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Statistic identifier: St000770
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Collection: Integer partitions
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Description: The major index of an integer partition when read from bottom to top.
This is the sum of the positions of the corners of the shape of an integer partition when reading from bottom to top.
For example, the partition $\lambda = (8,6,6,4,3,3)$ has corners at positions 3,6,9, and 13, giving a major index of 31.
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References:
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Code:
def statistic(L):
L = list(L) + [0]
return sum( len(L)-i-2+L[i] for i in range(len(L)-1) if L[i] > L[i+1] )
-----------------------------------------------------------------------------
Statistic values:
[2] => 2
[1,1] => 1
[3] => 3
[2,1] => 4
[1,1,1] => 1
[4] => 4
[3,1] => 5
[2,2] => 2
[2,1,1] => 5
[1,1,1,1] => 1
[5] => 5
[4,1] => 6
[3,2] => 6
[3,1,1] => 6
[2,2,1] => 4
[2,1,1,1] => 6
[1,1,1,1,1] => 1
[6] => 6
[5,1] => 7
[4,2] => 7
[4,1,1] => 7
[3,3] => 3
[3,2,1] => 9
[3,1,1,1] => 7
[2,2,2] => 2
[2,2,1,1] => 5
[2,1,1,1,1] => 7
[1,1,1,1,1,1] => 1
[7] => 7
[6,1] => 8
[5,2] => 8
[5,1,1] => 8
[4,3] => 8
[4,2,1] => 10
[4,1,1,1] => 8
[3,3,1] => 5
[3,2,2] => 7
[3,2,1,1] => 11
[3,1,1,1,1] => 8
[2,2,2,1] => 4
[2,2,1,1,1] => 6
[2,1,1,1,1,1] => 8
[1,1,1,1,1,1,1] => 1
[8] => 8
[7,1] => 9
[6,2] => 9
[6,1,1] => 9
[5,3] => 9
[5,2,1] => 11
[5,1,1,1] => 9
[4,4] => 4
[4,3,1] => 11
[4,2,2] => 8
[4,2,1,1] => 12
[4,1,1,1,1] => 9
[3,3,2] => 6
[3,3,1,1] => 6
[3,2,2,1] => 10
[3,2,1,1,1] => 13
[3,1,1,1,1,1] => 9
[2,2,2,2] => 2
[2,2,2,1,1] => 5
[2,2,1,1,1,1] => 7
[2,1,1,1,1,1,1] => 9
[1,1,1,1,1,1,1,1] => 1
[9] => 9
[8,1] => 10
[7,2] => 10
[7,1,1] => 10
[6,3] => 10
[6,2,1] => 12
[6,1,1,1] => 10
[5,4] => 10
[5,3,1] => 12
[5,2,2] => 9
[5,2,1,1] => 13
[5,1,1,1,1] => 10
[4,4,1] => 6
[4,3,2] => 12
[4,3,1,1] => 13
[4,2,2,1] => 11
[4,2,1,1,1] => 14
[4,1,1,1,1,1] => 10
[3,3,3] => 3
[3,3,2,1] => 9
[3,3,1,1,1] => 7
[3,2,2,2] => 8
[3,2,2,1,1] => 12
[3,2,1,1,1,1] => 15
[3,1,1,1,1,1,1] => 10
[2,2,2,2,1] => 4
[2,2,2,1,1,1] => 6
[2,2,1,1,1,1,1] => 8
[2,1,1,1,1,1,1,1] => 10
[1,1,1,1,1,1,1,1,1] => 1
[10] => 10
[9,1] => 11
[8,2] => 11
[8,1,1] => 11
[7,3] => 11
[7,2,1] => 13
[7,1,1,1] => 11
[6,4] => 11
[6,3,1] => 13
[6,2,2] => 10
[6,2,1,1] => 14
[6,1,1,1,1] => 11
[5,5] => 5
[5,4,1] => 13
[5,3,2] => 13
[5,3,1,1] => 14
[5,2,2,1] => 12
[5,2,1,1,1] => 15
[5,1,1,1,1,1] => 11
[4,4,2] => 7
[4,4,1,1] => 7
[4,3,3] => 9
[4,3,2,1] => 16
[4,3,1,1,1] => 15
[4,2,2,2] => 9
[4,2,2,1,1] => 13
[4,2,1,1,1,1] => 16
[4,1,1,1,1,1,1] => 11
[3,3,3,1] => 5
[3,3,2,2] => 7
[3,3,2,1,1] => 11
[3,3,1,1,1,1] => 8
[3,2,2,2,1] => 11
[3,2,2,1,1,1] => 14
