- FindStat

Identifier

Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000290: Binary words ⟶ ℤ

Values

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

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

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

[3] => [1,1,1] => 1110 => 3

[2,1] => [2,1] => 1010 => 4

[1,1,1] => [3] => 1000 => 1

[4] => [1,1,1,1] => 11110 => 4

[3,1] => [2,1,1] => 10110 => 5

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

[2,1,1] => [3,1] => 10010 => 5

[1,1,1,1] => [4] => 10000 => 1

[5] => [1,1,1,1,1] => 111110 => 5

[4,1] => [2,1,1,1] => 101110 => 6

[3,2] => [2,2,1] => 11010 => 6

[3,1,1] => [3,1,1] => 100110 => 6

[2,2,1] => [3,2] => 10100 => 4

[2,1,1,1] => [4,1] => 100010 => 6

[1,1,1,1,1] => [5] => 100000 => 1

[6] => [1,1,1,1,1,1] => 1111110 => 6

[5,1] => [2,1,1,1,1] => 1011110 => 7

[4,2] => [2,2,1,1] => 110110 => 7

[4,1,1] => [3,1,1,1] => 1001110 => 7

[3,3] => [2,2,2] => 11100 => 3

[3,2,1] => [3,2,1] => 101010 => 9

[3,1,1,1] => [4,1,1] => 1000110 => 7

[2,2,2] => [3,3] => 11000 => 2

[2,2,1,1] => [4,2] => 100100 => 5

[2,1,1,1,1] => [5,1] => 1000010 => 7

[1,1,1,1,1,1] => [6] => 1000000 => 1

[7] => [1,1,1,1,1,1,1] => 11111110 => 7

[6,1] => [2,1,1,1,1,1] => 10111110 => 8

[5,2] => [2,2,1,1,1] => 1101110 => 8

[5,1,1] => [3,1,1,1,1] => 10011110 => 8

[4,3] => [2,2,2,1] => 111010 => 8

[4,2,1] => [3,2,1,1] => 1010110 => 10

[4,1,1,1] => [4,1,1,1] => 10001110 => 8

[3,3,1] => [3,2,2] => 101100 => 5

[3,2,2] => [3,3,1] => 110010 => 7

[3,2,1,1] => [4,2,1] => 1001010 => 11

[3,1,1,1,1] => [5,1,1] => 10000110 => 8

[2,2,2,1] => [4,3] => 101000 => 4

[2,2,1,1,1] => [5,2] => 1000100 => 6

[2,1,1,1,1,1] => [6,1] => 10000010 => 8

[1,1,1,1,1,1,1] => [7] => 10000000 => 1

[8] => [1,1,1,1,1,1,1,1] => 111111110 => 8

[7,1] => [2,1,1,1,1,1,1] => 101111110 => 9

[6,2] => [2,2,1,1,1,1] => 11011110 => 9

[6,1,1] => [3,1,1,1,1,1] => 100111110 => 9

[5,3] => [2,2,2,1,1] => 1110110 => 9

[5,2,1] => [3,2,1,1,1] => 10101110 => 11

[5,1,1,1] => [4,1,1,1,1] => 100011110 => 9

[4,4] => [2,2,2,2] => 111100 => 4

[4,3,1] => [3,2,2,1] => 1011010 => 11

[4,2,2] => [3,3,1,1] => 1100110 => 8

[4,2,1,1] => [4,2,1,1] => 10010110 => 12

[4,1,1,1,1] => [5,1,1,1] => 100001110 => 9

[3,3,2] => [3,3,2] => 110100 => 6

[3,3,1,1] => [4,2,2] => 1001100 => 6

[3,2,2,1] => [4,3,1] => 1010010 => 10

[3,2,1,1,1] => [5,2,1] => 10001010 => 13

[3,1,1,1,1,1] => [6,1,1] => 100000110 => 9

[2,2,2,2] => [4,4] => 110000 => 2

[2,2,2,1,1] => [5,3] => 1001000 => 5

[2,2,1,1,1,1] => [6,2] => 10000100 => 7

[2,1,1,1,1,1,1] => [7,1] => 100000010 => 9

[1,1,1,1,1,1,1,1] => [8] => 100000000 => 1

[9] => [1,1,1,1,1,1,1,1,1] => 1111111110 => 9

[8,1] => [2,1,1,1,1,1,1,1] => 1011111110 => 10

[7,2] => [2,2,1,1,1,1,1] => 110111110 => 10

[7,1,1] => [3,1,1,1,1,1,1] => 1001111110 => 10

