Identifier
Values
[1] => 1 => [1,1] => 0
[1,1] => 11 => [1,1,1] => 0
[2] => 10 => [1,2] => 1
[1,1,1] => 111 => [1,1,1,1] => 0
[1,2] => 110 => [1,1,2] => 1
[2,1] => 101 => [1,2,1] => 1
[3] => 100 => [1,3] => 2
[1,1,1,1] => 1111 => [1,1,1,1,1] => 0
[1,1,2] => 1110 => [1,1,1,2] => 1
[1,2,1] => 1101 => [1,1,2,1] => 1
[1,3] => 1100 => [1,1,3] => 2
[2,1,1] => 1011 => [1,2,1,1] => 1
[2,2] => 1010 => [1,2,2] => 1
[3,1] => 1001 => [1,3,1] => 2
[4] => 1000 => [1,4] => 3
[1,1,1,1,1] => 11111 => [1,1,1,1,1,1] => 0
[1,1,1,2] => 11110 => [1,1,1,1,2] => 1
[1,1,2,1] => 11101 => [1,1,1,2,1] => 1
[1,1,3] => 11100 => [1,1,1,3] => 2
[1,2,1,1] => 11011 => [1,1,2,1,1] => 1
[1,2,2] => 11010 => [1,1,2,2] => 1
[1,3,1] => 11001 => [1,1,3,1] => 2
[1,4] => 11000 => [1,1,4] => 3
[2,1,1,1] => 10111 => [1,2,1,1,1] => 1
[2,1,2] => 10110 => [1,2,1,2] => 2
[2,2,1] => 10101 => [1,2,2,1] => 1
[2,3] => 10100 => [1,2,3] => 2
[3,1,1] => 10011 => [1,3,1,1] => 2
[3,2] => 10010 => [1,3,2] => 2
[4,1] => 10001 => [1,4,1] => 3
[5] => 10000 => [1,5] => 4
[1,1,1,1,1,1] => 111111 => [1,1,1,1,1,1,1] => 0
[1,1,1,1,2] => 111110 => [1,1,1,1,1,2] => 1
[1,1,1,2,1] => 111101 => [1,1,1,1,2,1] => 1
[1,1,1,3] => 111100 => [1,1,1,1,3] => 2
[1,1,2,1,1] => 111011 => [1,1,1,2,1,1] => 1
[1,1,2,2] => 111010 => [1,1,1,2,2] => 1
[1,1,3,1] => 111001 => [1,1,1,3,1] => 2
[1,1,4] => 111000 => [1,1,1,4] => 3
[1,2,1,1,1] => 110111 => [1,1,2,1,1,1] => 1
[1,2,1,2] => 110110 => [1,1,2,1,2] => 2
[1,2,2,1] => 110101 => [1,1,2,2,1] => 1
[1,2,3] => 110100 => [1,1,2,3] => 2
[1,3,1,1] => 110011 => [1,1,3,1,1] => 2
[1,3,2] => 110010 => [1,1,3,2] => 2
[1,4,1] => 110001 => [1,1,4,1] => 3
[1,5] => 110000 => [1,1,5] => 4
[2,1,1,1,1] => 101111 => [1,2,1,1,1,1] => 1
[2,1,1,2] => 101110 => [1,2,1,1,2] => 2
[2,1,2,1] => 101101 => [1,2,1,2,1] => 2
[2,1,3] => 101100 => [1,2,1,3] => 3
[2,2,1,1] => 101011 => [1,2,2,1,1] => 1
[2,2,2] => 101010 => [1,2,2,2] => 1
[2,3,1] => 101001 => [1,2,3,1] => 2
[2,4] => 101000 => [1,2,4] => 3
[3,1,1,1] => 100111 => [1,3,1,1,1] => 2
[3,1,2] => 100110 => [1,3,1,2] => 3
[3,2,1] => 100101 => [1,3,2,1] => 2
[3,3] => 100100 => [1,3,3] => 2
[4,1,1] => 100011 => [1,4,1,1] => 3
[4,2] => 100010 => [1,4,2] => 3
[5,1] => 100001 => [1,5,1] => 4
[6] => 100000 => [1,6] => 5
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Description
The descent variation of a composition.
Defined in [1].
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 $0$.