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-----------------------------------------------------------------------------
Statistic identifier: St000380
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
Description: Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition.
Put differently, this is the smallest number $n$ such that the partition fits into the triangular partition $(n-1,n-2,\dots,1)$.
-----------------------------------------------------------------------------
References:
-----------------------------------------------------------------------------
Code:
def statistic(p):
if p:
return max( p[i]+i+1 for i in range(len(p)) )
return 0
-----------------------------------------------------------------------------
Statistic values:
[] => 0
[1] => 2
[2] => 3
[1,1] => 3
[3] => 4
[2,1] => 3
[1,1,1] => 4
[4] => 5
[3,1] => 4
[2,2] => 4
[2,1,1] => 4
[1,1,1,1] => 5
[5] => 6
[4,1] => 5
[3,2] => 4
[3,1,1] => 4
[2,2,1] => 4
[2,1,1,1] => 5
[1,1,1,1,1] => 6
[6] => 7
[5,1] => 6
[4,2] => 5
[4,1,1] => 5
[3,3] => 5
[3,2,1] => 4
[3,1,1,1] => 5
[2,2,2] => 5
[2,2,1,1] => 5
[2,1,1,1,1] => 6
[1,1,1,1,1,1] => 7
[7] => 8
[6,1] => 7
[5,2] => 6
[5,1,1] => 6
[4,3] => 5
[4,2,1] => 5
[4,1,1,1] => 5
[3,3,1] => 5
[3,2,2] => 5
[3,2,1,1] => 5
[3,1,1,1,1] => 6
[2,2,2,1] => 5
[2,2,1,1,1] => 6
[2,1,1,1,1,1] => 7
[1,1,1,1,1,1,1] => 8
[8] => 9
[7,1] => 8
[6,2] => 7
[6,1,1] => 7
[5,3] => 6
[5,2,1] => 6
[5,1,1,1] => 6
[4,4] => 6
[4,3,1] => 5
[4,2,2] => 5
[4,2,1,1] => 5
[4,1,1,1,1] => 6
[3,3,2] => 5
[3,3,1,1] => 5
[3,2,2,1] => 5
[3,2,1,1,1] => 6
[3,1,1,1,1,1] => 7
[2,2,2,2] => 6
[2,2,2,1,1] => 6
[2,2,1,1,1,1] => 7
[2,1,1,1,1,1,1] => 8
[1,1,1,1,1,1,1,1] => 9
[9] => 10
[8,1] => 9
[7,2] => 8
[7,1,1] => 8
[6,3] => 7
[6,2,1] => 7
[6,1,1,1] => 7
[5,4] => 6
[5,3,1] => 6
[5,2,2] => 6
[5,2,1,1] => 6
[5,1,1,1,1] => 6
[4,4,1] => 6
[4,3,2] => 5
[4,3,1,1] => 5
[4,2,2,1] => 5
[4,2,1,1,1] => 6
[4,1,1,1,1,1] => 7
[3,3,3] => 6
[3,3,2,1] => 5
[3,3,1,1,1] => 6
[3,2,2,2] => 6
[3,2,2,1,1] => 6
[3,2,1,1,1,1] => 7
[3,1,1,1,1,1,1] => 8
[2,2,2,2,1] => 6
[2,2,2,1,1,1] => 7
[2,2,1,1,1,1,1] => 8
[2,1,1,1,1,1,1,1] => 9
[1,1,1,1,1,1,1,1,1] => 10
[10] => 11
[9,1] => 10
[8,2] => 9
[8,1,1] => 9
[7,3] => 8
[7,2,1] => 8
[7,1,1,1] => 8
[6,4] => 7
[6,3,1] => 7
[6,2,2] => 7
[6,2,1,1] => 7
