0,0] => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0

[1,0,1,1,0,0,1,0] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0

[1,0,1,1,0,1,0,0] => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0

[1,0,1,1,1,0,0,0] => [1,1,1] => [1,0,1,1,1,0,0,0] => 10111000 => 0

[1,1,0,0,1,0,1,0] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0

[1,1,0,0,1,1,0,0] => [2,2] => [1,1,0,0,1,1,0,0] => 11001100 => 0

[1,1,0,1,0,0,1,0] => [3,1] => [1,1,0,1,0,0,1,0] => 11010010 => 0

[1,1,0,1,0,1,0,0] => [2,1] => [1,0,1,0,1,0] => 101010 => 0

[1,1,0,1,1,0,0,0] => [1,1] => [1,0,1,1,0,0] => 101100 => 0

[1,1,1,0,0,0,1,0] => [3] => [1,1,1,0,0,0,1,0] => 11100010 => 0

[1,1,1,0,0,1,0,0] => [2] => [1,1,0,0,1,0] => 110010 => 0

[1,1,1,0,1,0,0,0] => [1] => [1,0,1,0] => 1010 => 0

[1,1,0,1,0,1,0,1,0,0] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0

[1,1,0,1,0,1,1,0,0,0] => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0

[1,1,0,1,1,0,0,1,0,0] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0

[1,1,0,1,1,0,1,0,0,0] => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0

[1,1,0,1,1,1,0,0,0,0] => [1,1,1] => [1,0,1,1,1,0,0,0] => 10111000 => 0

[1,1,1,0,0,1,0,1,0,0] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0

[1,1,1,0,0,1,1,0,0,0] => [2,2] => [1,1,0,0,1,1,0,0] => 11001100 => 0

[1,1,1,0,1,0,0,1,0,0] => [3,1] => [1,1,0,1,0,0,1,0] => 11010010 => 0

[1,1,1,0,1,0,1,0,0,0] => [2,1] => [1,0,1,0,1,0] => 101010 => 0

[1,1,1,0,1,1,0,0,0,0] => [1,1] => [1,0,1,1,0,0] => 101100 => 0

[1,1,1,1,0,0,0,1,0,0] => [3] => [1,1,1,0,0,0,1,0] => 11100010 => 0

[1,1,1,1,0,0,1,0,0,0] => [2] => [1,1,0,0,1,0] => 110010 => 0

[1,1,1,1,0,1,0,0,0,0] => [1] => [1,0,1,0] => 1010 => 0

[1,1,1,0,1,0,1,0,1,0,0,0] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0

[1,1,1,0,1,0,1,1,0,0,0,0] => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0

[1,1,1,0,1,1,0,0,1,0,0,0] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0

[1,1,1,0,1,1,0,1,0,0,0,0] => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0

[1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,1] => [1,0,1,1,1,0,0,0] => 10111000 => 0

[1,1,1,1,0,0,1,0,1,0,0,0] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0

[1,1,1,1,0,0,1,1,0,0,0,0] => [2,2] => [1,1,0,0,1,1,0,0] => 11001100 => 0

[1,1,1,1,0,1,0,0,1,0,0,0] => [3,1] => [1,1,0,1,0,0,1,0] => 11010010 => 0

[1,1,1,1,0,1,0,1,0,0,0,0] => [2,1] => [1,0,1,0,1,0] => 101010 => 0

[1,1,1,1,0,1,1,0,0,0,0,0] => [1,1] => [1,0,1,1,0,0] => 101100 => 0

[1,1,1,1,1,0,0,0,1,0,0,0] => [3] => [1,1,1,0,0,0,1,0] => 11100010 => 0

[1,1,1,1,1,0,0,1,0,0,0,0] => [2] => [1,1,0,0,1,0] => 110010 => 0

[1,1,1,1,1,0,1,0,0,0,0,0] => [1] => [1,0,1,0] => 1010 => 0

[1,1,1,1,0,1,0,1,0,1,0,0,0,0] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0

[1,1,1,1,0,1,0,1,1,0,0,0,0,0] => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0

[1,1,1,1,0,1,1,0,0,1,0,0,0,0] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0

