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Identifier

Mp00094: Integer compositions —to binary word⟶ Binary words

Images

[1] => 1

[1,1] => 11

[2] => 10

[1,1,1] => 111

[1,2] => 110

[2,1] => 101

[3] => 100

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

[1,1,2] => 1110

[1,2,1] => 1101

[1,3] => 1100

[2,1,1] => 1011

[2,2] => 1010

[3,1] => 1001

[4] => 1000

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

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

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

[1,1,3] => 11100

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

[1,2,2] => 11010

[1,3,1] => 11001

[1,4] => 11000

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

[2,1,2] => 10110

[2,2,1] => 10101

[2,3] => 10100

[3,1,1] => 10011

[3,2] => 10010

[4,1] => 10001

[5] => 10000

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

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

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

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

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

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

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

[1,1,4] => 111000

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

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

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

[1,2,3] => 110100

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

[1,3,2] => 110010

[1,4,1] => 110001

[1,5] => 110000

[2,1,1,1,1] => 101111

[2,1,1,2] => 101110

[2,1,2,1] => 101101

[2,1,3] => 101100

[2,2,1,1] => 101011

[2,2,2] => 101010

[2,3,1] => 101001

[2,4] => 101000

[3,1,1,1] => 100111

[3,1,2] => 100110

[3,2,1] => 100101

[3,3] => 100100

[4,1,1] => 100011

[4,2] => 100010

[5,1] => 100001

[6] => 100000

[1,1,1,1,1,1,1] => 1111111

[1,1,1,1,1,2] => 1111110

[1,1,1,1,2,1] => 1111101

[1,1,1,1,3] => 1111100

[1,1,1,2,1,1] => 1111011

[1,1,1,2,2] => 1111010

[1,1,1,3,1] => 1111001

[1,1,1,4] => 1111000

[1,1,2,1,1,1] => 1110111

[1,1,2,1,2] => 1110110

[1,1,2,2,1] => 1110101

[1,1,2,3] => 1110100

[1,1,3,1,1] => 1110011

[1,1,3,2] => 1110010

[1,1,4,1] => 1110001

[1,1,5] => 1110000

[1,2,1,1,1,1] => 1101111

[1,2,1,1,2] => 1101110

[1,2,1,2,1] => 1101101

[1,2,1,3] => 1101100

[1,2,2,1,1] => 1101011

[1,2,2,2] => 1101010

[1,2,3,1] => 1101001

[1,2,4] => 1101000

[1,3,1,1,1] => 1100111

[1,3,1,2] => 1100110

[1,3,2,1] => 1100101

[1,3,3] => 1100100

[1,4,1,1] => 1100011

[1,4,2] => 1100010

[1,5,1] => 1100001

[1,6] => 1100000

[2,1,1,1,1,1] => 1011111

[2,1,1,1,2] => 1011110

[2,1,1,2,1] => 1011101

[2,1,1,3] => 1011100

[2,1,2,1,1] => 1011011

[2,1,2,2] => 1011010

>>> Load all 2305 entries. <<<

click to show experimental identities (only identities of compositions of up to three maps are shown)

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 $0$.

Properties

injective, graded

Sage code

def mapping(w):
    # we diverge from the SageMath convention for the empty
    # composition to make the map graded in the strict sense
    if w:
        return Words([0,1])(w.to_code())
    return Words([0,1])([])

Weight

Created

Sep 22, 2020 at 11:07 by FindStatCrew

Updated

Sep 22, 2020 at 11:07 by Martin Rubey