Identifier
-
Mp00269:
Binary words
—flag zeros to zeros⟶
Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000383: Integer compositions ⟶ ℤ
Values
0 => 0 => [2] => [1,1] => 1
1 => 1 => [1,1] => [2] => 2
00 => 01 => [2,1] => [2,1] => 1
01 => 10 => [1,2] => [1,2] => 2
10 => 00 => [3] => [1,1,1] => 1
11 => 11 => [1,1,1] => [3] => 3
000 => 011 => [2,1,1] => [3,1] => 1
001 => 101 => [1,2,1] => [2,2] => 2
010 => 000 => [4] => [1,1,1,1] => 1
011 => 110 => [1,1,2] => [1,3] => 3
100 => 010 => [2,2] => [1,2,1] => 1
101 => 100 => [1,3] => [1,1,2] => 2
110 => 001 => [3,1] => [2,1,1] => 1
111 => 111 => [1,1,1,1] => [4] => 4
0000 => 0111 => [2,1,1,1] => [4,1] => 1
0001 => 1011 => [1,2,1,1] => [3,2] => 2
0010 => 0001 => [4,1] => [2,1,1,1] => 1
0011 => 1101 => [1,1,2,1] => [2,3] => 3
0100 => 0100 => [2,3] => [1,1,2,1] => 1
0101 => 1000 => [1,4] => [1,1,1,2] => 2
0110 => 0010 => [3,2] => [1,2,1,1] => 1
0111 => 1110 => [1,1,1,2] => [1,4] => 4
1000 => 0110 => [2,1,2] => [1,3,1] => 1
1001 => 1010 => [1,2,2] => [1,2,2] => 2
1010 => 0000 => [5] => [1,1,1,1,1] => 1
1011 => 1100 => [1,1,3] => [1,1,3] => 3
1100 => 0101 => [2,2,1] => [2,2,1] => 1
1101 => 1001 => [1,3,1] => [2,1,2] => 2
1110 => 0011 => [3,1,1] => [3,1,1] => 1
1111 => 1111 => [1,1,1,1,1] => [5] => 5
00000 => 01111 => [2,1,1,1,1] => [5,1] => 1
00001 => 10111 => [1,2,1,1,1] => [4,2] => 2
00010 => 00011 => [4,1,1] => [3,1,1,1] => 1
00011 => 11011 => [1,1,2,1,1] => [3,3] => 3
00100 => 01001 => [2,3,1] => [2,1,2,1] => 1
00101 => 10001 => [1,4,1] => [2,1,1,2] => 2
00110 => 00101 => [3,2,1] => [2,2,1,1] => 1
00111 => 11101 => [1,1,1,2,1] => [2,4] => 4
01000 => 01100 => [2,1,3] => [1,1,3,1] => 1
01001 => 10100 => [1,2,3] => [1,1,2,2] => 2
01010 => 00000 => [6] => [1,1,1,1,1,1] => 1
01011 => 11000 => [1,1,4] => [1,1,1,3] => 3
01100 => 01010 => [2,2,2] => [1,2,2,1] => 1
01101 => 10010 => [1,3,2] => [1,2,1,2] => 2
01110 => 00110 => [3,1,2] => [1,3,1,1] => 1
01111 => 11110 => [1,1,1,1,2] => [1,5] => 5
10000 => 01110 => [2,1,1,2] => [1,4,1] => 1
10001 => 10110 => [1,2,1,2] => [1,3,2] => 2
10010 => 00010 => [4,2] => [1,2,1,1,1] => 1
10011 => 11010 => [1,1,2,2] => [1,2,3] => 3
10100 => 01000 => [2,4] => [1,1,1,2,1] => 1
10101 => 10000 => [1,5] => [1,1,1,1,2] => 2
10110 => 00100 => [3,3] => [1,1,2,1,1] => 1
10111 => 11100 => [1,1,1,3] => [1,1,4] => 4
11000 => 01101 => [2,1,2,1] => [2,3,1] => 1
11001 => 10101 => [1,2,2,1] => [2,2,2] => 2
11010 => 00001 => [5,1] => [2,1,1,1,1] => 1
11011 => 11001 => [1,1,3,1] => [2,1,3] => 3
11100 => 01011 => [2,2,1,1] => [3,2,1] => 1
11101 => 10011 => [1,3,1,1] => [3,1,2] => 2
11110 => 00111 => [3,1,1,1] => [4,1,1] => 1
11111 => 11111 => [1,1,1,1,1,1] => [6] => 6
000000 => 011111 => [2,1,1,1,1,1] => [6,1] => 1
