- FindStat

Identifier

Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000288: Binary words ⟶ ℤ

Values

[1] => [1] => 10 => 01 => 1

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

[2] => [2] => 100 => 011 => 2

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

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

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

[3] => [3] => 1000 => 0111 => 3

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

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

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

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

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

[2,2] => [2,2] => 1100 => 0011 => 2

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

[4] => [4] => 10000 => 01111 => 4

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[5] => [5] => 100000 => 011111 => 5

[1,1,1,1,1,1] => [1,1,1,1,1,1] => 1111110 => 0000001 => 1

[1,1,1,1,2] => [2,1,1,1,1] => 1011110 => 0100001 => 2

[1,1,1,2,1] => [2,1,1,1,1] => 1011110 => 0100001 => 2

[1,1,1,3] => [3,1,1,1] => 1001110 => 0110001 => 3

[1,1,2,1,1] => [2,1,1,1,1] => 1011110 => 0100001 => 2

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

[1,1,3,1] => [3,1,1,1] => 1001110 => 0110001 => 3

[1,1,4] => [4,1,1] => 1000110 => 0111001 => 4

[1,2,1,1,1] => [2,1,1,1,1] => 1011110 => 0100001 => 2

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

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

[1,2,3] => [3,2,1] => 101010 => 010101 => 3

[1,3,1,1] => [3,1,1,1] => 1001110 => 0110001 => 3

[1,3,2] => [3,2,1] => 101010 => 010101 => 3

[1,4,1] => [4,1,1] => 1000110 => 0111001 => 4

[1,5] => [5,1] => 1000010 => 0111101 => 5

[2,1,1,1,1] => [2,1,1,1,1] => 1011110 => 0100001 => 2

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

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

[2,1,3] => [3,2,1] => 101010 => 010101 => 3

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

[2,2,2] => [2,2,2] => 11100 => 00011 => 2

[2,3,1] => [3,2,1] => 101010 => 010101 => 3

[2,4] => [4,2] => 100100 => 011011 => 4

[3,1,1,1] => [3,1,1,1] => 1001110 => 0110001 => 3

[3,1,2] => [3,2,1] => 101010 => 010101 => 3

[3,2,1] => [3,2,1] => 101010 => 010101 => 3

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

[4,1,1] => [4,1,1] => 1000110 => 0111001 => 4

[4,2] => [4,2] => 100100 => 011011 => 4

[5,1] => [5,1] => 1000010 => 0111101 => 5

[6] => [6] => 1000000 => 0111111 => 6

[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 11111110 => 00000001 => 1

[1,1,1,1,1,2] => [2,1,1,1,1,1] => 10111110 => 01000001 => 2

[1,1,1,1,2,1] => [2,1,1,1,1,1] => 10111110 => 01000001 => 2

[1,1,1,1,3] => [3,1,1,1,1] => 10011110 => 01100001 => 3

[1,1,1,2,1,1] => [2,1,1,1,1,1] => 10111110 => 01000001 => 2

[1,1,1,2,2] => [2,2,1,1,1] => 1101110 => 0010001 => 2

[1,1,1,3,1] => [3,1,1,1,1] => 10011110 => 01100001 => 3

[1,1,1,4] => [4,1,1,1] => 10001110 => 01110001 => 4

[1,1,2,1,1,1] => [2,1,1,1,1,1] => 10111110 => 01000001 => 2

[1,1,2,1,2] => [2,2,1,1,1] => 1101110 => 0010001 => 2

[1,1,2,2,1] => [2,2,1,1,1] => 1101110 => 0010001 => 2

[1,1,2,3] => [3,2,1,1] => 1010110 => 0101001 => 3

[1,1,3,1,1] => [3,1,1,1,1] => 10011110 => 01100001 => 3

[1,1,3,2] => [3,2,1,1] => 1010110 => 0101001 => 3

[1,1,4,1] => [4,1,1,1] => 10001110 => 01110001 => 4

[1,1,5] => [5,1,1] => 10000110 => 01111001 => 5

[1,2,1,1,1,1] => [2,1,1,1,1,1] => 10111110 => 01000001 => 2

[1,2,1,1,2] => [2,2,1,1,1] => 1101110 => 0010001 => 2

[1,2,1,2,1] => [2,2,1,1,1] => 1101110 => 0010001 => 2

[1,2,1,3] => [3,2,1,1] => 1010110 => 0101001 => 3

[1,2,2,1,1] => [2,2,1,1,1] => 1101110 => 0010001 => 2

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

[1,2,3,1] => [3,2,1,1] => 1010110 => 0101001 => 3

[1,2,4] => [4,2,1] => 1001010 => 0110101 => 4

[1,3,1,1,1] => [3,1,1,1,1] => 10011110 => 01100001 => 3

[1,3,1,2] => [3,2,1,1] => 1010110 => 0101001 => 3

[1,3,2,1] => [3,2,1,1] => 1010110 => 0101001 => 3

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

[1,4,1,1] => [4,1,1,1] => 10001110 => 01110001 => 4

[1,4,2] => [4,2,1] => 1001010 => 0110101 => 4

[1,5,1] => [5,1,1] => 10000110 => 01111001 => 5

[1,6] => [6,1] => 10000010 => 01111101 => 6

[2,1,1,1,1,1] => [2,1,1,1,1,1] => 10111110 => 01000001 => 2

[2,1,1,1,2] => [2,2,1,1,1] => 1101110 => 0010001 => 2

[2,1,1,2,1] => [2,2,1,1,1] => 1101110 => 0010001 => 2

[2,1,1,3] => [3,2,1,1] => 1010110 => 0101001 => 3

[2,1,2,1,1] => [2,2,1,1,1] => 1101110 => 0010001 => 2

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

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$F_{1} = q$

$F_{2} = q + q^{2}$

$F_{3} = q + 2\ q^{2} + q^{3}$

$F_{4} = q + 4\ q^{2} + 2\ q^{3} + q^{4}$

$F_{5} = q + 7\ q^{2} + 5\ q^{3} + 2\ q^{4} + q^{5}$

$F_{6} = q + 12\ q^{2} + 11\ q^{3} + 5\ q^{4} + 2\ q^{5} + q^{6}$

$F_{7} = q + 20\ q^{2} + 23\ q^{3} + 12\ q^{4} + 5\ q^{5} + 2\ q^{6} + q^{7}$

$F_{8} = q + 33\ q^{2} + 47\ q^{3} + 27\ q^{4} + 12\ q^{5} + 5\ q^{6} + 2\ q^{7} + q^{8}$

$F_{9} = q + 54\ q^{2} + 94\ q^{3} + 59\ q^{4} + 28\ q^{5} + 12\ q^{6} + 5\ q^{7} + 2\ q^{8} + q^{9}$

$F_{10} = q + 88\ q^{2} + 185\ q^{3} + 127\ q^{4} + 63\ q^{5} + 28\ q^{6} + 12\ q^{7} + 5\ q^{8} + 2\ q^{9} + q^{10}$

Description

The number of ones in a binary word.
This is also known as the Hamming weight of the word.

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.

Map

to binary word

Description

Return the partition as binary word, by traversing its shape from the first row to the last row, down steps as 1 and left steps as 0.