- FindStat

Identifier

Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000983: Binary words ⟶ ℤ

Values

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

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

[1,1,0,0] => [2] => 10 => 01 => 2

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

[1,0,1,1,0,0] => [1,2] => 110 => 011 => 2

[1,1,0,0,1,0] => [2,1] => 101 => 001 => 2

[1,1,0,1,0,0] => [2,1] => 101 => 001 => 2

[1,1,1,0,0,0] => [3] => 100 => 101 => 3

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

[1,0,1,0,1,1,0,0] => [1,1,2] => 1110 => 0111 => 2

[1,0,1,1,0,0,1,0] => [1,2,1] => 1101 => 0011 => 2

[1,0,1,1,0,1,0,0] => [1,2,1] => 1101 => 0011 => 2

[1,0,1,1,1,0,0,0] => [1,3] => 1100 => 1011 => 3

[1,1,0,0,1,0,1,0] => [2,1,1] => 1011 => 0001 => 2

[1,1,0,0,1,1,0,0] => [2,2] => 1010 => 1001 => 2

[1,1,0,1,0,0,1,0] => [2,1,1] => 1011 => 0001 => 2

[1,1,0,1,0,1,0,0] => [2,1,1] => 1011 => 0001 => 2

[1,1,0,1,1,0,0,0] => [2,2] => 1010 => 1001 => 2

[1,1,1,0,0,0,1,0] => [3,1] => 1001 => 1101 => 3

[1,1,1,0,0,1,0,0] => [3,1] => 1001 => 1101 => 3

[1,1,1,0,1,0,0,0] => [3,1] => 1001 => 1101 => 3

[1,1,1,1,0,0,0,0] => [4] => 1000 => 0101 => 4

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

[1,0,1,0,1,0,1,1,0,0] => [1,1,1,2] => 11110 => 01111 => 2

[1,0,1,0,1,1,0,0,1,0] => [1,1,2,1] => 11101 => 00111 => 2

[1,0,1,0,1,1,0,1,0,0] => [1,1,2,1] => 11101 => 00111 => 2

[1,0,1,0,1,1,1,0,0,0] => [1,1,3] => 11100 => 10111 => 3

[1,0,1,1,0,0,1,0,1,0] => [1,2,1,1] => 11011 => 00011 => 2

[1,0,1,1,0,0,1,1,0,0] => [1,2,2] => 11010 => 10011 => 2

[1,0,1,1,0,1,0,0,1,0] => [1,2,1,1] => 11011 => 00011 => 2

[1,0,1,1,0,1,0,1,0,0] => [1,2,1,1] => 11011 => 00011 => 2

[1,0,1,1,0,1,1,0,0,0] => [1,2,2] => 11010 => 10011 => 2

[1,0,1,1,1,0,0,0,1,0] => [1,3,1] => 11001 => 11011 => 3

[1,0,1,1,1,0,0,1,0,0] => [1,3,1] => 11001 => 11011 => 3

[1,0,1,1,1,0,1,0,0,0] => [1,3,1] => 11001 => 11011 => 3

[1,0,1,1,1,1,0,0,0,0] => [1,4] => 11000 => 01011 => 4

[1,1,0,0,1,0,1,0,1,0] => [2,1,1,1] => 10111 => 00001 => 2

[1,1,0,0,1,0,1,1,0,0] => [2,1,2] => 10110 => 10001 => 2

[1,1,0,0,1,1,0,0,1,0] => [2,2,1] => 10101 => 11001 => 2

[1,1,0,0,1,1,0,1,0,0] => [2,2,1] => 10101 => 11001 => 2

[1,1,0,0,1,1,1,0,0,0] => [2,3] => 10100 => 01001 => 3

[1,1,0,1,0,0,1,0,1,0] => [2,1,1,1] => 10111 => 00001 => 2

[1,1,0,1,0,0,1,1,0,0] => [2,1,2] => 10110 => 10001 => 2

[1,1,0,1,0,1,0,0,1,0] => [2,1,1,1] => 10111 => 00001 => 2

[1,1,0,1,0,1,0,1,0,0] => [2,1,1,1] => 10111 => 00001 => 2

[1,1,0,1,0,1,1,0,0,0] => [2,1,2] => 10110 => 10001 => 2

[1,1,0,1,1,0,0,0,1,0] => [2,2,1] => 10101 => 11001 => 2

[1,1,0,1,1,0,0,1,0,0] => [2,2,1] => 10101 => 11001 => 2

[1,1,0,1,1,0,1,0,0,0] => [2,2,1] => 10101 => 11001 => 2

[1,1,0,1,1,1,0,0,0,0] => [2,3] => 10100 => 01001 => 3

[1,1,1,0,0,0,1,0,1,0] => [3,1,1] => 10011 => 11101 => 3

[1,1,1,0,0,0,1,1,0,0] => [3,2] => 10010 => 01101 => 3

[1,1,1,0,0,1,0,0,1,0] => [3,1,1] => 10011 => 11101 => 3

[1,1,1,0,0,1,0,1,0,0] => [3,1,1] => 10011 => 11101 => 3

[1,1,1,0,0,1,1,0,0,0] => [3,2] => 10010 => 01101 => 3

[1,1,1,0,1,0,0,0,1,0] => [3,1,1] => 10011 => 11101 => 3

[1,1,1,0,1,0,0,1,0,0] => [3,1,1] => 10011 => 11101 => 3

[1,1,1,0,1,0,1,0,0,0] => [3,1,1] => 10011 => 11101 => 3

[1,1,1,0,1,1,0,0,0,0] => [3,2] => 10010 => 01101 => 3

[1,1,1,1,0,0,0,0,1,0] => [4,1] => 10001 => 00101 => 4

[1,1,1,1,0,0,0,1,0,0] => [4,1] => 10001 => 00101 => 4

[1,1,1,1,0,0,1,0,0,0] => [4,1] => 10001 => 00101 => 4

[1,1,1,1,0,1,0,0,0,0] => [4,1] => 10001 => 00101 => 4

[1,1,1,1,1,0,0,0,0,0] => [5] => 10000 => 10101 => 5

