- FindStat

Identifier

St000509: Integer partitions ⟶ ℤ

Values

[2] => 1

[1,1] => -1

[3] => 3

[2,1] => 0

[1,1,1] => -3

[4] => 6

[3,1] => 2

[2,2] => 0

[2,1,1] => -2

[1,1,1,1] => -6

[5] => 10

[4,1] => 5

[3,2] => 2

[3,1,1] => 0

[2,2,1] => -2

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

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

[6] => 15

[5,1] => 9

[4,2] => 5

[4,1,1] => 3

[3,3] => 3

[3,2,1] => 0

[3,1,1,1] => -3

[2,2,2] => -3

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

[2,1,1,1,1] => -9

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

[7] => 21

[6,1] => 14

[5,2] => 9

[5,1,1] => 7

[4,3] => 6

[4,2,1] => 3

[4,1,1,1] => 0

[3,3,1] => 1

[3,2,2] => -1

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

[3,1,1,1,1] => -7

[2,2,2,1] => -6

[2,2,1,1,1] => -9

[2,1,1,1,1,1] => -14

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

[8] => 28

[7,1] => 20

[6,2] => 14

[6,1,1] => 12

[5,3] => 10

[5,2,1] => 7

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

[4,4] => 8

[4,3,1] => 4

[4,2,2] => 2

[4,2,1,1] => 0

[4,1,1,1,1] => -4

[3,3,2] => 0

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

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

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

[3,1,1,1,1,1] => -12

[2,2,2,2] => -8

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

[2,2,1,1,1,1] => -14

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

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

[9] => 36

[8,1] => 27

[7,2] => 20

[7,1,1] => 18

[6,3] => 15

[6,2,1] => 12

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

[5,4] => 12

[5,3,1] => 8

[5,2,2] => 6

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

[5,1,1,1,1] => 0

[4,4,1] => 6

[4,3,2] => 3

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

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

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

[4,1,1,1,1,1] => -9

[3,3,3] => 0

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

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

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

[3,2,2,1,1] => -8

[3,2,1,1,1,1] => -12

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

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

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

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

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

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

[10] => 45

[9,1] => 35

[8,2] => 27

[8,1,1] => 25

[7,3] => 21

[7,2,1] => 18

>>> Load all 270 entries. <<<

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$F_{2} = q^{-1} + q$

$F_{3} = q^{-3} + 1 + q^{3}$

$F_{4} = q^{-6} + q^{-2} + 1 + q^{2} + q^{6}$

$F_{5} = q^{-10} + q^{-5} + q^{-2} + 1 + q^{2} + q^{5} + q^{10}$

$F_{6} = q^{-15} + q^{-9} + q^{-5} + 2\ q^{-3} + 1 + 2\ q^{3} + q^{5} + q^{9} + q^{15}$

$F_{7} = q^{-21} + q^{-14} + q^{-9} + q^{-7} + q^{-6} + q^{-3} + q^{-1} + 1 + q + q^{3} + q^{6} + q^{7} + q^{9} + q^{14} + q^{21}$

$F_{8} = q^{-28} + q^{-20} + q^{-14} + q^{-12} + q^{-10} + q^{-8} + q^{-7} + 2\ q^{-4} + q^{-2} + 2 + q^{2} + 2\ q^{4} + q^{7} + q^{8} + q^{10} + q^{12} + q^{14} + q^{20} + q^{28}$

$F_{9} = q^{-36} + q^{-27} + q^{-20} + q^{-18} + q^{-15} + 2\ q^{-12} + q^{-9} + q^{-8} + 2\ q^{-6} + q^{-4} + q^{-3} + q^{-1} + 2 + q + q^{3} + q^{4} + 2\ q^{6} + q^{8} + q^{9} + 2\ q^{12} + q^{15} + q^{18} + q^{20} + q^{27} + q^{36}$

$F_{10} = q^{-45} + q^{-35} + q^{-27} + q^{-25} + q^{-21} + q^{-18} + q^{-17} + 2\ q^{-15} + q^{-13} + q^{-11} + q^{-10} + q^{-9} + q^{-7} + 3\ q^{-5} + 3\ q^{-3} + 2 + 3\ q^{3} + 3\ q^{5} + q^{7} + q^{9} + q^{10} + q^{11} + q^{13} + 2\ q^{15} + q^{17} + q^{18} + q^{21} + q^{25} + q^{27} + q^{35} + q^{45}$

$F_{11} = q^{-55} + q^{-44} + q^{-35} + q^{-33} + q^{-28} + q^{-25} + q^{-23} + q^{-22} + q^{-20} + q^{-19} + q^{-17} + 2\ q^{-15} + q^{-13} + q^{-12} + q^{-11} + q^{-10} + q^{-9} + q^{-8} + 2\ q^{-7} + 2\ q^{-5} + q^{-4} + q^{-2} + 2\ q^{-1} + 2 + 2\ q + q^{2} + q^{4} + 2\ q^{5} + 2\ q^{7} + q^{8} + q^{9} + q^{10} + q^{11} + q^{12} + q^{13} + 2\ q^{15} + q^{17} + q^{19} + q^{20} + q^{22} + q^{23} + q^{25} + q^{28} + q^{33} + q^{35} + q^{44} + q^{55}$

$F_{12} = q^{-66} + q^{-54} + q^{-44} + q^{-42} + q^{-36} + q^{-33} + 2\ q^{-30} + 2\ q^{-26} + 2\ q^{-24} + q^{-22} + q^{-21} + 3\ q^{-18} + q^{-16} + 2\ q^{-14} + 3\ q^{-12} + q^{-11} + q^{-10} + 2\ q^{-9} + 5\ q^{-6} + 2\ q^{-4} + q^{-3} + 2\ q^{-2} + 3 + 2\ q^{2} + q^{3} + 2\ q^{4} + 5\ q^{6} + 2\ q^{9} + q^{10} + q^{11} + 3\ q^{12} + 2\ q^{14} + q^{16} + 3\ q^{18} + q^{21} + q^{22} + 2\ q^{24} + 2\ q^{26} + 2\ q^{30} + q^{33} + q^{36} + q^{42} + q^{44} + q^{54} + q^{66}$

Description

The diagonal index (content) of a partition.
The diagonal index of the cell at row $r$ and column $c$ of a partition is $c - r$; this is sometimes called the content of the cell. The diagonal index of a partition is the sum of the diagonal index of each cell of the partition.

Code

def statistic(la):
    return sum((j - i) for (i, j) in la.cells())

Created

May 25, 2016 at 20:28 by Franco Saliola

Updated

May 25, 2016 at 20:28 by Franco Saliola