*****************************************************************************
* www.FindStat.org - The Combinatorial Statistic Finder *
* *
* Copyright (C) 2019 The FindStatCrew <info@findstat.org> *
* *
* This information is distributed in the hope that it will be useful, *
* but WITHOUT ANY WARRANTY; without even the implied warranty of *
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. *
*****************************************************************************
-----------------------------------------------------------------------------
Statistic identifier: St001912
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
Description: The length of the preperiod in Bulgarian solitaire corresponding to an integer partition.
Bulgarian solitaire is the dynamical system where a move consists of removing the first column of the Ferrers diagram and inserting it as a row.
-----------------------------------------------------------------------------
References: [1] [[arXiv 2308.05321]]
-----------------------------------------------------------------------------
Code:
def statistic(la):
def move(la):
return Partition(sorted([p-1 for p in la] + [len(la)], reverse=True))
n = len(la)
B = DiscreteDynamicalSystem(Partitions(n), move)
return B.orbit(la, preperiod=True)[1]
-----------------------------------------------------------------------------
Statistic values:
[1] => 0
[2] => 0
[1,1] => 0
[3] => 1
[2,1] => 0
[1,1,1] => 2
[4] => 1
[3,1] => 0
[2,2] => 0
[2,1,1] => 0
[1,1,1,1] => 2
[5] => 2
[4,1] => 1
[3,2] => 0
[3,1,1] => 0
[2,2,1] => 0
[2,1,1,1] => 2
[1,1,1,1,1] => 3
[6] => 3
[5,1] => 2
[4,2] => 1
[4,1,1] => 5
[3,3] => 4
[3,2,1] => 0
[3,1,1,1] => 2
[2,2,2] => 3
[2,2,1,1] => 6
[2,1,1,1,1] => 3
[1,1,1,1,1,1] => 4
[7] => 3
[6,1] => 2
[5,2] => 1
[5,1,1] => 2
[4,3] => 1
[4,2,1] => 0
[4,1,1,1] => 2
[3,3,1] => 0
[3,2,2] => 0
[3,2,1,1] => 0
[3,1,1,1,1] => 2
[2,2,2,1] => 3
[2,2,1,1,1] => 3
[2,1,1,1,1,1] => 3
[1,1,1,1,1,1,1] => 4
[8] => 4
