- FindStat

Identifier

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000691: Binary words ⟶ ℤ

Values

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

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

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

[1,2,3] => [3] => 100 => 101 => 2

[1,3,2] => [2,1] => 101 => 001 => 1

[2,1,3] => [1,2] => 110 => 011 => 1

[2,3,1] => [2,1] => 101 => 001 => 1

[3,1,2] => [1,2] => 110 => 011 => 1

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

[1,2,3,4] => [4] => 1000 => 0101 => 3

[1,2,4,3] => [3,1] => 1001 => 1101 => 2

[1,3,2,4] => [2,2] => 1010 => 1001 => 2

[1,3,4,2] => [3,1] => 1001 => 1101 => 2

[1,4,2,3] => [2,2] => 1010 => 1001 => 2

[1,4,3,2] => [2,1,1] => 1011 => 0001 => 1

[2,1,3,4] => [1,3] => 1100 => 1011 => 2

[2,1,4,3] => [1,2,1] => 1101 => 0011 => 1

[2,3,1,4] => [2,2] => 1010 => 1001 => 2

[2,3,4,1] => [3,1] => 1001 => 1101 => 2

[2,4,1,3] => [2,2] => 1010 => 1001 => 2

[2,4,3,1] => [2,1,1] => 1011 => 0001 => 1

[3,1,2,4] => [1,3] => 1100 => 1011 => 2

[3,1,4,2] => [1,2,1] => 1101 => 0011 => 1

[3,2,1,4] => [1,1,2] => 1110 => 0111 => 1

[3,2,4,1] => [1,2,1] => 1101 => 0011 => 1

[3,4,1,2] => [2,2] => 1010 => 1001 => 2

[3,4,2,1] => [2,1,1] => 1011 => 0001 => 1

[4,1,2,3] => [1,3] => 1100 => 1011 => 2

[4,1,3,2] => [1,2,1] => 1101 => 0011 => 1

[4,2,1,3] => [1,1,2] => 1110 => 0111 => 1

[4,2,3,1] => [1,2,1] => 1101 => 0011 => 1

[4,3,1,2] => [1,1,2] => 1110 => 0111 => 1

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

[1,2,3,4,5] => [5] => 10000 => 10101 => 4

[1,2,3,5,4] => [4,1] => 10001 => 00101 => 3

[1,2,4,3,5] => [3,2] => 10010 => 01101 => 3

[1,2,4,5,3] => [4,1] => 10001 => 00101 => 3

[1,2,5,3,4] => [3,2] => 10010 => 01101 => 3

[1,2,5,4,3] => [3,1,1] => 10011 => 11101 => 2

[1,3,2,4,5] => [2,3] => 10100 => 01001 => 3

[1,3,2,5,4] => [2,2,1] => 10101 => 11001 => 2

[1,3,4,2,5] => [3,2] => 10010 => 01101 => 3

[1,3,4,5,2] => [4,1] => 10001 => 00101 => 3

[1,3,5,2,4] => [3,2] => 10010 => 01101 => 3

[1,3,5,4,2] => [3,1,1] => 10011 => 11101 => 2

[1,4,2,3,5] => [2,3] => 10100 => 01001 => 3

[1,4,2,5,3] => [2,2,1] => 10101 => 11001 => 2

[1,4,3,2,5] => [2,1,2] => 10110 => 10001 => 2

[1,4,3,5,2] => [2,2,1] => 10101 => 11001 => 2

[1,4,5,2,3] => [3,2] => 10010 => 01101 => 3

[1,4,5,3,2] => [3,1,1] => 10011 => 11101 => 2

[1,5,2,3,4] => [2,3] => 10100 => 01001 => 3

[1,5,2,4,3] => [2,2,1] => 10101 => 11001 => 2

[1,5,3,2,4] => [2,1,2] => 10110 => 10001 => 2

[1,5,3,4,2] => [2,2,1] => 10101 => 11001 => 2

[1,5,4,2,3] => [2,1,2] => 10110 => 10001 => 2

[1,5,4,3,2] => [2,1,1,1] => 10111 => 00001 => 1

[2,1,3,4,5] => [1,4] => 11000 => 01011 => 3

[2,1,3,5,4] => [1,3,1] => 11001 => 11011 => 2

