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Statistic identifier: St000477
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Collection: Integer partitions
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Description: The weight of a partition according to Alladi.
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References: [1] Alladi, K. Partition identities involving gaps and weights [[MathSciNet:1401759]]
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Code:
def statistic(pi):
"""
The weight according to Alladi.
sage: r=8; RR = [pi for pi in Partitions(r) if all(pi[i] - pi[i+1] >= 2 for i in range(len(pi)-1))]
sage: sum(weight(pi) for pi in RR) == Partitions(r).cardinality()
"""
return pi[-1]*prod(pi[i] - pi[i+1] -1 for i in range(len(pi)-1))
-----------------------------------------------------------------------------
Statistic values:
[2] => 2
[1,1] => -1
[3] => 3
[2,1] => 0
[1,1,1] => 1
[4] => 4
[3,1] => 1
[2,2] => -2
[2,1,1] => 0
[1,1,1,1] => -1
[5] => 5
[4,1] => 2
[3,2] => 0
[3,1,1] => -1
[2,2,1] => 0
[2,1,1,1] => 0
[1,1,1,1,1] => 1
[6] => 6
[5,1] => 3
[4,2] => 2
[4,1,1] => -2
[3,3] => -3
[3,2,1] => 0
[3,1,1,1] => 1
[2,2,2] => 2
[2,2,1,1] => 0
[2,1,1,1,1] => 0
[1,1,1,1,1,1] => -1
[7] => 7
[6,1] => 4
[5,2] => 4
[5,1,1] => -3
[4,3] => 0
[4,2,1] => 0
[4,1,1,1] => 2
[3,3,1] => -1
[3,2,2] => 0
[3,2,1,1] => 0
[3,1,1,1,1] => -1
[2,2,2,1] => 0
[2,2,1,1,1] => 0
[2,1,1,1,1,1] => 0
[1,1,1,1,1,1,1] => 1
[8] => 8
[7,1] => 5
[6,2] => 6
[6,1,1] => -4
[5,3] => 3
[5,2,1] => 0
[5,1,1,1] => 3
[4,4] => -4
[4,3,1] => 0
[4,2,2] => -2
[4,2,1,1] => 0
[4,1,1,1,1] => -2
[3,3,2] => 0
[3,3,1,1] => 1
[3,2,2,1] => 0
[3,2,1,1,1] => 0
[3,1,1,1,1,1] => 1
[2,2,2,2] => -2
[2,2,2,1,1] => 0
[2,2,1,1,1,1] => 0
[2,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1] => -1
[9] => 9
[8,1] => 6
[7,2] => 8
[7,1,1] => -5
[6,3] => 6
[6,2,1] => 0
[6,1,1,1] => 4
[5,4] => 0
[5,3,1] => 1
[5,2,2] => -4
[5,2,1,1] => 0
[5,1,1,1,1] => -3
[4,4,1] => -2
[4,3,2] => 0
[4,3,1,1] => 0
[4,2,2,1] => 0
[4,2,1,1,1] => 0
[4,1,1,1,1,1] => 2
[3,3,3] => 3
[3,3,2,1] => 0
[3,3,1,1,1] => -1
[3,2,2,2] => 0
[3,2,2,1,1] => 0
[3,2,1,1,1,1] => 0
[3,1,1,1,1,1,1] => -1
[2,2,2,2,1] => 0
[2,2,2,1,1,1] => 0
[2,2,1,1,1,1,1] => 0
[2,1,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1,1] => 1
[10] => 10
[9,1] => 7
[8,2] => 10
[8,1,1] => -6
[7,3] => 9
[7,2,1] => 0
[7,1,1,1] => 5
[6,4] => 4
[6,3,1] => 2
[6,2,2] => -6
[6,2,1,1] => 0
[6,1,1,1,1] => -4
[5,5] => -5
[5,4,1] => 0
[5,3,2] => 0
[5,3,1,1] => -1
[5,2,2,1] => 0
[5,2,1,1,1] => 0
[5,1,1,1,1,1] => 3
[4,4,2] => -2
[4,4,1,1] => 2
[4,3,3] => 0
[4,3,2,1] => 0
[4,3,1,1,1] => 0
[4,2,2,2] => 2
[4,2,2,1,1] => 0
[4,2,1,1,1,1] => 0
[4,1,1,1,1,1,1] => -2
[3,3,3,1] => 1
[3,3,2,2] => 0
[3,3,2,1,1] => 0
[3,3,1,1,1,1] => 1
[3,2,2,2,1] => 0
[3,2,2,1,1,1] => 0