000001 => 101111 => [1,2,1,1,1,1] => [5,2] => 2
000010 => 000111 => [4,1,1,1] => [4,1,1,1] => 1
000011 => 110111 => [1,1,2,1,1,1] => [4,3] => 3
000100 => 010011 => [2,3,1,1] => [3,1,2,1] => 1
000101 => 100011 => [1,4,1,1] => [3,1,1,2] => 2
000110 => 001011 => [3,2,1,1] => [3,2,1,1] => 1
000111 => 111011 => [1,1,1,2,1,1] => [3,4] => 4
001000 => 011001 => [2,1,3,1] => [2,1,3,1] => 1
001001 => 101001 => [1,2,3,1] => [2,1,2,2] => 2
001010 => 000001 => [6,1] => [2,1,1,1,1,1] => 1
001011 => 110001 => [1,1,4,1] => [2,1,1,3] => 3
001100 => 010101 => [2,2,2,1] => [2,2,2,1] => 1
001101 => 100101 => [1,3,2,1] => [2,2,1,2] => 2
001110 => 001101 => [3,1,2,1] => [2,3,1,1] => 1
001111 => 111101 => [1,1,1,1,2,1] => [2,5] => 5
010000 => 011100 => [2,1,1,3] => [1,1,4,1] => 1
010001 => 101100 => [1,2,1,3] => [1,1,3,2] => 2
010010 => 000100 => [4,3] => [1,1,2,1,1,1] => 1
010011 => 110100 => [1,1,2,3] => [1,1,2,3] => 3
010100 => 010000 => [2,5] => [1,1,1,1,2,1] => 1
010101 => 100000 => [1,6] => [1,1,1,1,1,2] => 2
010110 => 001000 => [3,4] => [1,1,1,2,1,1] => 1
010111 => 111000 => [1,1,1,4] => [1,1,1,4] => 4
011000 => 011010 => [2,1,2,2] => [1,2,3,1] => 1
011001 => 101010 => [1,2,2,2] => [1,2,2,2] => 2
011010 => 000010 => [5,2] => [1,2,1,1,1,1] => 1
011011 => 110010 => [1,1,3,2] => [1,2,1,3] => 3
011100 => 010110 => [2,2,1,2] => [1,3,2,1] => 1
011101 => 100110 => [1,3,1,2] => [1,3,1,2] => 2
011110 => 001110 => [3,1,1,2] => [1,4,1,1] => 1
011111 => 111110 => [1,1,1,1,1,2] => [1,6] => 6
100000 => 011110 => [2,1,1,1,2] => [1,5,1] => 1
100001 => 101110 => [1,2,1,1,2] => [1,4,2] => 2
100010 => 000110 => [4,1,2] => [1,3,1,1,1] => 1
100011 => 110110 => [1,1,2,1,2] => [1,3,3] => 3
100100 => 010010 => [2,3,2] => [1,2,1,2,1] => 1
100101 => 100010 => [1,4,2] => [1,2,1,1,2] => 2
100110 => 001010 => [3,2,2] => [1,2,2,1,1] => 1
>>> Load all 312 entries. <<<
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Description
The last part of an integer composition.
Map
conjugate
Description
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.
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
flag zeros to zeros
Description
Return a binary word of the same length, such that the number of zeros equals the number of occurrences of $10$ in the word obtained from the original word by prepending the reverse of the complement.
For example, the image of the word $w=1\dots 1$ is $1\dots 1$, because $0\dots 01\dots 1$ has no occurrences of $10$. The words $10\dots 10$ and $010\dots 10$ have image $0\dots 0$.
For example, the image of the word $w=1\dots 1$ is $1\dots 1$, because $0\dots 01\dots 1$ has no occurrences of $10$. The words $10\dots 10$ and $010\dots 10$ have image $0\dots 0$.
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