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Statistic identifier: St000867
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Collection: Integer partitions
-----------------------------------------------------------------------------
Description: The sum of the hook lengths in the first row 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 plus one. This statistic is the sum of the hook lengths of the first row of a partition.
Put differently, for a partition of size $n$ with first parth $\lambda_1$, this is $\binom{\lambda_1}{2} + n$.
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References:
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Code:
def statistic(L):
return L.size() + binomial(L[0], 2)
-----------------------------------------------------------------------------
Statistic values:
[] => 0
[1] => 1
[2] => 3
[1,1] => 2
[3] => 6
[2,1] => 4
[1,1,1] => 3
[4] => 10
[3,1] => 7
[2,2] => 5
[2,1,1] => 5
[1,1,1,1] => 4
[5] => 15
[4,1] => 11
[3,2] => 8
[3,1,1] => 8
[2,2,1] => 6
[2,1,1,1] => 6
[1,1,1,1,1] => 5
[6] => 21
[5,1] => 16
[4,2] => 12
[4,1,1] => 12
[3,3] => 9
[3,2,1] => 9
[3,1,1,1] => 9
[2,2,2] => 7
[2,2,1,1] => 7
[2,1,1,1,1] => 7
[1,1,1,1,1,1] => 6
[7] => 28
[6,1] => 22
[5,2] => 17
[5,1,1] => 17
[4,3] => 13
[4,2,1] => 13
[4,1,1,1] => 13
[3,3,1] => 10
[3,2,2] => 10
[3,2,1,1] => 10
[3,1,1,1,1] => 10
[2,2,2,1] => 8
[2,2,1,1,1] => 8
[2,1,1,1,1,1] => 8
[1,1,1,1,1,1,1] => 7
[8] => 36
[7,1] => 29
[6,2] => 23
[6,1,1] => 23
[5,3] => 18
[5,2,1] => 18
[5,1,1,1] => 18
[4,4] => 14
[4,3,1] => 14
[4,2,2] => 14
[4,2,1,1] => 14
[4,1,1,1,1] => 14
[3,3,2] => 11
[3,3,1,1] => 11
[3,2,2,1] => 11
[3,2,1,1,1] => 11
[3,1,1,1,1,1] => 11
[2,2,2,2] => 9
[2,2,2,1,1] => 9
[2,2,1,1,1,1] => 9
[2,1,1,1,1,1,1] => 9
[1,1,1,1,1,1,1,1] => 8
[9] => 45
[8,1] => 37
[7,2] => 30
[7,1,1] => 30
[6,3] => 24
[6,2,1] => 24
[6,1,1,1] => 24
[5,4] => 19
[5,3,1] => 19
[5,2,2] => 19
[5,2,1,1] => 19
[5,1,1,1,1] => 19
[4,4,1] => 15
[4,3,2] => 15
[4,3,1,1] => 15
[4,2,2,1] => 15
[4,2,1,1,1] => 15
[4,1,1,1,1,1] => 15
[3,3,3] => 12
[3,3,2,1] => 12
[3,3,1,1,1] => 12
[3,2,2,2] => 12
[3,2,2,1,1] => 12
[3,2,1,1,1,1] => 12
[3,1,1,1,1,1,1] => 12
[2,2,2,2,1] => 10
[2,2,2,1,1,1] => 10
[2,2,1,1,1,1,1] => 10
[2,1,1,1,1,1,1,1] => 10
[1,1,1,1,1,1,1,1,1] => 9
[10] => 55
[9,1] => 46
[8,2] => 38
[8,1,1] => 38
[7,3] => 31
[7,2,1] => 31
[7,1,1,1] => 31
[6,4] => 25
[6,3,1] => 25
[6,2,2] => 25
[6,2,1,1] => 25
[6,1,1,1,1] => 25
[5,5] => 20
[5,4,1] => 20
[5,3,2] => 20
[5,3,1,1] => 20
[5,2,2,1] => 20
[5,2,1,1,1] => 20
[5,1,1,1,1,1] => 20
[4,4,2] => 16
[4,4,1,1] => 16
[4,3,3] => 16
[4,3,2,1] => 16
[4,3,1,1,1] => 16
[4,2,2,2] => 16
[4,2,2,1,1] => 16
[4,2,1,1,1,1] => 16
[4,1,1,1,1,1,1] => 16
[3,3,3,1] => 13
[3,3,2,2] => 13
[3,3,2,1,1] => 13
[3,3,1,1,1,1] => 13
[3,2,2,2,1] => 13
[3,2,2,1,1,1] => 13
[3,2,1,1,1,1,1] => 13
[3,1,1,1,1,1,1,1] => 13
[2,2,2,2,2] => 11
[2,2,2,2,1,1] => 11
[2,2,2,1,1,1,1] => 11
[2,2,1,1,1,1,1,1] => 11