[6,1,1,1,1] => 7
[5,5] => 7
[5,4,1] => 6
[5,3,2] => 6
[5,3,1,1] => 6
[5,2,2,1] => 6
[5,2,1,1,1] => 6
[5,1,1,1,1,1] => 7
[4,4,2] => 6
[4,4,1,1] => 6
[4,3,3] => 6
[4,3,2,1] => 5
[4,3,1,1,1] => 6
[4,2,2,2] => 6
[4,2,2,1,1] => 6
[4,2,1,1,1,1] => 7
[4,1,1,1,1,1,1] => 8
[3,3,3,1] => 6
[3,3,2,2] => 6
[3,3,2,1,1] => 6
[3,3,1,1,1,1] => 7
[3,2,2,2,1] => 6
[3,2,2,1,1,1] => 7
[3,2,1,1,1,1,1] => 8
[3,1,1,1,1,1,1,1] => 9
[2,2,2,2,2] => 7
[2,2,2,2,1,1] => 7
[2,2,2,1,1,1,1] => 8
[2,2,1,1,1,1,1,1] => 9
[2,1,1,1,1,1,1,1,1] => 10
[1,1,1,1,1,1,1,1,1,1] => 11
[11] => 12
[10,1] => 11
[9,2] => 10
[9,1,1] => 10
[8,3] => 9
[8,2,1] => 9
[8,1,1,1] => 9
[7,4] => 8
[7,3,1] => 8
[7,2,2] => 8
[7,2,1,1] => 8
[7,1,1,1,1] => 8
[6,5] => 7
[6,4,1] => 7
[6,3,2] => 7
[6,3,1,1] => 7
[6,2,2,1] => 7
[6,2,1,1,1] => 7
[6,1,1,1,1,1] => 7
[5,5,1] => 7
[5,4,2] => 6
[5,4,1,1] => 6
[5,3,3] => 6
[5,3,2,1] => 6
[5,3,1,1,1] => 6
[5,2,2,2] => 6
[5,2,2,1,1] => 6
[5,2,1,1,1,1] => 7
[5,1,1,1,1,1,1] => 8
[4,4,3] => 6
[4,4,2,1] => 6
[4,4,1,1,1] => 6
[4,3,3,1] => 6
[4,3,2,2] => 6
[4,3,2,1,1] => 6
[4,3,1,1,1,1] => 7
[4,2,2,2,1] => 6
[4,2,2,1,1,1] => 7
[4,2,1,1,1,1,1] => 8
[4,1,1,1,1,1,1,1] => 9
[3,3,3,2] => 6
[3,3,3,1,1] => 6
[3,3,2,2,1] => 6
[3,3,2,1,1,1] => 7
[3,3,1,1,1,1,1] => 8
[3,2,2,2,2] => 7
[3,2,2,2,1,1] => 7
[3,2,2,1,1,1,1] => 8
[3,2,1,1,1,1,1,1] => 9
[3,1,1,1,1,1,1,1,1] => 10
[2,2,2,2,2,1] => 7
[2,2,2,2,1,1,1] => 8
[2,2,2,1,1,1,1,1] => 9
[2,2,1,1,1,1,1,1,1] => 10
[2,1,1,1,1,1,1,1,1,1] => 11
[1,1,1,1,1,1,1,1,1,1,1] => 12
[12] => 13
[11,1] => 12
[10,2] => 11
[10,1,1] => 11
[9,3] => 10
[9,2,1] => 10
[9,1,1,1] => 10
[8,4] => 9
[8,3,1] => 9
[8,2,2] => 9
[8,2,1,1] => 9
[8,1,1,1,1] => 9
[7,5] => 8
[7,4,1] => 8
[7,3,2] => 8
[7,3,1,1] => 8
[7,2,2,1] => 8
[7,2,1,1,1] => 8
[7,1,1,1,1,1] => 8
[6,6] => 8
[6,5,1] => 7
[6,4,2] => 7
[6,4,1,1] => 7
[6,3,3] => 7
[6,3,2,1] => 7
[6,3,1,1,1] => 7
[6,2,2,2] => 7
[6,2,2,1,1] => 7
[6,2,1,1,1,1] => 7
[6,1,1,1,1,1,1] => 8
[5,5,2] => 7
[5,5,1,1] => 7
[5,4,3] => 6
[5,4,2,1] => 6
[5,4,1,1,1] => 6
[5,3,3,1] => 6
[5,3,2,2] => 6
[5,3,2,1,1] => 6
[5,3,1,1,1,1] => 7
[5,2,2,2,1] => 6
[5,2,2,1,1,1] => 7
[5,2,1,1,1,1,1] => 8
[5,1,1,1,1,1,1,1] => 9
[4,4,4] => 7
[4,4,3,1] => 6
[4,4,2,2] => 6
[4,4,2,1,1] => 6
[4,4,1,1,1,1] => 7
[4,3,3,2] => 6
[4,3,3,1,1] => 6
