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Identifier

St000869: Integer partitions ⟶ ℤ

Values

[] => 0

[1] => 1

[2] => 3

[1,1] => 3

[3] => 6

[2,1] => 5

[1,1,1] => 6

[4] => 10

[3,1] => 8

[2,2] => 8

[2,1,1] => 8

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

[5] => 15

[4,1] => 12

[3,2] => 11

[3,1,1] => 11

[2,2,1] => 11

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

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

[6] => 21

[5,1] => 17

[4,2] => 15

[4,1,1] => 15

[3,3] => 15

[3,2,1] => 14

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

[2,2,2] => 15

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

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

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

[7] => 28

[6,1] => 23

[5,2] => 20

[5,1,1] => 20

[4,3] => 19

[4,2,1] => 18

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

[3,3,1] => 18

[3,2,2] => 18

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

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

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

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

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

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

[8] => 36

[7,1] => 30

[6,2] => 26

[6,1,1] => 26

[5,3] => 24

[5,2,1] => 23

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

[4,4] => 24

[4,3,1] => 22

[4,2,2] => 22

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

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

[3,3,2] => 22

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

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

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

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

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

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

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

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

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

[9] => 45

[8,1] => 38

[7,2] => 33

[7,1,1] => 33

[6,3] => 30

[6,2,1] => 29

[6,1,1,1] => 30

[5,4] => 29

[5,3,1] => 27

[5,2,2] => 27

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

[5,1,1,1,1] => 29

[4,4,1] => 27

[4,3,2] => 26

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

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

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

[4,1,1,1,1,1] => 30

[3,3,3] => 27

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

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

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

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

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

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

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

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

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

[2,1,1,1,1,1,1,1] => 38

[1,1,1,1,1,1,1,1,1] => 45

[10] => 55

[9,1] => 47

[8,2] => 41

[8,1,1] => 41

>>> Load all 757 entries. <<<

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$F_{1} = q$

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

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

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

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

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

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

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

$F_{9} = 4\ q^{26} + 9\ q^{27} + 5\ q^{29} + 4\ q^{30} + 4\ q^{33} + 2\ q^{38} + 2\ q^{45}$

$F_{10} = q^{30} + 12\ q^{31} + 3\ q^{32} + 6\ q^{33} + 6\ q^{35} + 2\ q^{36} + 4\ q^{37} + 4\ q^{41} + 2\ q^{47} + 2\ q^{55}$

$F_{11} = 5\ q^{35} + 12\ q^{36} + 6\ q^{37} + 6\ q^{38} + 6\ q^{40} + 3\ q^{41} + 4\ q^{42} + 2\ q^{44} + 4\ q^{45} + 4\ q^{50} + 2\ q^{57} + 2\ q^{66}$

$F_{12} = 10\ q^{40} + 6\ q^{41} + 18\ q^{42} + 9\ q^{44} + 4\ q^{45} + 12\ q^{48} + 4\ q^{50} + 2\ q^{53} + 4\ q^{54} + 4\ q^{60} + 2\ q^{68} + 2\ q^{78}$

Description

The sum of the hook lengths of an integer partition.
For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below + 1. This statistic is the sum of all hook lengths of a partition.

References

[1] Amdeberhan, T. Average on sum of hooks in a rectangle MathOverflow:273095

Code

def statistic(L):
    return sum(L.hooks())

Created

Jun 27, 2017 at 09:02 by Christian Stump

Updated

Mar 11, 2024 at 11:26 by Martin Rubey