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

[1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,2] => 111110 => 011111 => 2

[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,2,1] => 111101 => 001111 => 2

[1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,2,1] => 111101 => 001111 => 2

[1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,3] => 111100 => 101111 => 3

[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,2,1,1] => 111011 => 000111 => 2

[1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,2,2] => 111010 => 100111 => 2

[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,2,1,1] => 111011 => 000111 => 2

[1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,2,1,1] => 111011 => 000111 => 2

[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,2,2] => 111010 => 100111 => 2

[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,3,1] => 111001 => 110111 => 3

[1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,3,1] => 111001 => 110111 => 3

[1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,3,1] => 111001 => 110111 => 3

[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,4] => 111000 => 010111 => 4

[1,0,1,1,0,0,1,0,1,0,1,0] => [1,2,1,1,1] => 110111 => 000011 => 2

[1,0,1,1,0,0,1,0,1,1,0,0] => [1,2,1,2] => 110110 => 100011 => 2

[1,0,1,1,0,0,1,1,0,0,1,0] => [1,2,2,1] => 110101 => 110011 => 2

[1,0,1,1,0,0,1,1,0,1,0,0] => [1,2,2,1] => 110101 => 110011 => 2

[1,0,1,1,0,0,1,1,1,0,0,0] => [1,2,3] => 110100 => 010011 => 3

[1,0,1,1,0,1,0,0,1,0,1,0] => [1,2,1,1,1] => 110111 => 000011 => 2

[1,0,1,1,0,1,0,0,1,1,0,0] => [1,2,1,2] => 110110 => 100011 => 2

[1,0,1,1,0,1,0,1,0,0,1,0] => [1,2,1,1,1] => 110111 => 000011 => 2

[1,0,1,1,0,1,0,1,0,1,0,0] => [1,2,1,1,1] => 110111 => 000011 => 2

[1,0,1,1,0,1,0,1,1,0,0,0] => [1,2,1,2] => 110110 => 100011 => 2

[1,0,1,1,0,1,1,0,0,0,1,0] => [1,2,2,1] => 110101 => 110011 => 2

[1,0,1,1,0,1,1,0,0,1,0,0] => [1,2,2,1] => 110101 => 110011 => 2

[1,0,1,1,0,1,1,0,1,0,0,0] => [1,2,2,1] => 110101 => 110011 => 2

[1,0,1,1,0,1,1,1,0,0,0,0] => [1,2,3] => 110100 => 010011 => 3

[1,0,1,1,1,0,0,0,1,0,1,0] => [1,3,1,1] => 110011 => 111011 => 3

[1,0,1,1,1,0,0,0,1,1,0,0] => [1,3,2] => 110010 => 011011 => 3

[1,0,1,1,1,0,0,1,0,0,1,0] => [1,3,1,1] => 110011 => 111011 => 3

[1,0,1,1,1,0,0,1,0,1,0,0] => [1,3,1,1] => 110011 => 111011 => 3

[1,0,1,1,1,0,0,1,1,0,0,0] => [1,3,2] => 110010 => 011011 => 3

[1,0,1,1,1,0,1,0,0,0,1,0] => [1,3,1,1] => 110011 => 111011 => 3

[1,0,1,1,1,0,1,0,0,1,0,0] => [1,3,1,1] => 110011 => 111011 => 3

[1,0,1,1,1,0,1,0,1,0,0,0] => [1,3,1,1] => 110011 => 111011 => 3

[1,0,1,1,1,0,1,1,0,0,0,0] => [1,3,2] => 110010 => 011011 => 3

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Description

The length of the longest alternating subword.
This is the length of the longest consecutive subword of the form $010...$ or of the form $101...$.

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 $0$.

Map

zeros to flag zeros

Description

Return a binary word of the same length, such that the number of occurrences of $10$ in the word obtained by prepending the reverse of the complement equals the number of $0$s in the original word.
For example, the image of the word $w=1\dots1$ is $1\dots1$, because $w$ has no zeros, and $1\dots1$ is the only word such that prepending the reverse of its complement has no occurrence of the factor $10$.
On the other hand, $0\dots0$ must be mapped to $10\dots10$ if the length is even, and $010\dots10$ if it is odd.

Map

rise composition

Description

Send a Dyck path to the composition of sizes of its rises.