- FindStat

Identifier

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤ

Values

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

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

00 => [3] => [1,1,1,0,0,0] => [] => 0

01 => [2,1] => [1,1,0,0,1,0] => [2] => 1

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

11 => [1,1,1] => [1,0,1,0,1,0] => [2,1] => 2

000 => [4] => [1,1,1,1,0,0,0,0] => [] => 0

001 => [3,1] => [1,1,1,0,0,0,1,0] => [3] => 1

010 => [2,2] => [1,1,0,0,1,1,0,0] => [2,2] => 2

011 => [2,1,1] => [1,1,0,0,1,0,1,0] => [3,2] => 2

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

101 => [1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,1] => 3

110 => [1,1,2] => [1,0,1,0,1,1,0,0] => [2,2,1] => 3

111 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [3,2,1] => 3

0000 => [5] => [1,1,1,1,1,0,0,0,0,0] => [] => 0

0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [4] => 1

0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [3,3] => 2

0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [4,3] => 2

0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,2,2] => 3

0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [4,2,2] => 3

0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [3,3,2] => 3

0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [4,3,2] => 3

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

1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => 4

1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,3,1,1] => 4

1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [4,3,1,1] => 4

1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => 4

1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,2,2,1] => 4

1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [3,3,2,1] => 4

1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [4,3,2,1] => 4

00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [] => 0

00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [5] => 1

00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [4,4] => 2

00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [5,4] => 2

00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [3,3,3] => 3

00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [5,3,3] => 3

00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [4,4,3] => 3

00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [5,4,3] => 3

01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [2,2,2,2] => 4

01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [5,2,2,2] => 4

01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [4,4,2,2] => 4

01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [5,4,2,2] => 4

01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [3,3,3,2] => 4

01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [5,3,3,2] => 4

01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [4,4,3,2] => 4

01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [5,4,3,2] => 4

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

10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [5,1,1,1,1] => 5

10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [4,4,1,1,1] => 5

10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [5,4,1,1,1] => 5

10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [3,3,3,1,1] => 5

10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [5,3,3,1,1] => 5

10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [4,4,3,1,1] => 5

10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,1] => 5

11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [2,2,2,2,1] => 5

11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [5,2,2,2,1] => 5

11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [4,4,2,2,1] => 5

11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [5,4,2,2,1] => 5

11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [3,3,3,2,1] => 5

11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [5,3,3,2,1] => 5

11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [4,4,3,2,1] => 5

11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => 5

000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [] => 0

000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [6] => 1

000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [5,5] => 2

000011 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [6,5] => 2

000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [4,4,4] => 3

000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => [6,4,4] => 3

000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [5,5,4] => 3

000111 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [6,5,4] => 3

001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [3,3,3,3] => 4

001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [6,3,3,3] => 4

001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [5,5,3,3] => 4

001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [6,5,3,3] => 4

001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0] => [4,4,4,3] => 4

001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0] => [6,4,4,3] => 4

001110 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0] => [5,5,4,3] => 4

010000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [2,2,2,2,2] => 5

010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => [6,2,2,2,2] => 5

010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => [5,5,2,2,2] => 5

010011 => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,2,2] => 5

010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => [4,4,4,2,2] => 5

011000 => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0] => [3,3,3,3,2] => 5

011001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0] => [6,3,3,3,2] => 5

011100 => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [4,4,4,3,2] => 5

100000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,1] => 6

100001 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [6,1,1,1,1,1] => 6

100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => [5,5,1,1,1,1] => 6

100011 => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0] => [6,5,1,1,1,1] => 6

100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => [4,4,4,1,1,1] => 6

100101 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [6,4,4,1,1,1] => 6

100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => [5,5,4,1,1,1] => 6

101000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => [3,3,3,3,1,1] => 6

101001 => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [6,3,3,3,1,1] => 6

101100 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0] => [4,4,4,3,1,1] => 6

110000 => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [2,2,2,2,2,1] => 6

110001 => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [6,2,2,2,2,1] => 6

110010 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => [5,5,2,2,2,1] => 6

110100 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0] => [4,4,4,2,2,1] => 6

111000 => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [3,3,3,3,2,1] => 6

0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [] => 0

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Description

The length of the partition.

Map

bounce path

Description

The bounce path determined by 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 $(n-1,\ldots,2,1)$.