[6,3] => [2,2,2,1,1,1] => 11101110 => 10

[6,2,1] => [3,2,1,1,1,1] => 101011110 => 12

[6,1,1,1] => [4,1,1,1,1,1] => 1000111110 => 10

[5,4] => [2,2,2,2,1] => 1111010 => 10

[5,3,1] => [3,2,2,1,1] => 10110110 => 12

[5,2,2] => [3,3,1,1,1] => 11001110 => 9

[5,2,1,1] => [4,2,1,1,1] => 100101110 => 13

[5,1,1,1,1] => [5,1,1,1,1] => 1000011110 => 10

[4,4,1] => [3,2,2,2] => 1011100 => 6

[4,3,2] => [3,3,2,1] => 1101010 => 12

[4,3,1,1] => [4,2,2,1] => 10011010 => 13

[4,2,2,1] => [4,3,1,1] => 10100110 => 11

[4,2,1,1,1] => [5,2,1,1] => 100010110 => 14

[4,1,1,1,1,1] => [6,1,1,1] => 1000001110 => 10

[3,3,3] => [3,3,3] => 111000 => 3

[3,3,2,1] => [4,3,2] => 1010100 => 9

[3,3,1,1,1] => [5,2,2] => 10001100 => 7

[3,2,2,2] => [4,4,1] => 1100010 => 8

[3,2,2,1,1] => [5,3,1] => 10010010 => 12

[3,2,1,1,1,1] => [6,2,1] => 100001010 => 15

[3,1,1,1,1,1,1] => [7,1,1] => 1000000110 => 10

[2,2,2,2,1] => [5,4] => 1010000 => 4

[2,2,2,1,1,1] => [6,3] => 10001000 => 6

[2,2,1,1,1,1,1] => [7,2] => 100000100 => 8

[2,1,1,1,1,1,1,1] => [8,1] => 1000000010 => 10

[1,1,1,1,1,1,1,1,1] => [9] => 1000000000 => 1

[10] => [1,1,1,1,1,1,1,1,1,1] => 11111111110 => 10

[9,1] => [2,1,1,1,1,1,1,1,1] => 10111111110 => 11

[8,2] => [2,2,1,1,1,1,1,1] => 1101111110 => 11

[8,1,1] => [3,1,1,1,1,1,1,1] => 10011111110 => 11

[7,3] => [2,2,2,1,1,1,1] => 111011110 => 11

>>> Load all 352 entries. <<<

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$F_{1} = q$

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

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

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

$F_{5} = q + q^{4} + q^{5} + 4\ q^{6}$

$F_{6} = q + q^{2} + q^{3} + q^{5} + q^{6} + 5\ q^{7} + q^{9}$

$F_{7} = q + q^{4} + q^{5} + q^{6} + 2\ q^{7} + 7\ q^{8} + q^{10} + q^{11}$

$F_{8} = q + q^{2} + q^{4} + q^{5} + 2\ q^{6} + q^{7} + 2\ q^{8} + 8\ q^{9} + q^{10} + 2\ q^{11} + q^{12} + q^{13}$

$F_{9} = q + q^{3} + q^{4} + 2\ q^{6} + q^{7} + 2\ q^{8} + 3\ q^{9} + 10\ q^{10} + q^{11} + 4\ q^{12} + 2\ q^{13} + q^{14} + q^{15}$

$F_{10} = q + q^{2} + 3\ q^{5} + 4\ q^{7} + q^{8} + 3\ q^{9} + 2\ q^{10} + 13\ q^{11} + q^{12} + 5\ q^{13} + 3\ q^{14} + 2\ q^{15} + 2\ q^{16} + q^{17}$

Description

The major index of a binary word.
This is the sum of the positions of descents, i.e., a one followed by a zero.
For words of length

$n$ with

$a$ zeros, the generating function for the major index is the

$q$ -binomial coefficient

$\binom{n}{a}_q$ .

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.

Map

conjugate

Description

Return the conjugate partition of the partition.
The conjugate partition of the partition

$\lambda$ of

$n$ is the partition

$\lambda^*$ whose Ferrers diagram is obtained from the diagram of

$\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.