[2,1,1,1,1,1,1,1,1] => 11
[1,1,1,1,1,1,1,1,1,1] => 10
[11] => 66
[10,1] => 56
[9,2] => 47
[9,1,1] => 47
[8,3] => 39
[8,2,1] => 39
[8,1,1,1] => 39
[7,4] => 32
[7,3,1] => 32
[7,2,2] => 32
[7,2,1,1] => 32
[7,1,1,1,1] => 32
[6,5] => 26
[6,4,1] => 26
[6,3,2] => 26
[6,3,1,1] => 26
[6,2,2,1] => 26
[6,2,1,1,1] => 26
[6,1,1,1,1,1] => 26
[5,5,1] => 21
[5,4,2] => 21
[5,4,1,1] => 21
[5,3,3] => 21
[5,3,2,1] => 21
[5,3,1,1,1] => 21
[5,2,2,2] => 21
[5,2,2,1,1] => 21
[5,2,1,1,1,1] => 21
[5,1,1,1,1,1,1] => 21
[4,4,3] => 17
[4,4,2,1] => 17
[4,4,1,1,1] => 17
[4,3,3,1] => 17
[4,3,2,2] => 17
[4,3,2,1,1] => 17
[4,3,1,1,1,1] => 17
[4,2,2,2,1] => 17
[4,2,2,1,1,1] => 17
[4,2,1,1,1,1,1] => 17
[4,1,1,1,1,1,1,1] => 17
[3,3,3,2] => 14
[3,3,3,1,1] => 14
[3,3,2,2,1] => 14
[3,3,2,1,1,1] => 14
[3,3,1,1,1,1,1] => 14
[3,2,2,2,2] => 14
[3,2,2,2,1,1] => 14
[3,2,2,1,1,1,1] => 14
[3,2,1,1,1,1,1,1] => 14
[3,1,1,1,1,1,1,1,1] => 14
[2,2,2,2,2,1] => 12
[2,2,2,2,1,1,1] => 12
[2,2,2,1,1,1,1,1] => 12
[2,2,1,1,1,1,1,1,1] => 12
[2,1,1,1,1,1,1,1,1,1] => 12
[1,1,1,1,1,1,1,1,1,1,1] => 11
[12] => 78
[11,1] => 67
[10,2] => 57
[10,1,1] => 57
[9,3] => 48
[9,2,1] => 48
[9,1,1,1] => 48
[8,4] => 40
[8,3,1] => 40
[8,2,2] => 40
[8,2,1,1] => 40
[8,1,1,1,1] => 40
[7,5] => 33
[7,4,1] => 33
[7,3,2] => 33
[7,3,1,1] => 33
[7,2,2,1] => 33
[7,2,1,1,1] => 33
[7,1,1,1,1,1] => 33
[6,6] => 27
[6,5,1] => 27
[6,4,2] => 27
[6,4,1,1] => 27
[6,3,3] => 27
[6,3,2,1] => 27
[6,3,1,1,1] => 27
[6,2,2,2] => 27
[6,2,2,1,1] => 27
[6,2,1,1,1,1] => 27
[6,1,1,1,1,1,1] => 27
[5,5,2] => 22
[5,5,1,1] => 22
[5,4,3] => 22
[5,4,2,1] => 22
[5,4,1,1,1] => 22
[5,3,3,1] => 22
[5,3,2,2] => 22
[5,3,2,1,1] => 22
[5,3,1,1,1,1] => 22
[5,2,2,2,1] => 22
[5,2,2,1,1,1] => 22
[5,2,1,1,1,1,1] => 22
[5,1,1,1,1,1,1,1] => 22
[4,4,4] => 18
[4,4,3,1] => 18
[4,4,2,2] => 18
[4,4,2,1,1] => 18
[4,4,1,1,1,1] => 18
[4,3,3,2] => 18
[4,3,3,1,1] => 18
[4,3,2,2,1] => 18
[4,3,2,1,1,1] => 18
[4,3,1,1,1,1,1] => 18
[4,2,2,2,2] => 18
[4,2,2,2,1,1] => 18
[4,2,2,1,1,1,1] => 18
[4,2,1,1,1,1,1,1] => 18
[4,1,1,1,1,1,1,1,1] => 18
[3,3,3,3] => 15
[3,3,3,2,1] => 15
[3,3,3,1,1,1] => 15
[3,3,2,2,2] => 15
[3,3,2,2,1,1] => 15
[3,3,2,1,1,1,1] => 15
[3,3,1,1,1,1,1,1] => 15
[3,2,2,2,2,1] => 15
[3,2,2,2,1,1,1] => 15
[3,2,2,1,1,1,1,1] => 15
[3,2,1,1,1,1,1,1,1] => 15
[3,1,1,1,1,1,1,1,1,1] => 15
[2,2,2,2,2,2] => 13
[2,2,2,2,2,1,1] => 13
[2,2,2,2,1,1,1,1] => 13
[2,2,2,1,1,1,1,1,1] => 13
[2,2,1,1,1,1,1,1,1,1] => 13
[2,1,1,1,1,1,1,1,1,1,1] => 13
[1,1,1,1,1,1,1,1,1,1,1,1] => 12
[5,4,3,1] => 23
[5,4,2,2] => 23
[5,4,2,1,1] => 23
[5,3,3,2] => 23
[5,3,3,1,1] => 23
[5,3,2,2,1] => 23
[4,4,3,2] => 19
[4,4,3,1,1] => 19
[4,4,2,2,1] => 19
[4,3,3,2,1] => 19
[5,4,3,2] => 24
[5,4,3,1,1] => 24
[5,4,2,2,1] => 24
[5,3,3,2,1] => 24
[4,4,3,2,1] => 20
[5,4,3,2,1] => 25
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Created: Jun 27, 2017 at 09:08 by Christian Stump
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Last Updated: Jun 19, 2020 at 13:57 by Martin Rubey