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-----------------------------------------------------------------------------
Statistic identifier: St000473
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
Description: The number of parts of a partition that are strictly bigger than the number of ones.
This is part of the definition of Dyson's crank of a partition, see [[St000474]].
-----------------------------------------------------------------------------
References: [1] Andrews, G. E., Garvan, F. G. Dyson's crank of a partition [[MathSciNet:0929094]]
-----------------------------------------------------------------------------
Code:
def statistic(L):
ones = list(L).count(1)
return sum(1 for part in L if part > ones)
-----------------------------------------------------------------------------
Statistic values:
[] => 0
[1] => 0
[2] => 1
[1,1] => 0
[3] => 1
[2,1] => 1
[1,1,1] => 0
[4] => 1
[3,1] => 1
[2,2] => 2
[2,1,1] => 0
[1,1,1,1] => 0
[5] => 1
[4,1] => 1
[3,2] => 2
[3,1,1] => 1
[2,2,1] => 2
[2,1,1,1] => 0
[1,1,1,1,1] => 0
[6] => 1
[5,1] => 1
[4,2] => 2
[4,1,1] => 1
[3,3] => 2
[3,2,1] => 2
[3,1,1,1] => 0
[2,2,2] => 3
[2,2,1,1] => 0
[2,1,1,1,1] => 0
[1,1,1,1,1,1] => 0
[7] => 1
[6,1] => 1
[5,2] => 2
[5,1,1] => 1
[4,3] => 2
[4,2,1] => 2
[4,1,1,1] => 1
[3,3,1] => 2
[3,2,2] => 3
[3,2,1,1] => 1
[3,1,1,1,1] => 0
[2,2,2,1] => 3
[2,2,1,1,1] => 0
[2,1,1,1,1,1] => 0
[1,1,1,1,1,1,1] => 0
[8] => 1
[7,1] => 1
[6,2] => 2
[6,1,1] => 1
[5,3] => 2
[5,2,1] => 2
[5,1,1,1] => 1
[4,4] => 2
[4,3,1] => 2
[4,2,2] => 3
[4,2,1,1] => 1
[4,1,1,1,1] => 0
[3,3,2] => 3
[3,3,1,1] => 2
[3,2,2,1] => 3
[3,2,1,1,1] => 0
[3,1,1,1,1,1] => 0
[2,2,2,2] => 4
[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] => 0
[9] => 1
[8,1] => 1
[7,2] => 2
[7,1,1] => 1
[6,3] => 2
[6,2,1] => 2
[6,1,1,1] => 1
[5,4] => 2
[5,3,1] => 2
[5,2,2] => 3
[5,2,1,1] => 1
[5,1,1,1,1] => 1
[4,4,1] => 2
[4,3,2] => 3
[4,3,1,1] => 2
[4,2,2,1] => 3
[4,2,1,1,1] => 1
[4,1,1,1,1,1] => 0
[3,3,3] => 3
[3,3,2,1] => 3
[3,3,1,1,1] => 0
[3,2,2,2] => 4
[3,2,2,1,1] => 1
[3,2,1,1,1,1] => 0
[3,1,1,1,1,1,1] => 0
[2,2,2,2,1] => 4
[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] => 0
[10] => 1
[9,1] => 1
[8,2] => 2
[8,1,1] => 1
[7,3] => 2
[7,2,1] => 2
[7,1,1,1] => 1
[6,4] => 2
[6,3,1] => 2
[6,2,2] => 3
[6,2,1,1] => 1
[6,1,1,1,1] => 1
[5,5] => 2
[5,4,1] => 2
[5,3,2] => 3
[5,3,1,1] => 2
[5,2,2,1] => 3
[5,2,1,1,1] => 1
[5,1,1,1,1,1] => 0
[4,4,2] => 3
[4,4,1,1] => 2
[4,3,3] => 3
[4,3,2,1] => 3
[4,3,1,1,1] => 1
[4,2,2,2] => 4
[4,2,2,1,1] => 1
[4,2,1,1,1,1] => 0
