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Identifier

Mp00178: Binary words —to composition⟶ Integer compositions

Images

=> [1]

0 => [2]

1 => [1,1]

00 => [3]

01 => [2,1]

10 => [1,2]

11 => [1,1,1]

000 => [4]

001 => [3,1]

010 => [2,2]

011 => [2,1,1]

100 => [1,3]

101 => [1,2,1]

110 => [1,1,2]

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

0000 => [5]

0001 => [4,1]

0010 => [3,2]

0011 => [3,1,1]

0100 => [2,3]

0101 => [2,2,1]

0110 => [2,1,2]

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

1000 => [1,4]

1001 => [1,3,1]

1010 => [1,2,2]

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

1100 => [1,1,3]

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

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

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

00000 => [6]

00001 => [5,1]

00010 => [4,2]

00011 => [4,1,1]

00100 => [3,3]

00101 => [3,2,1]

00110 => [3,1,2]

00111 => [3,1,1,1]

01000 => [2,4]

01001 => [2,3,1]

01010 => [2,2,2]

01011 => [2,2,1,1]

01100 => [2,1,3]

01101 => [2,1,2,1]

01110 => [2,1,1,2]

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

10000 => [1,5]

10001 => [1,4,1]

10010 => [1,3,2]

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

10100 => [1,2,3]

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

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

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

11000 => [1,1,4]

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

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

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

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

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

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

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

000000 => [7]

000001 => [6,1]

000010 => [5,2]

000011 => [5,1,1]

000100 => [4,3]

000101 => [4,2,1]

000110 => [4,1,2]

000111 => [4,1,1,1]

001000 => [3,4]

001001 => [3,3,1]

001010 => [3,2,2]

001011 => [3,2,1,1]

001100 => [3,1,3]

001101 => [3,1,2,1]

001110 => [3,1,1,2]

001111 => [3,1,1,1,1]

010000 => [2,5]

010001 => [2,4,1]

010010 => [2,3,2]

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

010100 => [2,2,3]

010101 => [2,2,2,1]

010110 => [2,2,1,2]

010111 => [2,2,1,1,1]

011000 => [2,1,4]

011001 => [2,1,3,1]

011010 => [2,1,2,2]

011011 => [2,1,2,1,1]

011100 => [2,1,1,3]

011101 => [2,1,1,2,1]

011110 => [2,1,1,1,2]

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

100000 => [1,6]

100001 => [1,5,1]

100010 => [1,4,2]

100011 => [1,4,1,1]

100100 => [1,3,3]

100101 => [1,3,2,1]

>>> Load all 6009 entries. <<<

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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.

Properties

injective, graded

Sage code

def mapping(w):
    return Compositions().from_code([1] + list(w))

Weight

Created

Sep 21, 2020 at 21:21 by Martin Rubey

Updated

Sep 21, 2020 at 21:21 by Martin Rubey