[1,1,1,1,0,1,1,0,1,0,0,0,0,0] => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0

[1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [1,1,1] => [1,0,1,1,1,0,0,0] => 10111000 => 0

[1,1,1,1,1,0,0,1,0,1,0,0,0,0] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0

[1,1,1,1,1,0,0,1,1,0,0,0,0,0] => [2,2] => [1,1,0,0,1,1,0,0] => 11001100 => 0

[1,1,1,1,1,0,1,0,0,1,0,0,0,0] => [3,1] => [1,1,0,1,0,0,1,0] => 11010010 => 0

[1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [2,1] => [1,0,1,0,1,0] => 101010 => 0

[1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [1,1] => [1,0,1,1,0,0] => 101100 => 0

[1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [3] => [1,1,1,0,0,0,1,0] => 11100010 => 0

[1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [2] => [1,1,0,0,1,0] => 110010 => 0

[1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [1] => [1,0,1,0] => 1010 => 0

[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0

[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0] => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0

[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0

[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0] => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0

[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0] => [1,1,1] => [1,0,1,1,1,0,0,0] => 10111000 => 0

[1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0

[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0] => [2,2] => [1,1,0,0,1,1,0,0] => 11001100 => 0

[1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0] => [3,1] => [1,1,0,1,0,0,1,0] => 11010010 => 0

[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [2,1] => [1,0,1,0,1,0] => 101010 => 0

[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0] => [1,1] => [1,0,1,1,0,0] => 101100 => 0

[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0] => [3] => [1,1,1,0,0,0,1,0] => 11100010 => 0

[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [2] => [1,1,0,0,1,0] => 110010 => 0

[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [1] => [1,0,1,0] => 1010 => 0

[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0

[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0] => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0

[1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0

[1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0] => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0

[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1] => [1,0,1,1,1,0,0,0] => 10111000 => 0

[1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0

[1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0] => [2,2] => [1,1,0,0,1,1,0,0] => 11001100 => 0

[1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0] => [3,1] => [1,1,0,1,0,0,1,0] => 11010010 => 0

[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0] => [2,1] => [1,0,1,0,1,0] => 101010 => 0

[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0] => [1,1] => [1,0,1,1,0,0] => 101100 => 0

[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0] => [3] => [1,1,1,0,0,0,1,0] => 11100010 => 0

[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0] => [2] => [1,1,0,0,1,0] => 110010 => 0

[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0] => [1] => [1,0,1,0] => 1010 => 0

[1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0] => [1,1] => [1,0,1,1,0,0] => 101100 => 0

[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0] => [2] => [1,1,0,0,1,0] => 110010 => 0

[1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0] => [1,1] => [1,0,1,1,0,0] => 101100 => 0

[1,1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,0] => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0

[1,1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0

[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0] => [2] => [1,1,0,0,1,0] => 110010 => 0

[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0] => [1] => [1,0,1,0] => 1010 => 0

[1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0] => [1,1,1] => [1,0,1,1,1,0,0,0] => 10111000 => 0

[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0] => [1] => [1,0,1,0] => 1010 => 0

[1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0] => [3] => [1,1,1,0,0,0,1,0] => 11100010 => 0

[1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0] => [2,2] => [1,1,0,0,1,1,0,0] => 11001100 => 0

[1,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0] => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0

[1,1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,0] => [3,1] => [1,1,0,1,0,0,1,0] => 11010010 => 0

[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0] => [2,1] => [1,0,1,0,1,0] => 101010 => 0

[1,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,0] => [2,1] => [1,0,1,0,1,0] => 101010 => 0

[1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0

[1,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,0] => [3] => [1,1,1,0,0,0,1,0] => 11100010 => 0

[1,1,1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,0,0] => [3,1] => [1,1,0,1,0,0,1,0] => 11010010 => 0

>>> Load all 107 entries. <<<

search for individual values

searching the database for the individual values of this statistic

Description

The number of times the path corresponding to a binary word crosses the base line.
Interpret each

$0$ as a step

$(1,-1)$ and

$1$ as a step

$(1,1)$ . Then this statistic counts the number of times the path crosses the

$x$ -axis.

Map

to binary word

Description

Return the Dyck word as binary word.

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

Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.