[7,1] => 3
[6,2] => 2
[6,1,1] => 2
[5,3] => 1
[5,2,1] => 1
[5,1,1,1] => 2
[4,4] => 1
[4,3,1] => 0
[4,2,2] => 0
[4,2,1,1] => 0
[4,1,1,1,1] => 2
[3,3,2] => 0
[3,3,1,1] => 0
[3,2,2,1] => 0
[3,2,1,1,1] => 2
[3,1,1,1,1,1] => 3
[2,2,2,2] => 3
[2,2,2,1,1] => 3
[2,2,1,1,1,1] => 3
[2,1,1,1,1,1,1] => 4
[1,1,1,1,1,1,1,1] => 5
[9] => 5
[8,1] => 4
[7,2] => 3
[7,1,1] => 3
[6,3] => 2
[6,2,1] => 2
[6,1,1,1] => 2
[5,4] => 1
[5,3,1] => 1
[5,2,2] => 1
[5,2,1,1] => 6
[5,1,1,1,1] => 2
[4,4,1] => 5
[4,3,2] => 0
[4,3,1,1] => 0
[4,2,2,1] => 0
[4,2,1,1,1] => 2
[4,1,1,1,1,1] => 3
[3,3,3] => 4
[3,3,2,1] => 0
[3,3,1,1,1] => 2
[3,2,2,2] => 3
[3,2,2,1,1] => 7
[3,2,1,1,1,1] => 3
[3,1,1,1,1,1,1] => 4
[2,2,2,2,1] => 3
[2,2,2,1,1,1] => 3
[2,2,1,1,1,1,1] => 4
[2,1,1,1,1,1,1,1] => 5
[1,1,1,1,1,1,1,1,1] => 6
[10] => 6
[9,1] => 5
[8,2] => 4
[8,1,1] => 9
[7,3] => 8
[7,2,1] => 3
[7,1,1,1] => 3
[6,4] => 2
[6,3,1] => 2
[6,2,2] => 7
[6,2,1,1] => 11
[6,1,1,1,1] => 7
[5,5] => 6
[5,4,1] => 10
[5,3,2] => 1
[5,3,1,1] => 6
[5,2,2,1] => 11
[5,2,1,1,1] => 11
[5,1,1,1,1,1] => 3
[4,4,2] => 5
[4,4,1,1] => 10
[4,3,3] => 9
[4,3,2,1] => 0
[4,3,1,1,1] => 2
[4,2,2,2] => 3
[4,2,2,1,1] => 7
[4,2,1,1,1,1] => 3
[4,1,1,1,1,1,1] => 9
[3,3,3,1] => 4
[3,3,2,2] => 8
[3,3,2,1,1] => 12
[3,3,1,1,1,1] => 8
[3,2,2,2,1] => 12
[3,2,2,1,1,1] => 12
[3,2,1,1,1,1,1] => 4
[3,1,1,1,1,1,1,1] => 5
[2,2,2,2,2] => 4
[2,2,2,2,1,1] => 8
[2,2,2,1,1,1,1] => 4
[2,2,1,1,1,1,1,1] => 10
[2,1,1,1,1,1,1,1,1] => 6
[1,1,1,1,1,1,1,1,1,1] => 7
[11] => 6
[10,1] => 5
[9,2] => 4
[9,1,1] => 5
[8,3] => 4
[8,2,1] => 3
[8,1,1,1] => 3
[7,4] => 2
[7,3,1] => 2
[7,2,2] => 3
[7,2,1,1] => 3
[7,1,1,1,1] => 3
[6,5] => 2
[6,4,1] => 2
[6,3,2] => 1
[6,3,1,1] => 2
[6,2,2,1] => 7
[6,2,1,1,1] => 7
[6,1,1,1,1,1] => 3
[5,5,1] => 6
[5,4,2] => 1
[5,4,1,1] => 6
[5,3,3] => 1
[5,3,2,1] => 0
[5,3,1,1,1] => 2
[5,2,2,2] => 3
[5,2,2,1,1] => 7
[5,2,1,1,1,1] => 3
[5,1,1,1,1,1,1] => 3
[4,4,3] => 5
[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] => 2
