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-----------------------------------------------------------------------------
Statistic identifier: St000997
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Collection: Integer partitions
-----------------------------------------------------------------------------
Description: The even-odd crank of an integer partition.
This is the largest even part minus the number of odd parts.
-----------------------------------------------------------------------------
References: [1] Chern, S. On partitions with even parts below odd parts [[arXiv:1710.08507]]
-----------------------------------------------------------------------------
Code:
def largest_even_part(L):
L += [0]
for a in L:
if is_even(a):
return a
def statistic(L):
return largest_even_part(L) - sum( 1 for a in L if is_odd(a) )
-----------------------------------------------------------------------------
Statistic values:
[2] => 2
[1,1] => -2
[3] => -1
[2,1] => 1
[1,1,1] => -3
[4] => 4
[3,1] => -2
[2,2] => 2
[2,1,1] => 0
[1,1,1,1] => -4
[5] => -1
[4,1] => 3
[3,2] => 1
[3,1,1] => -3
[2,2,1] => 1
[2,1,1,1] => -1
[1,1,1,1,1] => -5
[6] => 6
[5,1] => -2
[4,2] => 4
[4,1,1] => 2
[3,3] => -2
[3,2,1] => 0
[3,1,1,1] => -4
[2,2,2] => 2
[2,2,1,1] => 0
[2,1,1,1,1] => -2
[1,1,1,1,1,1] => -6
[7] => -1
[6,1] => 5
[5,2] => 1
[5,1,1] => -3
[4,3] => 3
[4,2,1] => 3
[4,1,1,1] => 1
[3,3,1] => -3
[3,2,2] => 1
[3,2,1,1] => -1
[3,1,1,1,1] => -5
[2,2,2,1] => 1
[2,2,1,1,1] => -1
[2,1,1,1,1,1] => -3
[1,1,1,1,1,1,1] => -7
[8] => 8
[7,1] => -2
[6,2] => 6
[6,1,1] => 4
[5,3] => -2
[5,2,1] => 0
[5,1,1,1] => -4
[4,4] => 4
[4,3,1] => 2
[4,2,2] => 4
[4,2,1,1] => 2
[4,1,1,1,1] => 0
[3,3,2] => 0
[3,3,1,1] => -4
[3,2,2,1] => 0
[3,2,1,1,1] => -2
[3,1,1,1,1,1] => -6
[2,2,2,2] => 2
[2,2,2,1,1] => 0
[2,2,1,1,1,1] => -2
[2,1,1,1,1,1,1] => -4
[1,1,1,1,1,1,1,1] => -8
[9] => -1
[8,1] => 7
[7,2] => 1
[7,1,1] => -3
[6,3] => 5
[6,2,1] => 5
[6,1,1,1] => 3
[5,4] => 3
[5,3,1] => -3
[5,2,2] => 1
[5,2,1,1] => -1
[5,1,1,1,1] => -5
[4,4,1] => 3
[4,3,2] => 3
[4,3,1,1] => 1
[4,2,2,1] => 3
[4,2,1,1,1] => 1
[4,1,1,1,1,1] => -1
[3,3,3] => -3
[3,3,2,1] => -1
[3,3,1,1,1] => -5
[3,2,2,2] => 1
[3,2,2,1,1] => -1
[3,2,1,1,1,1] => -3
[3,1,1,1,1,1,1] => -7
[2,2,2,2,1] => 1
[2,2,2,1,1,1] => -1
[2,2,1,1,1,1,1] => -3
[2,1,1,1,1,1,1,1] => -5
[1,1,1,1,1,1,1,1,1] => -9
[10] => 10
[9,1] => -2
[8,2] => 8
[8,1,1] => 6
[7,3] => -2
[7,2,1] => 0
[7,1,1,1] => -4
[6,4] => 6
[6,3,1] => 4
[6,2,2] => 6
[6,2,1,1] => 4
[6,1,1,1,1] => 2
[5,5] => -2
[5,4,1] => 2
[5,3,2] => 0
[5,3,1,1] => -4
[5,2,2,1] => 0
[5,2,1,1,1] => -2
[5,1,1,1,1,1] => -6
[4,4,2] => 4
[4,4,1,1] => 2
[4,3,3] => 2
[4,3,2,1] => 2
[4,3,1,1,1] => 0
[4,2,2,2] => 4
[4,2,2,1,1] => 2
[4,2,1,1,1,1] => 0
[4,1,1,1,1,1,1] => -2
[3,3,3,1] => -4
[3,3,2,2] => 0
[3,3,2,1,1] => -2
[3,3,1,1,1,1] => -6