[3,2,1,1,1,1,1] => 17
[3,1,1,1,1,1,1,1] => 11
[2,2,2,2,2] => 2
[2,2,2,2,1,1] => 5
[2,2,2,1,1,1,1] => 7
[2,2,1,1,1,1,1,1] => 9
[2,1,1,1,1,1,1,1,1] => 11
[1,1,1,1,1,1,1,1,1,1] => 1
[11] => 11
[10,1] => 12
[9,2] => 12
[9,1,1] => 12
[8,3] => 12
[8,2,1] => 14
[8,1,1,1] => 12
[7,4] => 12
[7,3,1] => 14
[7,2,2] => 11
[7,2,1,1] => 15
[7,1,1,1,1] => 12
[6,5] => 12
[6,4,1] => 14
[6,3,2] => 14
[6,3,1,1] => 15
[6,2,2,1] => 13
[6,2,1,1,1] => 16
[6,1,1,1,1,1] => 12
[5,5,1] => 7
[5,4,2] => 14
[5,4,1,1] => 15
[5,3,3] => 10
[5,3,2,1] => 17
[5,3,1,1,1] => 16
[5,2,2,2] => 10
[5,2,2,1,1] => 14
[5,2,1,1,1,1] => 17
[5,1,1,1,1,1,1] => 12
[4,4,3] => 8
[4,4,2,1] => 10
[4,4,1,1,1] => 8
[4,3,3,1] => 12
[4,3,2,2] => 14
[4,3,2,1,1] => 19
[4,3,1,1,1,1] => 17
[4,2,2,2,1] => 12
[4,2,2,1,1,1] => 15
[4,2,1,1,1,1,1] => 18
[4,1,1,1,1,1,1,1] => 12
[3,3,3,2] => 6
[3,3,3,1,1] => 6
[3,3,2,2,1] => 10
[3,3,2,1,1,1] => 13
[3,3,1,1,1,1,1] => 9
[3,2,2,2,2] => 9
[3,2,2,2,1,1] => 13
[3,2,2,1,1,1,1] => 16
[3,2,1,1,1,1,1,1] => 19
[3,1,1,1,1,1,1,1,1] => 12
[2,2,2,2,2,1] => 4
[2,2,2,2,1,1,1] => 6
[2,2,2,1,1,1,1,1] => 8
[2,2,1,1,1,1,1,1,1] => 10
[2,1,1,1,1,1,1,1,1,1] => 12
[1,1,1,1,1,1,1,1,1,1,1] => 1
[12] => 12
[11,1] => 13
[10,2] => 13
[10,1,1] => 13
[9,3] => 13
[9,2,1] => 15
[9,1,1,1] => 13
[8,4] => 13
[8,3,1] => 15
[8,2,2] => 12
[8,2,1,1] => 16
[8,1,1,1,1] => 13
[7,5] => 13
[7,4,1] => 15
[7,3,2] => 15
[7,3,1,1] => 16
[7,2,2,1] => 14
[7,2,1,1,1] => 17
[7,1,1,1,1,1] => 13
[6,6] => 6
[6,5,1] => 15
[6,4,2] => 15
[6,4,1,1] => 16
[6,3,3] => 11
[6,3,2,1] => 18
[6,3,1,1,1] => 17
[6,2,2,2] => 11
[6,2,2,1,1] => 15
[6,2,1,1,1,1] => 18
[6,1,1,1,1,1,1] => 13
[5,5,2] => 8
[5,5,1,1] => 8
[5,4,3] => 15
[5,4,2,1] => 18
[5,4,1,1,1] => 17
[5,3,3,1] => 13
[5,3,2,2] => 15
[5,3,2,1,1] => 20
[5,3,1,1,1,1] => 18
[5,2,2,2,1] => 13
[5,2,2,1,1,1] => 16
[5,2,1,1,1,1,1] => 19
[5,1,1,1,1,1,1,1] => 13
[4,4,4] => 4
[4,4,3,1] => 11
[4,4,2,2] => 8
[4,4,2,1,1] => 12
[4,4,1,1,1,1] => 9
[4,3,3,2] => 13
[4,3,3,1,1] => 14
[4,3,2,2,1] => 18
[4,3,2,1,1,1] => 22
[4,3,1,1,1,1,1] => 19
[4,2,2,2,2] => 10
[4,2,2,2,1,1] => 14
[4,2,2,1,1,1,1] => 17
[4,2,1,1,1,1,1,1] => 20
[4,1,1,1,1,1,1,1,1] => 13
[3,3,3,3] => 3
[3,3,3,2,1] => 9
[3,3,3,1,1,1] => 7
[3,3,2,2,2] => 8
[3,3,2,2,1,1] => 12
[3,3,2,1,1,1,1] => 15
[3,3,1,1,1,1,1,1] => 10
[3,2,2,2,2,1] => 12
[3,2,2,2,1,1,1] => 15
[3,2,2,1,1,1,1,1] => 18
[3,2,1,1,1,1,1,1,1] => 21
[3,1,1,1,1,1,1,1,1,1] => 13
[2,2,2,2,2,2] => 2
[2,2,2,2,2,1,1] => 5
[2,2,2,2,1,1,1,1] => 7
[2,2,2,1,1,1,1,1,1] => 9
[2,2,1,1,1,1,1,1,1,1] => 11
[2,1,1,1,1,1,1,1,1,1,1] => 13
[1,1,1,1,1,1,1,1,1,1,1,1] => 1
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Created: Apr 10, 2017 at 15:37 by Christian Stump
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Last Updated: Apr 10, 2017 at 15:37 by Christian Stump