[4,2,2,2,1] => 3
[4,2,2,1,1,1] => 3
[4,2,1,1,1,1,1] => 3
[4,1,1,1,1,1,1,1] => 5
[3,3,3,2] => 4
[3,3,3,1,1] => 4
[3,3,2,2,1] => 8
[3,3,2,1,1,1] => 8
[3,3,1,1,1,1,1] => 4
[3,2,2,2,2] => 4
[3,2,2,2,1,1] => 8
[3,2,2,1,1,1,1] => 4
[3,2,1,1,1,1,1,1] => 4
[3,1,1,1,1,1,1,1,1] => 5
[2,2,2,2,2,1] => 4
[2,2,2,2,1,1,1] => 4
[2,2,2,1,1,1,1,1] => 4
[2,2,1,1,1,1,1,1,1] => 6
[2,1,1,1,1,1,1,1,1,1] => 6
[1,1,1,1,1,1,1,1,1,1,1] => 7
[12] => 7
[11,1] => 6
[10,2] => 5
[10,1,1] => 5
[9,3] => 4
[9,2,1] => 4
[9,1,1,1] => 4
[8,4] => 3
[8,3,1] => 3
[8,2,2] => 3
[8,2,1,1] => 3
[8,1,1,1,1] => 3
[7,5] => 2
[7,4,1] => 2
[7,3,2] => 2
[7,3,1,1] => 2
[7,2,2,1] => 3
[7,2,1,1,1] => 3
[7,1,1,1,1,1] => 3
[6,6] => 2
[6,5,1] => 2
[6,4,2] => 1
[6,4,1,1] => 2
[6,3,3] => 1
[6,3,2,1] => 1
[6,3,1,1,1] => 2
[6,2,2,2] => 3
[6,2,2,1,1] => 7
[6,2,1,1,1,1] => 3
[6,1,1,1,1,1,1] => 3
[5,5,2] => 1
[5,5,1,1] => 6
[5,4,3] => 1
[5,4,2,1] => 0
[5,4,1,1,1] => 2
[5,3,3,1] => 0
[5,3,2,2] => 0
[5,3,2,1,1] => 0
[5,3,1,1,1,1] => 2
[5,2,2,2,1] => 3
[5,2,2,1,1,1] => 3
[5,2,1,1,1,1,1] => 3
[5,1,1,1,1,1,1,1] => 4
[4,4,4] => 5
[4,4,3,1] => 0
[4,4,2,2] => 0
[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] => 2
[4,3,1,1,1,1,1] => 3
[4,2,2,2,2] => 3
[4,2,2,2,1,1] => 3
[4,2,2,1,1,1,1] => 3
[4,2,1,1,1,1,1,1] => 4
[4,1,1,1,1,1,1,1,1] => 5
[3,3,3,3] => 4
[3,3,3,2,1] => 4
[3,3,3,1,1,1] => 4
[3,3,2,2,2] => 4
[3,3,2,2,1,1] => 8
[3,3,2,1,1,1,1] => 4
[3,3,1,1,1,1,1,1] => 4
[3,2,2,2,2,1] => 4
[3,2,2,2,1,1,1] => 4
[3,2,2,1,1,1,1,1] => 4
[3,2,1,1,1,1,1,1,1] => 5
[3,1,1,1,1,1,1,1,1,1] => 6
[2,2,2,2,2,2] => 4
[2,2,2,2,2,1,1] => 4
[2,2,2,2,1,1,1,1] => 4
[2,2,2,1,1,1,1,1,1] => 5
[2,2,1,1,1,1,1,1,1,1] => 6
[2,1,1,1,1,1,1,1,1,1,1] => 7
[1,1,1,1,1,1,1,1,1,1,1,1] => 8
[13] => 8
[12,1] => 7
[11,2] => 6
[11,1,1] => 6
[10,3] => 5
[10,2,1] => 5
[10,1,1,1] => 5
[9,4] => 4
[9,3,1] => 4
[9,2,2] => 4
[9,2,1,1] => 4
[9,1,1,1,1] => 4
[8,5] => 3
[8,4,1] => 3
[8,3,2] => 3
[8,3,1,1] => 3
[8,2,2,1] => 3
[8,2,1,1,1] => 3
[8,1,1,1,1,1] => 3
