- FindStat

Identifier

Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1] => 111111 => [1,1,1,1,1,1,1] => 1

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

[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,2,1] => 111101 => [1,1,1,1,2,1] => 1

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

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

[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,2,1,1] => 111011 => [1,1,1,2,1,1] => 1

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

[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,3,1] => 111001 => [1,1,1,3,1] => 1

[1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,4] => 111000 => [1,1,1,4] => 4

[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,4] => 111000 => [1,1,1,4] => 4

[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,3,1] => 111001 => [1,1,1,3,1] => 1

[1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,4] => 111000 => [1,1,1,4] => 4

[1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,4] => 111000 => [1,1,1,4] => 4

[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,4] => 111000 => [1,1,1,4] => 4

[1,0,1,1,0,0,1,0,1,0,1,0] => [1,2,1,1,1] => 110111 => [1,1,2,1,1,1] => 1

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

[1,0,1,1,0,0,1,1,0,0,1,0] => [1,2,2,1] => 110101 => [1,1,2,2,1] => 1

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

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

[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,1,1] => 110011 => [1,1,3,1,1] => 1

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

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

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

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

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

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

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

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

[1,0,1,1,1,0,0,0,1,0,1,0] => [1,3,1,1] => 110011 => [1,1,3,1,1] => 1

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

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

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

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

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

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

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

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

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Description

The last part of an integer composition.

Map

touch composition

Description

Sends a Dyck path to its touch composition given by the composition of lengths of its touch points.

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 binary word

Description

Return the composition as a binary word, treating ones as separators.
Encoding a positive integer $i$ as the word $10\dots 0$ consisting of a one followed by $i-1$ zeros, the binary word of a composition $(i_1,\dots,i_k)$ is the concatenation of of words for $i_1,\dots,i_k$.
The image of this map contains precisely the words which do not begin with a $0$.