- FindStat

Identifier

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

1111 => [1,1,1,1,1] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

01111 => [2,1,1,1,1] => [2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => 2

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

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

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

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

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

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

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

10111 => [1,2,1,1,1] => [2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => 2

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

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

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

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

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

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

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

11111 => [1,1,1,1,1,1] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

001111 => [3,1,1,1,1] => [3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => 3

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

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

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

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

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

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

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

010111 => [2,2,1,1,1] => [2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => 2

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

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

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

011011 => [2,1,2,1,1] => [2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => 2

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

011101 => [2,1,1,2,1] => [2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => 2

011110 => [2,1,1,1,2] => [2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => 2

011111 => [2,1,1,1,1,1] => [2,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => 2

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

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

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

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

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

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

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

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Description

The number of odd rises of a Dyck path.
This is the number of ones at an odd position, with the initial position equal to 1.
The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].

Map

to partition

Description

Sends a composition to the partition obtained by sorting the entries.

Map

parallelogram polyomino

Description

Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.

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.