[7,6] => 2
[7,5,1] => 2
[7,4,2] => 2
[7,4,1,1] => 2
[7,3,3] => 2
[7,3,2,1] => 2
[7,3,1,1,1] => 2
[7,2,2,2] => 3
[7,2,2,1,1] => 7
[7,2,1,1,1,1] => 3
[7,1,1,1,1,1,1] => 3
[6,6,1] => 2
[6,5,2] => 1
[6,5,1,1] => 6
[6,4,3] => 1
[6,4,2,1] => 1
[6,4,1,1,1] => 2
[6,3,3,1] => 1
[6,3,2,2] => 1
[6,3,2,1,1] => 7
[6,3,1,1,1,1] => 2
[6,2,2,2,1] => 7
[6,2,2,1,1,1] => 7
[6,2,1,1,1,1,1] => 3
[6,1,1,1,1,1,1,1] => 4
[5,5,3] => 1
[5,5,2,1] => 6
[5,5,1,1,1] => 6
[5,4,4] => 5
[5,4,3,1] => 0
[5,4,2,2] => 0
[5,4,2,1,1] => 0
[5,4,1,1,1,1] => 2
[5,3,3,2] => 0
[5,3,3,1,1] => 0
[5,3,2,2,1] => 0
[5,3,2,1,1,1] => 2
[5,3,1,1,1,1,1] => 3
[5,2,2,2,2] => 3
[5,2,2,2,1,1] => 3
[5,2,2,1,1,1,1] => 3
[5,2,1,1,1,1,1,1] => 4
[5,1,1,1,1,1,1,1,1] => 5
[4,4,4,1] => 5
[4,4,3,2] => 0
[4,4,3,1,1] => 0
[4,4,2,2,1] => 0
[4,4,2,1,1,1] => 2
[4,4,1,1,1,1,1] => 3
[4,3,3,3] => 4
[4,3,3,2,1] => 0
[4,3,3,1,1,1] => 2
[4,3,2,2,2] => 3
[4,3,2,2,1,1] => 8
[4,3,2,1,1,1,1] => 3
[4,3,1,1,1,1,1,1] => 4
[4,2,2,2,2,1] => 3
[4,2,2,2,1,1,1] => 3
[4,2,2,1,1,1,1,1] => 4
[4,2,1,1,1,1,1,1,1] => 5
[4,1,1,1,1,1,1,1,1,1] => 6
[3,3,3,3,1] => 4
[3,3,3,2,2] => 4
[3,3,3,2,1,1] => 8
[3,3,3,1,1,1,1] => 4
[3,3,2,2,2,1] => 8
[3,3,2,2,1,1,1] => 8
[3,3,2,1,1,1,1,1] => 4
[3,3,1,1,1,1,1,1,1] => 5
[3,2,2,2,2,2] => 4
[3,2,2,2,2,1,1] => 4
[3,2,2,2,1,1,1,1] => 4
[3,2,2,1,1,1,1,1,1] => 5
[3,2,1,1,1,1,1,1,1,1] => 6
[3,1,1,1,1,1,1,1,1,1,1] => 7
[2,2,2,2,2,2,1] => 4
[2,2,2,2,2,1,1,1] => 4
[2,2,2,2,1,1,1,1,1] => 5
[2,2,2,1,1,1,1,1,1,1] => 6
[2,2,1,1,1,1,1,1,1,1,1] => 7
[2,1,1,1,1,1,1,1,1,1,1,1] => 8
[1,1,1,1,1,1,1,1,1,1,1,1,1] => 9
[14] => 9
[13,1] => 8
[12,2] => 7
[12,1,1] => 7
[11,3] => 6
[11,2,1] => 6
[11,1,1,1] => 6
[10,4] => 5
[10,3,1] => 5
[10,2,2] => 5
[10,2,1,1] => 11
[10,1,1,1,1] => 5
[9,5] => 4
[9,4,1] => 10
[9,3,2] => 4
[9,3,1,1] => 4
[9,2,2,1] => 4
[9,2,1,1,1] => 4
[9,1,1,1,1,1] => 4
[8,6] => 3
[8,5,1] => 3
[8,4,2] => 3
[8,4,1,1] => 3
[8,3,3] => 9
[8,3,2,1] => 3
[8,3,1,1,1] => 3
[8,2,2,2] => 3
[8,2,2,1,1] => 7
