- FindStat

Identifier

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤ

Values

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

[1,0,1,0] => [1] => 10 => [1,2] => 1

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

[1,0,1,0,1,0] => [2,1] => 1010 => [1,2,2] => 1

[1,0,1,1,0,0] => [1,1] => 110 => [1,1,2] => 1

[1,1,0,0,1,0] => [2] => 100 => [1,3] => 1

[1,1,0,1,0,0] => [1] => 10 => [1,2] => 1

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

[1,0,1,0,1,0,1,0] => [3,2,1] => 101010 => [1,2,2,2] => 1

[1,0,1,0,1,1,0,0] => [2,2,1] => 11010 => [1,1,2,2] => 1

[1,0,1,1,0,0,1,0] => [3,1,1] => 100110 => [1,3,1,2] => 1

[1,0,1,1,0,1,0,0] => [2,1,1] => 10110 => [1,2,1,2] => 1

[1,0,1,1,1,0,0,0] => [1,1,1] => 1110 => [1,1,1,2] => 1

[1,1,0,0,1,0,1,0] => [3,2] => 10100 => [1,2,3] => 1

[1,1,0,0,1,1,0,0] => [2,2] => 1100 => [1,1,3] => 1

[1,1,0,1,0,0,1,0] => [3,1] => 10010 => [1,3,2] => 1

[1,1,0,1,0,1,0,0] => [2,1] => 1010 => [1,2,2] => 1

[1,1,0,1,1,0,0,0] => [1,1] => 110 => [1,1,2] => 1

[1,1,1,0,0,0,1,0] => [3] => 1000 => [1,4] => 1

[1,1,1,0,0,1,0,0] => [2] => 100 => [1,3] => 1

[1,1,1,0,1,0,0,0] => [1] => 10 => [1,2] => 1

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

[1,0,1,0,1,0,1,0,1,0] => [4,3,2,1] => 10101010 => [1,2,2,2,2] => 1

[1,0,1,0,1,0,1,1,0,0] => [3,3,2,1] => 1101010 => [1,1,2,2,2] => 1

[1,0,1,0,1,1,0,0,1,0] => [4,2,2,1] => 10011010 => [1,3,1,2,2] => 1

[1,0,1,0,1,1,0,1,0,0] => [3,2,2,1] => 1011010 => [1,2,1,2,2] => 1

[1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => 111010 => [1,1,1,2,2] => 1

[1,0,1,1,0,0,1,0,1,0] => [4,3,1,1] => 10100110 => [1,2,3,1,2] => 1

[1,0,1,1,0,0,1,1,0,0] => [3,3,1,1] => 1100110 => [1,1,3,1,2] => 1

[1,0,1,1,0,1,0,0,1,0] => [4,2,1,1] => 10010110 => [1,3,2,1,2] => 1

[1,0,1,1,0,1,0,1,0,0] => [3,2,1,1] => 1010110 => [1,2,2,1,2] => 1

[1,0,1,1,0,1,1,0,0,0] => [2,2,1,1] => 110110 => [1,1,2,1,2] => 1

[1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => 10001110 => [1,4,1,1,2] => 1

[1,0,1,1,1,0,0,1,0,0] => [3,1,1,1] => 1001110 => [1,3,1,1,2] => 1

[1,0,1,1,1,0,1,0,0,0] => [2,1,1,1] => 101110 => [1,2,1,1,2] => 1

[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => 11110 => [1,1,1,1,2] => 1

