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Statistic identifier: St000870
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Collection: Integer partitions
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Description: The product of the hook lengths of the diagonal cells in 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 product of the hook lengths of the diagonal cells $(i,i)$ of a partition.
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References:
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Code:
def statistic(L):
return prod( L.hook_length(*c) for c in L.cells() if c[0] == c[1] )
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Statistic values:
[1] => 1
[2] => 2
[1,1] => 2
[3] => 3
[2,1] => 3
[1,1,1] => 3
[4] => 4
[3,1] => 4
[2,2] => 3
[2,1,1] => 4
[1,1,1,1] => 4
[5] => 5
[4,1] => 5
[3,2] => 4
[3,1,1] => 5
[2,2,1] => 4
[2,1,1,1] => 5
[1,1,1,1,1] => 5
[6] => 6
[5,1] => 6
[4,2] => 5
[4,1,1] => 6
[3,3] => 8
[3,2,1] => 5
[3,1,1,1] => 6
[2,2,2] => 8
[2,2,1,1] => 5
[2,1,1,1,1] => 6
[1,1,1,1,1,1] => 6
[7] => 7
[6,1] => 7
[5,2] => 6
[5,1,1] => 7
[4,3] => 10
[4,2,1] => 6
[4,1,1,1] => 7
[3,3,1] => 10
[3,2,2] => 10
[3,2,1,1] => 6
[3,1,1,1,1] => 7
[2,2,2,1] => 10
[2,2,1,1,1] => 6
[2,1,1,1,1,1] => 7
[1,1,1,1,1,1,1] => 7
[8] => 8
[7,1] => 8
[6,2] => 7
[6,1,1] => 8
[5,3] => 12
[5,2,1] => 7
[5,1,1,1] => 8
[4,4] => 15
[4,3,1] => 12
[4,2,2] => 12
[4,2,1,1] => 7
[4,1,1,1,1] => 8
[3,3,2] => 15
[3,3,1,1] => 12
[3,2,2,1] => 12
[3,2,1,1,1] => 7
[3,1,1,1,1,1] => 8
[2,2,2,2] => 15
[2,2,2,1,1] => 12
[2,2,1,1,1,1] => 7
[2,1,1,1,1,1,1] => 8
[1,1,1,1,1,1,1,1] => 8
[9] => 9
[8,1] => 9
[7,2] => 8
[7,1,1] => 9
[6,3] => 14
[6,2,1] => 8
[6,1,1,1] => 9
[5,4] => 18
[5,3,1] => 14
[5,2,2] => 14
[5,2,1,1] => 8
[5,1,1,1,1] => 9
[4,4,1] => 18
[4,3,2] => 18
[4,3,1,1] => 14
[4,2,2,1] => 14
[4,2,1,1,1] => 8
[4,1,1,1,1,1] => 9
[3,3,3] => 15
[3,3,2,1] => 18
[3,3,1,1,1] => 14
[3,2,2,2] => 18
[3,2,2,1,1] => 14
[3,2,1,1,1,1] => 8
[3,1,1,1,1,1,1] => 9
[2,2,2,2,1] => 18
[2,2,2,1,1,1] => 14
[2,2,1,1,1,1,1] => 8
[2,1,1,1,1,1,1,1] => 9
[1,1,1,1,1,1,1,1,1] => 9
[10] => 10
[9,1] => 10
[8,2] => 9
[8,1,1] => 10
[7,3] => 16
[7,2,1] => 9
[7,1,1,1] => 10
[6,4] => 21
[6,3,1] => 16
[6,2,2] => 16
[6,2,1,1] => 9
[6,1,1,1,1] => 10
[5,5] => 24
[5,4,1] => 21
[5,3,2] => 21
[5,3,1,1] => 16
[5,2,2,1] => 16
[5,2,1,1,1] => 9
[5,1,1,1,1,1] => 10
[4,4,2] => 24
[4,4,1,1] => 21
[4,3,3] => 18
[4,3,2,1] => 21
[4,3,1,1,1] => 16
[4,2,2,2] => 21
[4,2,2,1,1] => 16
[4,2,1,1,1,1] => 9
[4,1,1,1,1,1,1] => 10
[3,3,3,1] => 18
[3,3,2,2] => 24
[3,3,2,1,1] => 21
[3,3,1,1,1,1] => 16
[3,2,2,2,1] => 21
[3,2,2,1,1,1] => 16
[3,2,1,1,1,1,1] => 9
[3,1,1,1,1,1,1,1] => 10
[2,2,2,2,2] => 24
[2,2,2,2,1,1] => 21
