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Statistic identifier: St000509
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Collection: Integer partitions
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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.
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References:
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Code:
def statistic(la):
return sum((j - i) for (i, j) in la.cells())
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Statistic 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
[7,1,1,1] => 15
[6,4] => 17
[6,3,1] => 13
[6,2,2] => 11
[6,2,1,1] => 9
[6,1,1,1,1] => 5
[5,5] => 15
[5,4,1] => 10
[5,3,2] => 7
[5,3,1,1] => 5
[5,2,2,1] => 3
[5,2,1,1,1] => 0
[5,1,1,1,1,1] => -5
[4,4,2] => 5
[4,4,1,1] => 3
[4,3,3] => 3
[4,3,2,1] => 0
[4,3,1,1,1] => -3
[4,2,2,2] => -3
[4,2,2,1,1] => -5
[4,2,1,1,1,1] => -9
[4,1,1,1,1,1,1] => -15
[3,3,3,1] => -3
[3,3,2,2] => -5
[3,3,2,1,1] => -7
[3,3,1,1,1,1] => -11
[3,2,2,2,1] => -10
[3,2,2,1,1,1] => -13
[3,2,1,1,1,1,1] => -18
[3,1,1,1,1,1,1,1] => -25
[2,2,2,2,2] => -15
[2,2,2,2,1,1] => -17
[2,2,2,1,1,1,1] => -21
[2,2,1,1,1,1,1,1] => -27
[2,1,1,1,1,1,1,1,1] => -35
[1,1,1,1,1,1,1,1,1,1] => -45
[11] => 55
[10,1] => 44
[9,2] => 35
[9,1,1] => 33
[8,3] => 28
[8,2,1] => 25
[8,1,1,1] => 22
[7,4] => 23
[7,3,1] => 19
[7,2,2] => 17
[7,2,1,1] => 15
[7,1,1,1,1] => 11
[6,5] => 20
[6,4,1] => 15
[6,3,2] => 12
[6,3,1,1] => 10
[6,2,2,1] => 8
[6,2,1,1,1] => 5
[6,1,1,1,1,1] => 0
[5,5,1] => 13
[5,4,2] => 9
[5,4,1,1] => 7
[5,3,3] => 7
[5,3,2,1] => 4
[5,3,1,1,1] => 1
[5,2,2,2] => 1
[5,2,2,1,1] => -1
[5,2,1,1,1,1] => -5
[5,1,1,1,1,1,1] => -11
[4,4,3] => 5
[4,4,2,1] => 2
[4,4,1,1,1] => -1
[4,3,3,1] => 0
[4,3,2,2] => -2
[4,3,2,1,1] => -4
[4,3,1,1,1,1] => -8
[4,2,2,2,1] => -7
[4,2,2,1,1,1] => -10
[4,2,1,1,1,1,1] => -15
[4,1,1,1,1,1,1,1] => -22
[3,3,3,2] => -5
[3,3,3,1,1] => -7
[3,3,2,2,1] => -9
[3,3,2,1,1,1] => -12
[3,3,1,1,1,1,1] => -17
[3,2,2,2,2] => -13
[3,2,2,2,1,1] => -15
[3,2,2,1,1,1,1] => -19
[3,2,1,1,1,1,1,1] => -25
[3,1,1,1,1,1,1,1,1] => -33
[2,2,2,2,2,1] => -20
[2,2,2,2,1,1,1] => -23
[2,2,2,1,1,1,1,1] => -28
[2,2,1,1,1,1,1,1,1] => -35
[2,1,1,1,1,1,1,1,1,1] => -44
[1,1,1,1,1,1,1,1,1,1,1] => -55
[12] => 66
[11,1] => 54
[10,2] => 44
[10,1,1] => 42
[9,3] => 36
[9,2,1] => 33
[9,1,1,1] => 30
[8,4] => 30
[8,3,1] => 26
[8,2,2] => 24
[8,2,1,1] => 22
[8,1,1,1,1] => 18
[7,5] => 26
[7,4,1] => 21
[7,3,2] => 18
[7,3,1,1] => 16
[7,2,2,1] => 14
[7,2,1,1,1] => 11
[7,1,1,1,1,1] => 6
[6,6] => 24
[6,5,1] => 18
[6,4,2] => 14
[6,4,1,1] => 12
[6,3,3] => 12
[6,3,2,1] => 9
[6,3,1,1,1] => 6
[6,2,2,2] => 6
[6,2,2,1,1] => 4
[6,2,1,1,1,1] => 0
[6,1,1,1,1,1,1] => -6
[5,5,2] => 12
[5,5,1,1] => 10
[5,4,3] => 9
[5,4,2,1] => 6
[5,4,1,1,1] => 3
[5,3,3,1] => 4
[5,3,2,2] => 2
[5,3,2,1,1] => 0
[5,3,1,1,1,1] => -4
[5,2,2,2,1] => -3
[5,2,2,1,1,1] => -6
[5,2,1,1,1,1,1] => -11
[5,1,1,1,1,1,1,1] => -18
[4,4,4] => 6
[4,4,3,1] => 2
[4,4,2,2] => 0
[4,4,2,1,1] => -2
[4,4,1,1,1,1] => -6
[4,3,3,2] => -2
[4,3,3,1,1] => -4
[4,3,2,2,1] => -6
[4,3,2,1,1,1] => -9
[4,3,1,1,1,1,1] => -14
[4,2,2,2,2] => -10
[4,2,2,2,1,1] => -12
[4,2,2,1,1,1,1] => -16
[4,2,1,1,1,1,1,1] => -22
[4,1,1,1,1,1,1,1,1] => -30
[3,3,3,3] => -6
[3,3,3,2,1] => -9
[3,3,3,1,1,1] => -12
[3,3,2,2,2] => -12
[3,3,2,2,1,1] => -14
[3,3,2,1,1,1,1] => -18
[3,3,1,1,1,1,1,1] => -24
[3,2,2,2,2,1] => -18
[3,2,2,2,1,1,1] => -21
[3,2,2,1,1,1,1,1] => -26
[3,2,1,1,1,1,1,1,1] => -33
[3,1,1,1,1,1,1,1,1,1] => -42
[2,2,2,2,2,2] => -24
[2,2,2,2,2,1,1] => -26
[2,2,2,2,1,1,1,1] => -30
[2,2,2,1,1,1,1,1,1] => -36
[2,2,1,1,1,1,1,1,1,1] => -44
[2,1,1,1,1,1,1,1,1,1,1] => -54
[1,1,1,1,1,1,1,1,1,1,1,1] => -66
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Created: May 25, 2016 at 20:28 by Franco Saliola
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Last Updated: May 25, 2016 at 20:28 by Franco Saliola