[2,1,4,3,5] => [1,2,2] => 11010 => 10011 => 2

[2,1,4,5,3] => [1,3,1] => 11001 => 11011 => 2

[2,1,5,3,4] => [1,2,2] => 11010 => 10011 => 2

[2,1,5,4,3] => [1,2,1,1] => 11011 => 00011 => 1

[2,3,1,4,5] => [2,3] => 10100 => 01001 => 3

[2,3,1,5,4] => [2,2,1] => 10101 => 11001 => 2

[2,3,4,1,5] => [3,2] => 10010 => 01101 => 3

[2,3,4,5,1] => [4,1] => 10001 => 00101 => 3

[2,3,5,1,4] => [3,2] => 10010 => 01101 => 3

[2,3,5,4,1] => [3,1,1] => 10011 => 11101 => 2

[2,4,1,3,5] => [2,3] => 10100 => 01001 => 3

[2,4,1,5,3] => [2,2,1] => 10101 => 11001 => 2

[2,4,3,1,5] => [2,1,2] => 10110 => 10001 => 2

[2,4,3,5,1] => [2,2,1] => 10101 => 11001 => 2

[2,4,5,1,3] => [3,2] => 10010 => 01101 => 3

[2,4,5,3,1] => [3,1,1] => 10011 => 11101 => 2

[2,5,1,3,4] => [2,3] => 10100 => 01001 => 3

[2,5,1,4,3] => [2,2,1] => 10101 => 11001 => 2

[2,5,3,1,4] => [2,1,2] => 10110 => 10001 => 2

[2,5,3,4,1] => [2,2,1] => 10101 => 11001 => 2

[2,5,4,1,3] => [2,1,2] => 10110 => 10001 => 2

[2,5,4,3,1] => [2,1,1,1] => 10111 => 00001 => 1

[3,1,2,4,5] => [1,4] => 11000 => 01011 => 3

[3,1,2,5,4] => [1,3,1] => 11001 => 11011 => 2

[3,1,4,2,5] => [1,2,2] => 11010 => 10011 => 2

[3,1,4,5,2] => [1,3,1] => 11001 => 11011 => 2

[3,1,5,2,4] => [1,2,2] => 11010 => 10011 => 2

[3,1,5,4,2] => [1,2,1,1] => 11011 => 00011 => 1

[3,2,1,4,5] => [1,1,3] => 11100 => 10111 => 2

[3,2,1,5,4] => [1,1,2,1] => 11101 => 00111 => 1

[3,2,4,1,5] => [1,2,2] => 11010 => 10011 => 2

[3,2,4,5,1] => [1,3,1] => 11001 => 11011 => 2

[3,2,5,1,4] => [1,2,2] => 11010 => 10011 => 2

[3,2,5,4,1] => [1,2,1,1] => 11011 => 00011 => 1

[3,4,1,2,5] => [2,3] => 10100 => 01001 => 3

[3,4,1,5,2] => [2,2,1] => 10101 => 11001 => 2

[3,4,2,1,5] => [2,1,2] => 10110 => 10001 => 2

[3,4,2,5,1] => [2,2,1] => 10101 => 11001 => 2

[3,4,5,1,2] => [3,2] => 10010 => 01101 => 3

[3,4,5,2,1] => [3,1,1] => 10011 => 11101 => 2

[3,5,1,2,4] => [2,3] => 10100 => 01001 => 3

[3,5,1,4,2] => [2,2,1] => 10101 => 11001 => 2

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Description

The number of changes of a binary word.
This is the number of indices $i$ such that $w_i \neq w_{i+1}$.

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

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

descent composition

Description

The descent composition of a permutation.
The descent composition of a permutation $\pi$ of length $n$ is the integer composition of $n$ whose descent set equals the descent set of $\pi$. The descent set of a permutation $\pi$ is $\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$. The descent set of a composition $c = (i_1, i_2, \ldots, i_k)$ is the set $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.