- FindStat

Identifier

Mp00095: Integer partitions —to binary word⟶ Binary words
St000347: Binary words ⟶ ℤ

Values

[1] => 10 => 1

[2] => 100 => 3

[1,1] => 110 => 3

[3] => 1000 => 6

[2,1] => 1010 => 5

[1,1,1] => 1110 => 6

[4] => 10000 => 10

[3,1] => 10010 => 8

[2,2] => 1100 => 8

[2,1,1] => 10110 => 8

[1,1,1,1] => 11110 => 10

[5] => 100000 => 15

[4,1] => 100010 => 12

[3,2] => 10100 => 11

[3,1,1] => 100110 => 11

[2,2,1] => 11010 => 11

[2,1,1,1] => 101110 => 12

[1,1,1,1,1] => 111110 => 15

[6] => 1000000 => 21

[5,1] => 1000010 => 17

[4,2] => 100100 => 15

[4,1,1] => 1000110 => 15

[3,3] => 11000 => 15

[3,2,1] => 101010 => 14

[3,1,1,1] => 1001110 => 15

[2,2,2] => 11100 => 15

[2,2,1,1] => 110110 => 15

[2,1,1,1,1] => 1011110 => 17

[1,1,1,1,1,1] => 1111110 => 21

[7] => 10000000 => 28

[6,1] => 10000010 => 23

[5,2] => 1000100 => 20

[5,1,1] => 10000110 => 20

[4,3] => 101000 => 19

[4,2,1] => 1001010 => 18

[4,1,1,1] => 10001110 => 19

[3,3,1] => 110010 => 18

[3,2,2] => 101100 => 18

[3,2,1,1] => 1010110 => 18

[3,1,1,1,1] => 10011110 => 20

[2,2,2,1] => 111010 => 19

[2,2,1,1,1] => 1101110 => 20

[2,1,1,1,1,1] => 10111110 => 23

[1,1,1,1,1,1,1] => 11111110 => 28

[8] => 100000000 => 36

[7,1] => 100000010 => 30

[6,2] => 10000100 => 26

[6,1,1] => 100000110 => 26

[5,3] => 1001000 => 24

[5,2,1] => 10001010 => 23

[5,1,1,1] => 100001110 => 24

[4,4] => 110000 => 24

[4,3,1] => 1010010 => 22

[4,2,2] => 1001100 => 22

[4,2,1,1] => 10010110 => 22

[4,1,1,1,1] => 100011110 => 24

[3,3,2] => 110100 => 22

[3,3,1,1] => 1100110 => 22

[3,2,2,1] => 1011010 => 22

[3,2,1,1,1] => 10101110 => 23

[3,1,1,1,1,1] => 100111110 => 26

[2,2,2,2] => 111100 => 24

[2,2,2,1,1] => 1110110 => 24

[2,2,1,1,1,1] => 11011110 => 26

[2,1,1,1,1,1,1] => 101111110 => 30

[1,1,1,1,1,1,1,1] => 111111110 => 36

[7,2] => 100000100 => 33

[6,3] => 10001000 => 30

[6,2,1] => 100001010 => 29

[5,4] => 1010000 => 29

[5,3,1] => 10010010 => 27

[5,2,2] => 10001100 => 27

[5,2,1,1] => 100010110 => 27

[4,4,1] => 1100010 => 27

[4,3,2] => 1010100 => 26

[4,3,1,1] => 10100110 => 26

[4,2,2,1] => 10011010 => 26

[4,2,1,1,1] => 100101110 => 27

[3,3,3] => 111000 => 27

[3,3,2,1] => 1101010 => 26

[3,3,1,1,1] => 11001110 => 27

[3,2,2,2] => 1011100 => 27

[3,2,2,1,1] => 10110110 => 27

[3,2,1,1,1,1] => 101011110 => 29

[2,2,2,2,1] => 1111010 => 29

[2,2,2,1,1,1] => 11101110 => 30

[2,2,1,1,1,1,1] => 110111110 => 33

[7,3] => 100001000 => 37

[6,4] => 10010000 => 35

[6,3,1] => 100010010 => 33

[6,2,2] => 100001100 => 33

[5,5] => 1100000 => 35

[5,4,1] => 10100010 => 32

[5,3,2] => 10010100 => 31

[5,3,1,1] => 100100110 => 31

[5,2,2,1] => 100011010 => 31

[4,4,2] => 1100100 => 31

[4,4,1,1] => 11000110 => 31

[4,3,3] => 1011000 => 31

[4,3,2,1] => 10101010 => 30

[4,3,1,1,1] => 101001110 => 31

>>> Load all 250 entries. <<<

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$F_{1} = q$

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

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

$F_{4} = 3\ q^{8} + 2\ q^{10}$

$F_{5} = 3\ q^{11} + 2\ q^{12} + 2\ q^{15}$

$F_{6} = q^{14} + 6\ q^{15} + 2\ q^{17} + 2\ q^{21}$

$F_{7} = 4\ q^{18} + 3\ q^{19} + 4\ q^{20} + 2\ q^{23} + 2\ q^{28}$

$F_{8} = 6\ q^{22} + 2\ q^{23} + 6\ q^{24} + 4\ q^{26} + 2\ q^{30} + 2\ q^{36}$

Description

The inversion sum of a binary word.
A pair

$a < b$ is an inversion of a binary word

$w = w_1 \cdots w_n$ if

$w_a = 1 > 0 = w_b$ . The inversion sum is given by

$\sum(b-a)$ over all inversions of

$\pi$ .

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.