[1,1,0,0,1,0,1,0,1,0] => [4,3,2] => 1010100 => [1,2,2,3] => 1

[1,1,0,0,1,0,1,1,0,0] => [3,3,2] => 110100 => [1,1,2,3] => 1

[1,1,0,0,1,1,0,0,1,0] => [4,2,2] => 1001100 => [1,3,1,3] => 1

[1,1,0,0,1,1,0,1,0,0] => [3,2,2] => 101100 => [1,2,1,3] => 1

[1,1,0,0,1,1,1,0,0,0] => [2,2,2] => 11100 => [1,1,1,3] => 1

[1,1,0,1,0,0,1,0,1,0] => [4,3,1] => 1010010 => [1,2,3,2] => 1

[1,1,0,1,0,0,1,1,0,0] => [3,3,1] => 110010 => [1,1,3,2] => 1

[1,1,0,1,0,1,0,0,1,0] => [4,2,1] => 1001010 => [1,3,2,2] => 1

[1,1,0,1,0,1,0,1,0,0] => [3,2,1] => 101010 => [1,2,2,2] => 1

[1,1,0,1,0,1,1,0,0,0] => [2,2,1] => 11010 => [1,1,2,2] => 1

[1,1,0,1,1,0,0,0,1,0] => [4,1,1] => 1000110 => [1,4,1,2] => 1

[1,1,0,1,1,0,0,1,0,0] => [3,1,1] => 100110 => [1,3,1,2] => 1

[1,1,0,1,1,0,1,0,0,0] => [2,1,1] => 10110 => [1,2,1,2] => 1

[1,1,0,1,1,1,0,0,0,0] => [1,1,1] => 1110 => [1,1,1,2] => 1

[1,1,1,0,0,0,1,0,1,0] => [4,3] => 101000 => [1,2,4] => 1

[1,1,1,0,0,0,1,1,0,0] => [3,3] => 11000 => [1,1,4] => 1

[1,1,1,0,0,1,0,0,1,0] => [4,2] => 100100 => [1,3,3] => 1

[1,1,1,0,0,1,0,1,0,0] => [3,2] => 10100 => [1,2,3] => 1

[1,1,1,0,0,1,1,0,0,0] => [2,2] => 1100 => [1,1,3] => 1

[1,1,1,0,1,0,0,0,1,0] => [4,1] => 100010 => [1,4,2] => 1

[1,1,1,0,1,0,0,1,0,0] => [3,1] => 10010 => [1,3,2] => 1

[1,1,1,0,1,0,1,0,0,0] => [2,1] => 1010 => [1,2,2] => 1

[1,1,1,0,1,1,0,0,0,0] => [1,1] => 110 => [1,1,2] => 1

[1,1,1,1,0,0,0,0,1,0] => [4] => 10000 => [1,5] => 1

[1,1,1,1,0,0,0,1,0,0] => [3] => 1000 => [1,4] => 1

[1,1,1,1,0,0,1,0,0,0] => [2] => 100 => [1,3] => 1

[1,1,1,1,0,1,0,0,0,0] => [1] => 10 => [1,2] => 1

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

[1,0,1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => 1010101010 => [1,2,2,2,2,2] => 1