[4,1,1,1,1,1,1] => 0
[3,3,3,1] => 3
[3,3,2,2] => 4
[3,3,2,1,1] => 2
[3,3,1,1,1,1] => 0
[3,2,2,2,1] => 4
[3,2,2,1,1,1] => 0
[3,2,1,1,1,1,1] => 0
[3,1,1,1,1,1,1,1] => 0
[2,2,2,2,2] => 5
[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] => 0
[11] => 1
[10,1] => 1
[9,2] => 2
[9,1,1] => 1
[8,3] => 2
[8,2,1] => 2
[8,1,1,1] => 1
[7,4] => 2
[7,3,1] => 2
[7,2,2] => 3
[7,2,1,1] => 1
[7,1,1,1,1] => 1
[6,5] => 2
[6,4,1] => 2
[6,3,2] => 3
[6,3,1,1] => 2
[6,2,2,1] => 3
[6,2,1,1,1] => 1
[6,1,1,1,1,1] => 1
[5,5,1] => 2
[5,4,2] => 3
[5,4,1,1] => 2
[5,3,3] => 3
[5,3,2,1] => 3
[5,3,1,1,1] => 1
[5,2,2,2] => 4
[5,2,2,1,1] => 1
[5,2,1,1,1,1] => 1
[5,1,1,1,1,1,1] => 0
[4,4,3] => 3
[4,4,2,1] => 3
[4,4,1,1,1] => 2
[4,3,3,1] => 3
[4,3,2,2] => 4
[4,3,2,1,1] => 2
[4,3,1,1,1,1] => 0
[4,2,2,2,1] => 4
[4,2,2,1,1,1] => 1
[4,2,1,1,1,1,1] => 0
[4,1,1,1,1,1,1,1] => 0
[3,3,3,2] => 4
[3,3,3,1,1] => 3
[3,3,2,2,1] => 4
[3,3,2,1,1,1] => 0
[3,3,1,1,1,1,1] => 0
[3,2,2,2,2] => 5
[3,2,2,2,1,1] => 1
[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] => 0
[2,2,2,2,2,1] => 5
[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] => 0
[12] => 1
[11,1] => 1
[10,2] => 2
[10,1,1] => 1
[9,3] => 2
[9,2,1] => 2
[9,1,1,1] => 1
[8,4] => 2
[8,3,1] => 2
[8,2,2] => 3
[8,2,1,1] => 1
[8,1,1,1,1] => 1
[7,5] => 2
[7,4,1] => 2
[7,3,2] => 3
[7,3,1,1] => 2
[7,2,2,1] => 3
[7,2,1,1,1] => 1
[7,1,1,1,1,1] => 1
[6,6] => 2
[6,5,1] => 2
[6,4,2] => 3
[6,4,1,1] => 2
[6,3,3] => 3
[6,3,2,1] => 3
[6,3,1,1,1] => 1
[6,2,2,2] => 4
[6,2,2,1,1] => 1
[6,2,1,1,1,1] => 1
[6,1,1,1,1,1,1] => 0
[5,5,2] => 3
[5,5,1,1] => 2
[5,4,3] => 3
[5,4,2,1] => 3
[5,4,1,1,1] => 2
[5,3,3,1] => 3
[5,3,2,2] => 4
[5,3,2,1,1] => 2
[5,3,1,1,1,1] => 1
[5,2,2,2,1] => 4
[5,2,2,1,1,1] => 1
[5,2,1,1,1,1,1] => 0
[5,1,1,1,1,1,1,1] => 0
[4,4,4] => 3
[4,4,3,1] => 3
[4,4,2,2] => 4
[4,4,2,1,1] => 2
[4,4,1,1,1,1] => 0
[4,3,3,2] => 4
[4,3,3,1,1] => 3
[4,3,2,2,1] => 4
[4,3,2,1,1,1] => 1
[4,3,1,1,1,1,1] => 0
[4,2,2,2,2] => 5
[4,2,2,2,1,1] => 1
[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] => 0
[3,3,3,3] => 4
[3,3,3,2,1] => 4
[3,3,3,1,1,1] => 0
[3,3,2,2,2] => 5
[3,3,2,2,1,1] => 2
[3,3,2,1,1,1,1] => 0
[3,3,1,1,1,1,1,1] => 0
[3,2,2,2,2,1] => 5
[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] => 0
[2,2,2,2,2,2] => 6
[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] => 0
-----------------------------------------------------------------------------
Created: Apr 19, 2016 at 10:40 by Christian Stump
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Last Updated: Oct 29, 2017 at 21:13 by Martin Rubey