[3,2,2,2,1] => 0
[3,2,2,1,1,1] => -2
[3,2,1,1,1,1,1] => -4
[3,1,1,1,1,1,1,1] => -8
[2,2,2,2,2] => 2
[2,2,2,2,1,1] => 0
[2,2,2,1,1,1,1] => -2
[2,2,1,1,1,1,1,1] => -4
[2,1,1,1,1,1,1,1,1] => -6
[1,1,1,1,1,1,1,1,1,1] => -10
[11] => -1
[10,1] => 9
[9,2] => 1
[9,1,1] => -3
[8,3] => 7
[8,2,1] => 7
[8,1,1,1] => 5
[7,4] => 3
[7,3,1] => -3
[7,2,2] => 1
[7,2,1,1] => -1
[7,1,1,1,1] => -5
[6,5] => 5
[6,4,1] => 5
[6,3,2] => 5
[6,3,1,1] => 3
[6,2,2,1] => 5
[6,2,1,1,1] => 3
[6,1,1,1,1,1] => 1
[5,5,1] => -3
[5,4,2] => 3
[5,4,1,1] => 1
[5,3,3] => -3
[5,3,2,1] => -1
[5,3,1,1,1] => -5
[5,2,2,2] => 1
[5,2,2,1,1] => -1
[5,2,1,1,1,1] => -3
[5,1,1,1,1,1,1] => -7
[4,4,3] => 3
[4,4,2,1] => 3
[4,4,1,1,1] => 1
[4,3,3,1] => 1
[4,3,2,2] => 3
[4,3,2,1,1] => 1
[4,3,1,1,1,1] => -1
[4,2,2,2,1] => 3
[4,2,2,1,1,1] => 1
[4,2,1,1,1,1,1] => -1
[4,1,1,1,1,1,1,1] => -3
[3,3,3,2] => -1
[3,3,3,1,1] => -5
[3,3,2,2,1] => -1
[3,3,2,1,1,1] => -3
[3,3,1,1,1,1,1] => -7
[3,2,2,2,2] => 1
[3,2,2,2,1,1] => -1
[3,2,2,1,1,1,1] => -3
[3,2,1,1,1,1,1,1] => -5
[3,1,1,1,1,1,1,1,1] => -9
[2,2,2,2,2,1] => 1
[2,2,2,2,1,1,1] => -1
[2,2,2,1,1,1,1,1] => -3
[2,2,1,1,1,1,1,1,1] => -5
[2,1,1,1,1,1,1,1,1,1] => -7
[1,1,1,1,1,1,1,1,1,1,1] => -11
[12] => 12
[11,1] => -2
[10,2] => 10
[10,1,1] => 8
[9,3] => -2
[9,2,1] => 0
[9,1,1,1] => -4
[8,4] => 8
[8,3,1] => 6
[8,2,2] => 8
[8,2,1,1] => 6
[8,1,1,1,1] => 4
[7,5] => -2
[7,4,1] => 2
[7,3,2] => 0
[7,3,1,1] => -4
[7,2,2,1] => 0
[7,2,1,1,1] => -2
[7,1,1,1,1,1] => -6
[6,6] => 6
[6,5,1] => 4
[6,4,2] => 6
[6,4,1,1] => 4
[6,3,3] => 4
[6,3,2,1] => 4
[6,3,1,1,1] => 2
[6,2,2,2] => 6
[6,2,2,1,1] => 4
[6,2,1,1,1,1] => 2
[6,1,1,1,1,1,1] => 0
[5,5,2] => 0
[5,5,1,1] => -4
[5,4,3] => 2
[5,4,2,1] => 2
[5,4,1,1,1] => 0
[5,3,3,1] => -4
[5,3,2,2] => 0
[5,3,2,1,1] => -2
[5,3,1,1,1,1] => -6
[5,2,2,2,1] => 0
[5,2,2,1,1,1] => -2
[5,2,1,1,1,1,1] => -4
[5,1,1,1,1,1,1,1] => -8
[4,4,4] => 4
[4,4,3,1] => 2
[4,4,2,2] => 4
[4,4,2,1,1] => 2
[4,4,1,1,1,1] => 0
[4,3,3,2] => 2
[4,3,3,1,1] => 0
[4,3,2,2,1] => 2
[4,3,2,1,1,1] => 0
[4,3,1,1,1,1,1] => -2
[4,2,2,2,2] => 4
[4,2,2,2,1,1] => 2
[4,2,2,1,1,1,1] => 0
[4,2,1,1,1,1,1,1] => -2
[4,1,1,1,1,1,1,1,1] => -4
[3,3,3,3] => -4
[3,3,3,2,1] => -2
[3,3,3,1,1,1] => -6
[3,3,2,2,2] => 0
[3,3,2,2,1,1] => -2
[3,3,2,1,1,1,1] => -4
[3,3,1,1,1,1,1,1] => -8
[3,2,2,2,2,1] => 0
[3,2,2,2,1,1,1] => -2
[3,2,2,1,1,1,1,1] => -4
[3,2,1,1,1,1,1,1,1] => -6
[3,1,1,1,1,1,1,1,1,1] => -10
[2,2,2,2,2,2] => 2
[2,2,2,2,2,1,1] => 0
[2,2,2,2,1,1,1,1] => -2
[2,2,2,1,1,1,1,1,1] => -4
[2,2,1,1,1,1,1,1,1,1] => -6
[2,1,1,1,1,1,1,1,1,1,1] => -8
[1,1,1,1,1,1,1,1,1,1,1,1] => -12
-----------------------------------------------------------------------------
Created: Oct 25, 2017 at 21:43 by Christian Stump
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Last Updated: Oct 25, 2017 at 21:43 by Christian Stump