- FindStat

Identifier

Mp00105: Binary words —complement⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤ

Values

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

1 => 0 => [2] => [2] => 2

00 => 11 => [1,1,1] => [1,1,1] => 1

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

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

11 => 00 => [3] => [3] => 3

000 => 111 => [1,1,1,1] => [1,1,1,1] => 1

001 => 110 => [1,1,2] => [2,1,1] => 2

010 => 101 => [1,2,1] => [2,1,1] => 2

011 => 100 => [1,3] => [3,1] => 3

100 => 011 => [2,1,1] => [2,1,1] => 2

101 => 010 => [2,2] => [2,2] => 2

110 => 001 => [3,1] => [3,1] => 3

111 => 000 => [4] => [4] => 4

0000 => 1111 => [1,1,1,1,1] => [1,1,1,1,1] => 1

0001 => 1110 => [1,1,1,2] => [2,1,1,1] => 2

0010 => 1101 => [1,1,2,1] => [2,1,1,1] => 2

0011 => 1100 => [1,1,3] => [3,1,1] => 3

0100 => 1011 => [1,2,1,1] => [2,1,1,1] => 2

0101 => 1010 => [1,2,2] => [2,2,1] => 2

0110 => 1001 => [1,3,1] => [3,1,1] => 3

0111 => 1000 => [1,4] => [4,1] => 4

1000 => 0111 => [2,1,1,1] => [2,1,1,1] => 2

1001 => 0110 => [2,1,2] => [2,2,1] => 2

1010 => 0101 => [2,2,1] => [2,2,1] => 2

1011 => 0100 => [2,3] => [3,2] => 3

1100 => 0011 => [3,1,1] => [3,1,1] => 3

1101 => 0010 => [3,2] => [3,2] => 3

1110 => 0001 => [4,1] => [4,1] => 4

1111 => 0000 => [5] => [5] => 5

00000 => 11111 => [1,1,1,1,1,1] => [1,1,1,1,1,1] => 1

00001 => 11110 => [1,1,1,1,2] => [2,1,1,1,1] => 2

00010 => 11101 => [1,1,1,2,1] => [2,1,1,1,1] => 2

00011 => 11100 => [1,1,1,3] => [3,1,1,1] => 3

00100 => 11011 => [1,1,2,1,1] => [2,1,1,1,1] => 2

00101 => 11010 => [1,1,2,2] => [2,2,1,1] => 2

00110 => 11001 => [1,1,3,1] => [3,1,1,1] => 3

00111 => 11000 => [1,1,4] => [4,1,1] => 4

01000 => 10111 => [1,2,1,1,1] => [2,1,1,1,1] => 2

01001 => 10110 => [1,2,1,2] => [2,2,1,1] => 2

01010 => 10101 => [1,2,2,1] => [2,2,1,1] => 2

01011 => 10100 => [1,2,3] => [3,2,1] => 3

01100 => 10011 => [1,3,1,1] => [3,1,1,1] => 3

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

01110 => 10001 => [1,4,1] => [4,1,1] => 4

01111 => 10000 => [1,5] => [5,1] => 5

10000 => 01111 => [2,1,1,1,1] => [2,1,1,1,1] => 2

10001 => 01110 => [2,1,1,2] => [2,2,1,1] => 2

10010 => 01101 => [2,1,2,1] => [2,2,1,1] => 2

10011 => 01100 => [2,1,3] => [3,2,1] => 3

10100 => 01011 => [2,2,1,1] => [2,2,1,1] => 2

10101 => 01010 => [2,2,2] => [2,2,2] => 2

10110 => 01001 => [2,3,1] => [3,2,1] => 3

10111 => 01000 => [2,4] => [4,2] => 4

11000 => 00111 => [3,1,1,1] => [3,1,1,1] => 3

11001 => 00110 => [3,1,2] => [3,2,1] => 3

11010 => 00101 => [3,2,1] => [3,2,1] => 3

11011 => 00100 => [3,3] => [3,3] => 3

11100 => 00011 => [4,1,1] => [4,1,1] => 4

11101 => 00010 => [4,2] => [4,2] => 4

11110 => 00001 => [5,1] => [5,1] => 5

11111 => 00000 => [6] => [6] => 6

000000 => 111111 => [1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 1

000001 => 111110 => [1,1,1,1,1,2] => [2,1,1,1,1,1] => 2

000010 => 111101 => [1,1,1,1,2,1] => [2,1,1,1,1,1] => 2

000011 => 111100 => [1,1,1,1,3] => [3,1,1,1,1] => 3

000100 => 111011 => [1,1,1,2,1,1] => [2,1,1,1,1,1] => 2

000101 => 111010 => [1,1,1,2,2] => [2,2,1,1,1] => 2

000110 => 111001 => [1,1,1,3,1] => [3,1,1,1,1] => 3

000111 => 111000 => [1,1,1,4] => [4,1,1,1] => 4

001000 => 110111 => [1,1,2,1,1,1] => [2,1,1,1,1,1] => 2

001001 => 110110 => [1,1,2,1,2] => [2,2,1,1,1] => 2

001010 => 110101 => [1,1,2,2,1] => [2,2,1,1,1] => 2

001011 => 110100 => [1,1,2,3] => [3,2,1,1] => 3

001100 => 110011 => [1,1,3,1,1] => [3,1,1,1,1] => 3

001101 => 110010 => [1,1,3,2] => [3,2,1,1] => 3

001110 => 110001 => [1,1,4,1] => [4,1,1,1] => 4

001111 => 110000 => [1,1,5] => [5,1,1] => 5

010000 => 101111 => [1,2,1,1,1,1] => [2,1,1,1,1,1] => 2

010001 => 101110 => [1,2,1,1,2] => [2,2,1,1,1] => 2

010010 => 101101 => [1,2,1,2,1] => [2,2,1,1,1] => 2

010011 => 101100 => [1,2,1,3] => [3,2,1,1] => 3

010100 => 101011 => [1,2,2,1,1] => [2,2,1,1,1] => 2

010101 => 101010 => [1,2,2,2] => [2,2,2,1] => 2

010110 => 101001 => [1,2,3,1] => [3,2,1,1] => 3

010111 => 101000 => [1,2,4] => [4,2,1] => 4

011000 => 100111 => [1,3,1,1,1] => [3,1,1,1,1] => 3

011001 => 100110 => [1,3,1,2] => [3,2,1,1] => 3

011010 => 100101 => [1,3,2,1] => [3,2,1,1] => 3

011011 => 100100 => [1,3,3] => [3,3,1] => 3

011100 => 100011 => [1,4,1,1] => [4,1,1,1] => 4

011101 => 100010 => [1,4,2] => [4,2,1] => 4

011110 => 100001 => [1,5,1] => [5,1,1] => 5

011111 => 100000 => [1,6] => [6,1] => 6

100000 => 011111 => [2,1,1,1,1,1] => [2,1,1,1,1,1] => 2

100001 => 011110 => [2,1,1,1,2] => [2,2,1,1,1] => 2

100010 => 011101 => [2,1,1,2,1] => [2,2,1,1,1] => 2

100011 => 011100 => [2,1,1,3] => [3,2,1,1] => 3

100100 => 011011 => [2,1,2,1,1] => [2,2,1,1,1] => 2

100101 => 011010 => [2,1,2,2] => [2,2,2,1] => 2

100110 => 011001 => [2,1,3,1] => [3,2,1,1] => 3

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Description

The largest part of an integer partition.

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 partition

Description

Sends a composition to the partition obtained by sorting the entries.

Map

complement

Description

Send a binary word to the word obtained by interchanging the two letters.