[3,2,1,1,1,1,1] => 0
[3,1,1,1,1,1,1,1] => 1
[2,2,2,2,2] => 2
[2,2,2,2,1,1] => 0
[2,2,2,1,1,1,1] => 0
[2,2,1,1,1,1,1,1] => 0
[2,1,1,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1,1,1] => -1
[11] => 11
[10,1] => 8
[9,2] => 12
[9,1,1] => -7
[8,3] => 12
[8,2,1] => 0
[8,1,1,1] => 6
[7,4] => 8
[7,3,1] => 3
[7,2,2] => -8
[7,2,1,1] => 0
[7,1,1,1,1] => -5
[6,5] => 0
[6,4,1] => 2
[6,3,2] => 0
[6,3,1,1] => -2
[6,2,2,1] => 0
[6,2,1,1,1] => 0
[6,1,1,1,1,1] => 4
[5,5,1] => -3
[5,4,2] => 0
[5,4,1,1] => 0
[5,3,3] => -3
[5,3,2,1] => 0
[5,3,1,1,1] => 1
[5,2,2,2] => 4
[5,2,2,1,1] => 0
[5,2,1,1,1,1] => 0
[5,1,1,1,1,1,1] => -3
[4,4,3] => 0
[4,4,2,1] => 0
[4,4,1,1,1] => -2
[4,3,3,1] => 0
[4,3,2,2] => 0
[4,3,2,1,1] => 0
[4,3,1,1,1,1] => 0
[4,2,2,2,1] => 0
[4,2,2,1,1,1] => 0
[4,2,1,1,1,1,1] => 0
[4,1,1,1,1,1,1,1] => 2
[3,3,3,2] => 0
[3,3,3,1,1] => -1
[3,3,2,2,1] => 0
[3,3,2,1,1,1] => 0
[3,3,1,1,1,1,1] => -1
[3,2,2,2,2] => 0
[3,2,2,2,1,1] => 0
[3,2,2,1,1,1,1] => 0
[3,2,1,1,1,1,1,1] => 0
[3,1,1,1,1,1,1,1,1] => -1
[2,2,2,2,2,1] => 0
[2,2,2,2,1,1,1] => 0
[2,2,2,1,1,1,1,1] => 0
[2,2,1,1,1,1,1,1,1] => 0
[2,1,1,1,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1,1,1,1] => 1
[12] => 12
[11,1] => 9
[10,2] => 14
[10,1,1] => -8
[9,3] => 15
[9,2,1] => 0
[9,1,1,1] => 7
[8,4] => 12
[8,3,1] => 4
[8,2,2] => -10
[8,2,1,1] => 0
[8,1,1,1,1] => -6
[7,5] => 5
[7,4,1] => 4
[7,3,2] => 0
[7,3,1,1] => -3
[7,2,2,1] => 0
[7,2,1,1,1] => 0
[7,1,1,1,1,1] => 5
[6,6] => -6
[6,5,1] => 0
[6,4,2] => 2
[6,4,1,1] => -2
[6,3,3] => -6
[6,3,2,1] => 0
[6,3,1,1,1] => 2
[6,2,2,2] => 6
[6,2,2,1,1] => 0
[6,2,1,1,1,1] => 0
[6,1,1,1,1,1,1] => -4
[5,5,2] => -4
[5,5,1,1] => 3
[5,4,3] => 0
[5,4,2,1] => 0
[5,4,1,1,1] => 0
[5,3,3,1] => -1
[5,3,2,2] => 0
[5,3,2,1,1] => 0
[5,3,1,1,1,1] => -1
[5,2,2,2,1] => 0
[5,2,2,1,1,1] => 0
[5,2,1,1,1,1,1] => 0
[5,1,1,1,1,1,1,1] => 3
[4,4,4] => 4
[4,4,3,1] => 0
[4,4,2,2] => 2
[4,4,2,1,1] => 0
[4,4,1,1,1,1] => 2
[4,3,3,2] => 0
[4,3,3,1,1] => 0
[4,3,2,2,1] => 0
[4,3,2,1,1,1] => 0
[4,3,1,1,1,1,1] => 0
[4,2,2,2,2] => -2
[4,2,2,2,1,1] => 0
[4,2,2,1,1,1,1] => 0
[4,2,1,1,1,1,1,1] => 0
[4,1,1,1,1,1,1,1,1] => -2
[3,3,3,3] => -3
[3,3,3,2,1] => 0
[3,3,3,1,1,1] => 1
[3,3,2,2,2] => 0
[3,3,2,2,1,1] => 0
[3,3,2,1,1,1,1] => 0
[3,3,1,1,1,1,1,1] => 1
[3,2,2,2,2,1] => 0
[3,2,2,2,1,1,1] => 0
[3,2,2,1,1,1,1,1] => 0
[3,2,1,1,1,1,1,1,1] => 0
[3,1,1,1,1,1,1,1,1,1] => 1
[2,2,2,2,2,2] => -2
[2,2,2,2,2,1,1] => 0
[2,2,2,2,1,1,1,1] => 0
[2,2,2,1,1,1,1,1,1] => 0
[2,2,1,1,1,1,1,1,1,1] => 0
[2,1,1,1,1,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1,1,1,1,1] => -1
-----------------------------------------------------------------------------
Created: May 03, 2016 at 08:01 by Martin Rubey
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Last Updated: May 03, 2016 at 11:59 by Martin Rubey