[2,2,2,1,1,1,1] => 16
[2,2,1,1,1,1,1,1] => 9
[2,1,1,1,1,1,1,1,1] => 10
[1,1,1,1,1,1,1,1,1,1] => 10
[11] => 11
[10,1] => 11
[9,2] => 10
[9,1,1] => 11
[8,3] => 18
[8,2,1] => 10
[8,1,1,1] => 11
[7,4] => 24
[7,3,1] => 18
[7,2,2] => 18
[7,2,1,1] => 10
[7,1,1,1,1] => 11
[6,5] => 28
[6,4,1] => 24
[6,3,2] => 24
[6,3,1,1] => 18
[6,2,2,1] => 18
[6,2,1,1,1] => 10
[6,1,1,1,1,1] => 11
[5,5,1] => 28
[5,4,2] => 28
[5,4,1,1] => 24
[5,3,3] => 21
[5,3,2,1] => 24
[5,3,1,1,1] => 18
[5,2,2,2] => 24
[5,2,2,1,1] => 18
[5,2,1,1,1,1] => 10
[5,1,1,1,1,1,1] => 11
[4,4,3] => 24
[4,4,2,1] => 28
[4,4,1,1,1] => 24
[4,3,3,1] => 21
[4,3,2,2] => 28
[4,3,2,1,1] => 24
[4,3,1,1,1,1] => 18
[4,2,2,2,1] => 24
[4,2,2,1,1,1] => 18
[4,2,1,1,1,1,1] => 10
[4,1,1,1,1,1,1,1] => 11
[3,3,3,2] => 24
[3,3,3,1,1] => 21
[3,3,2,2,1] => 28
[3,3,2,1,1,1] => 24
[3,3,1,1,1,1,1] => 18
[3,2,2,2,2] => 28
[3,2,2,2,1,1] => 24
[3,2,2,1,1,1,1] => 18
[3,2,1,1,1,1,1,1] => 10
[3,1,1,1,1,1,1,1,1] => 11
[2,2,2,2,2,1] => 28
[2,2,2,2,1,1,1] => 24
[2,2,2,1,1,1,1,1] => 18
[2,2,1,1,1,1,1,1,1] => 10
[2,1,1,1,1,1,1,1,1,1] => 11
[1,1,1,1,1,1,1,1,1,1,1] => 11
[12] => 12
[11,1] => 12
[10,2] => 11
[10,1,1] => 12
[9,3] => 20
[9,2,1] => 11
[9,1,1,1] => 12
[8,4] => 27
[8,3,1] => 20
[8,2,2] => 20
[8,2,1,1] => 11
[8,1,1,1,1] => 12
[7,5] => 32
[7,4,1] => 27
[7,3,2] => 27
[7,3,1,1] => 20
[7,2,2,1] => 20
[7,2,1,1,1] => 11
[7,1,1,1,1,1] => 12
[6,6] => 35
[6,5,1] => 32
[6,4,2] => 32
[6,4,1,1] => 27
[6,3,3] => 24
[6,3,2,1] => 27
[6,3,1,1,1] => 20
[6,2,2,2] => 27
[6,2,2,1,1] => 20
[6,2,1,1,1,1] => 11
[6,1,1,1,1,1,1] => 12
[5,5,2] => 35
[5,5,1,1] => 32
[5,4,3] => 28
[5,4,2,1] => 32
[5,4,1,1,1] => 27
[5,3,3,1] => 24
[5,3,2,2] => 32
[5,3,2,1,1] => 27
[5,3,1,1,1,1] => 20
[5,2,2,2,1] => 27
[5,2,2,1,1,1] => 20
[5,2,1,1,1,1,1] => 11
[5,1,1,1,1,1,1,1] => 12
[4,4,4] => 48
[4,4,3,1] => 28
[4,4,2,2] => 35
[4,4,2,1,1] => 32
[4,4,1,1,1,1] => 27
[4,3,3,2] => 28
[4,3,3,1,1] => 24
[4,3,2,2,1] => 32
[4,3,2,1,1,1] => 27
[4,3,1,1,1,1,1] => 20
[4,2,2,2,2] => 32
[4,2,2,2,1,1] => 27
[4,2,2,1,1,1,1] => 20
[4,2,1,1,1,1,1,1] => 11
[4,1,1,1,1,1,1,1,1] => 12
[3,3,3,3] => 48
[3,3,3,2,1] => 28
[3,3,3,1,1,1] => 24
[3,3,2,2,2] => 35
[3,3,2,2,1,1] => 32
[3,3,2,1,1,1,1] => 27
[3,3,1,1,1,1,1,1] => 20
[3,2,2,2,2,1] => 32
[3,2,2,2,1,1,1] => 27
[3,2,2,1,1,1,1,1] => 20
[3,2,1,1,1,1,1,1,1] => 11
[3,1,1,1,1,1,1,1,1,1] => 12
[2,2,2,2,2,2] => 35
[2,2,2,2,2,1,1] => 32
[2,2,2,2,1,1,1,1] => 27
[2,2,2,1,1,1,1,1,1] => 20
[2,2,1,1,1,1,1,1,1,1] => 11
[2,1,1,1,1,1,1,1,1,1,1] => 12
[1,1,1,1,1,1,1,1,1,1,1,1] => 12
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Created: Jun 27, 2017 at 09:05 by Christian Stump
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Last Updated: Jul 06, 2021 at 07:55 by Martin Rubey