[4,3,2,2,1] => 6
[4,3,2,1,1,1] => 7
[4,3,1,1,1,1,1] => 8
[4,2,2,2,2] => 7
[4,2,2,2,1,1] => 7
[4,2,2,1,1,1,1] => 8
[4,2,1,1,1,1,1,1] => 9
[4,1,1,1,1,1,1,1,1] => 10
[3,3,3,3] => 7
[3,3,3,2,1] => 6
[3,3,3,1,1,1] => 7
[3,3,2,2,2] => 7
[3,3,2,2,1,1] => 7
[3,3,2,1,1,1,1] => 8
[3,3,1,1,1,1,1,1] => 9
[3,2,2,2,2,1] => 7
[3,2,2,2,1,1,1] => 8
[3,2,2,1,1,1,1,1] => 9
[3,2,1,1,1,1,1,1,1] => 10
[3,1,1,1,1,1,1,1,1,1] => 11
[2,2,2,2,2,2] => 8
[2,2,2,2,2,1,1] => 8
[2,2,2,2,1,1,1,1] => 9
[2,2,2,1,1,1,1,1,1] => 10
[2,2,1,1,1,1,1,1,1,1] => 11
[2,1,1,1,1,1,1,1,1,1,1] => 12
[1,1,1,1,1,1,1,1,1,1,1,1] => 13
[8,5] => 9
[7,5,1] => 8
[7,4,2] => 8
[5,5,3] => 7
[5,4,4] => 7
[5,4,3,1] => 6
[5,4,2,2] => 6
[5,4,2,1,1] => 6
[5,3,3,2] => 6
[5,3,3,1,1] => 6
[5,3,2,2,1] => 6
[4,4,4,1] => 7
[4,4,3,2] => 6
[4,4,3,1,1] => 6
[4,4,2,2,1] => 6
[4,3,3,3] => 7
[4,3,3,2,1] => 6
[3,3,3,3,1] => 7
[3,3,3,2,2] => 7
[9,5] => 10
[8,5,1] => 9
[7,5,2] => 8
[7,4,3] => 8
[5,5,4] => 7
[5,4,3,2] => 6
[5,4,3,1,1] => 6
[5,4,2,2,1] => 6
[5,3,3,2,1] => 6
[5,3,2,2,2] => 7
[4,4,4,2] => 7
[4,4,3,3] => 7
[4,4,3,2,1] => 6
[3,3,3,3,2] => 7
[9,5,1] => 10
[8,5,2] => 9
[7,5,3] => 8
[5,5,5] => 8
[5,4,3,2,1] => 6
[5,3,2,2,2,1] => 7
[4,4,4,3] => 7
[3,3,3,3,3] => 8
[8,5,3] => 9
[7,5,3,1] => 8
[4,4,4,4] => 8
[8,6,3] => 9
[9,6,3] => 10
[8,6,4] => 9
[9,6,4] => 10
[8,5,4,2] => 9
[8,5,5,1] => 9
[7,5,4,3,1] => 8
[8,6,4,2] => 9
[10,6,4] => 11
[10,7,3] => 11
[9,7,4] => 10
[9,5,5,1] => 10
[6,5,4,3,2,1] => 7
[11,7,3] => 12
[9,6,4,3] => 10
[9,6,5,3] => 10
[8,6,5,3,1] => 9
[11,7,5,1] => 12
[9,7,5,3] => 10
[9,7,5,3,1] => 10
[10,7,5,3] => 11
[9,7,5,4,1] => 10
[7,6,5,4,3,2,1] => 8
[10,7,6,4,1] => 11
[9,7,6,4,2] => 10
[10,8,5,4,1] => 11
[10,8,6,4,1] => 11
[9,7,5,5,3,1] => 10
[11,8,6,4,1] => 12
[10,8,6,4,2] => 11
[11,8,6,5,1] => 12
[12,9,7,5,1] => 13
[13,9,7,5,1] => 14
[11,9,7,5,3,1] => 12
[11,8,7,5,4,1] => 12
[8,7,6,5,4,3,2,1] => 9
[11,9,7,5,5,3] => 12
[11,9,7,7,5,3,3] => 12
[11,9,7,6,5,3,1] => 12
[13,11,9,7,5,3,1] => 14
[13,11,9,7,7,5,3,1] => 14
[17,13,11,9,7,5,1] => 18
[15,13,11,9,7,5,3,1] => 16
[29,23,19,17,13,11,7,1] => 30
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Created: Feb 09, 2016 at 12:26 by Christian Stump
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Last Updated: Feb 25, 2021 at 20:07 by Martin Rubey