[1,0,1,0,1,0,1,0,1,1,0,0] => [4,4,3,2,1] => 110101010 => [1,1,2,2,2,2] => 1

[1,0,1,0,1,0,1,1,1,0,0,0] => [3,3,3,2,1] => 11101010 => [1,1,1,2,2,2] => 1

[1,0,1,0,1,1,0,1,1,0,0,0] => [3,3,2,2,1] => 11011010 => [1,1,2,1,2,2] => 1

[1,0,1,0,1,1,1,0,1,0,0,0] => [3,2,2,2,1] => 10111010 => [1,2,1,1,2,2] => 1

[1,0,1,0,1,1,1,1,0,0,0,0] => [2,2,2,2,1] => 1111010 => [1,1,1,1,2,2] => 1

[1,0,1,1,0,0,1,1,1,0,0,0] => [3,3,3,1,1] => 11100110 => [1,1,1,3,1,2] => 1

[1,0,1,1,0,1,0,0,1,1,0,0] => [4,4,2,1,1] => 110010110 => [1,1,3,2,1,2] => 1

[1,0,1,1,0,1,0,1,1,0,0,0] => [3,3,2,1,1] => 11010110 => [1,1,2,2,1,2] => 1

[1,0,1,1,0,1,1,0,0,1,0,0] => [4,2,2,1,1] => 100110110 => [1,3,1,2,1,2] => 1

[1,0,1,1,0,1,1,0,1,0,0,0] => [3,2,2,1,1] => 10110110 => [1,2,1,2,1,2] => 1

[1,0,1,1,0,1,1,1,0,0,0,0] => [2,2,2,1,1] => 1110110 => [1,1,1,2,1,2] => 1

[1,0,1,1,1,0,0,1,0,1,0,0] => [4,3,1,1,1] => 101001110 => [1,2,3,1,1,2] => 1

[1,0,1,1,1,0,0,1,1,0,0,0] => [3,3,1,1,1] => 11001110 => [1,1,3,1,1,2] => 1

[1,0,1,1,1,0,1,0,0,0,1,0] => [5,2,1,1,1] => 1000101110 => [1,4,2,1,1,2] => 1

[1,0,1,1,1,0,1,0,0,1,0,0] => [4,2,1,1,1] => 100101110 => [1,3,2,1,1,2] => 1

[1,0,1,1,1,0,1,0,1,0,0,0] => [3,2,1,1,1] => 10101110 => [1,2,2,1,1,2] => 1

[1,0,1,1,1,0,1,1,0,0,0,0] => [2,2,1,1,1] => 1101110 => [1,1,2,1,1,2] => 1

[1,0,1,1,1,1,0,0,0,0,1,0] => [5,1,1,1,1] => 1000011110 => [1,5,1,1,1,2] => 1

[1,0,1,1,1,1,0,0,0,1,0,0] => [4,1,1,1,1] => 100011110 => [1,4,1,1,1,2] => 1

[1,0,1,1,1,1,0,0,1,0,0,0] => [3,1,1,1,1] => 10011110 => [1,3,1,1,1,2] => 1

[1,0,1,1,1,1,0,1,0,0,0,0] => [2,1,1,1,1] => 1011110 => [1,2,1,1,1,2] => 1

[1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1] => 111110 => [1,1,1,1,1,2] => 1

[1,1,0,0,1,0,1,0,1,0,1,0] => [5,4,3,2] => 101010100 => [1,2,2,2,3] => 1

[1,1,0,0,1,0,1,0,1,1,0,0] => [4,4,3,2] => 11010100 => [1,1,2,2,3] => 1

[1,1,0,0,1,0,1,1,0,1,0,0] => [4,3,3,2] => 10110100 => [1,2,1,2,3] => 1

[1,1,0,0,1,0,1,1,1,0,0,0] => [3,3,3,2] => 1110100 => [1,1,1,2,3] => 1

[1,1,0,0,1,1,0,0,1,1,0,0] => [4,4,2,2] => 11001100 => [1,1,3,1,3] => 1

[1,1,0,0,1,1,0,1,0,1,0,0] => [4,3,2,2] => 10101100 => [1,2,2,1,3] => 1

[1,1,0,0,1,1,0,1,1,0,0,0] => [3,3,2,2] => 1101100 => [1,1,2,1,3] => 1

[1,1,0,0,1,1,1,0,0,1,0,0] => [4,2,2,2] => 10011100 => [1,3,1,1,3] => 1

[1,1,0,0,1,1,1,0,1,0,0,0] => [3,2,2,2] => 1011100 => [1,2,1,1,3] => 1

[1,1,0,0,1,1,1,1,0,0,0,0] => [2,2,2,2] => 111100 => [1,1,1,1,3] => 1

[1,1,0,1,0,0,1,0,1,0,1,0] => [5,4,3,1] => 101010010 => [1,2,2,3,2] => 1

[1,1,0,1,0,0,1,0,1,1,0,0] => [4,4,3,1] => 11010010 => [1,1,2,3,2] => 1

[1,1,0,1,0,0,1,1,0,1,0,0] => [4,3,3,1] => 10110010 => [1,2,1,3,2] => 1

[1,1,0,1,0,0,1,1,1,0,0,0] => [3,3,3,1] => 1110010 => [1,1,1,3,2] => 1

>>> Load all 902 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{1} = q$

$F_{2} = 2\ q$

$F_{3} = 5\ q$

$F_{4} = 14\ q$

$F_{5} = 42\ q$

Description

The first part of an integer composition.

Map

to composition

Description

The composition corresponding to a binary word.
Prepending

$1$ to a binary word

$w$ , the

$i$ -th part of the composition equals

$1$ plus the number of zeros after the

$i$ -th

$1$ in

$w$ .
This map is not surjective, since the empty composition does not have a preimage.

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 binary word

Description

Return the partition as binary word, by traversing its shape from the first row to the last row, down steps as 1 and left steps as 0.