- FindStat

Identifier

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000759: Integer partitions ⟶ ℤ

Values

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

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

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

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] => 3

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

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] => 1

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

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

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

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] => 4

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

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] => 1

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

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

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

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

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

1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => 2

1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => 2

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

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

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

1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,2,2,1] => 3

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] => 5

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

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] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1] => 2

10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [5,1,1,1,1] => 2

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

10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [5,4,1,1,1] => 2

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

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

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

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

11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [2,2,2,2,1] => 3

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

11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [4,4,2,2,1] => 3

11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [5,4,2,2,1] => 3

11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [3,3,3,2,1] => 4

11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [5,3,3,2,1] => 4

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] => 6

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

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] => 1

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

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

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

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

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

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

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

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

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

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

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

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

001111 => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [6,5,4,3] => 1

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

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

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

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

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

010101 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [6,4,4,2,2] => 1

010110 => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0] => [5,5,4,2,2] => 1

010111 => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,2] => 1

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

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

011010 => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [5,5,3,3,2] => 1

011011 => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [6,5,3,3,2] => 1

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

011101 => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [6,4,4,3,2] => 1

011110 => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [5,5,4,3,2] => 1

011111 => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2] => 1

100000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,1] => 2

100001 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [6,1,1,1,1,1] => 2

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

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] => 2

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

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] => 2

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] => 2

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Description

The smallest missing part in an integer partition.
In [3], this is referred to as the mex, the minimal excluded part of the partition.
For compositions, this is studied in [sec.3.2., 1].

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

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.