[8,2,1,1,1,1] => 3
[8,1,1,1,1,1,1] => 9
[7,7] => 8
[7,6,1] => 2
[7,5,2] => 2
[7,5,1,1] => 6
[7,4,3] => 2
[7,4,2,1] => 2
[7,4,1,1,1] => 2
[7,3,3,1] => 2
[7,3,2,2] => 8
[7,3,2,1,1] => 13
[7,3,1,1,1,1] => 8
[7,2,2,2,1] => 13
[7,2,2,1,1,1] => 13
[7,2,1,1,1,1,1] => 3
[7,1,1,1,1,1,1,1] => 4
[6,6,2] => 7
[6,6,1,1] => 12
[6,5,3] => 1
[6,5,2,1] => 12
[6,5,1,1,1] => 12
[6,4,4] => 5
[6,4,3,1] => 1
[6,4,2,2] => 1
[6,4,2,1,1] => 7
[6,4,1,1,1,1] => 2
[6,3,3,2] => 1
[6,3,3,1,1] => 7
[6,3,2,2,1] => 13
[6,3,2,1,1,1] => 13
[6,3,1,1,1,1,1] => 3
[6,2,2,2,2] => 7
[6,2,2,2,1,1] => 13
[6,2,2,1,1,1,1] => 7
[6,2,1,1,1,1,1,1] => 4
[6,1,1,1,1,1,1,1,1] => 5
[5,5,4] => 11
[5,5,3,1] => 6
[5,5,2,2] => 6
[5,5,2,1,1] => 12
[5,5,1,1,1,1] => 6
[5,4,4,1] => 11
[5,4,3,2] => 0
[5,4,3,1,1] => 0
[5,4,2,2,1] => 0
[5,4,2,1,1,1] => 2
[5,4,1,1,1,1,1] => 3
[5,3,3,3] => 4
[5,3,3,2,1] => 0
[5,3,3,1,1,1] => 2
[5,3,2,2,2] => 3
[5,3,2,2,1,1] => 8
[5,3,2,1,1,1,1] => 3
[5,3,1,1,1,1,1,1] => 4
[5,2,2,2,2,1] => 3
[5,2,2,2,1,1,1] => 3
[5,2,2,1,1,1,1,1] => 4
[5,2,1,1,1,1,1,1,1] => 11
[5,1,1,1,1,1,1,1,1,1] => 6
[4,4,4,2] => 5
[4,4,4,1,1] => 5
[4,4,3,3] => 10
[4,4,3,2,1] => 0
[4,4,3,1,1,1] => 2
[4,4,2,2,2] => 3
[4,4,2,2,1,1] => 8
[4,4,2,1,1,1,1] => 3
[4,4,1,1,1,1,1,1] => 10
[4,3,3,3,1] => 4
[4,3,3,2,2] => 9
[4,3,3,2,1,1] => 14
[4,3,3,1,1,1,1] => 9
[4,3,2,2,2,1] => 14
[4,3,2,2,1,1,1] => 14
[4,3,2,1,1,1,1,1] => 4
[4,3,1,1,1,1,1,1,1] => 5
[4,2,2,2,2,2] => 4
[4,2,2,2,2,1,1] => 9
[4,2,2,2,1,1,1,1] => 4
[4,2,2,1,1,1,1,1,1] => 5
[4,2,1,1,1,1,1,1,1,1] => 6
[4,1,1,1,1,1,1,1,1,1,1] => 7
[3,3,3,3,2] => 4
[3,3,3,3,1,1] => 8
[3,3,3,2,2,1] => 14
[3,3,3,2,1,1,1] => 14
[3,3,3,1,1,1,1,1] => 4
[3,3,2,2,2,2] => 8
[3,3,2,2,2,1,1] => 14
[3,3,2,2,1,1,1,1] => 8
[3,3,2,1,1,1,1,1,1] => 5
[3,3,1,1,1,1,1,1,1,1] => 6
[3,2,2,2,2,2,1] => 4
[3,2,2,2,2,1,1,1] => 4
[3,2,2,2,1,1,1,1,1] => 5
[3,2,2,1,1,1,1,1,1,1] => 12
[3,2,1,1,1,1,1,1,1,1,1] => 7
[3,1,1,1,1,1,1,1,1,1,1,1] => 8
[2,2,2,2,2,2,2] => 5
[2,2,2,2,2,2,1,1] => 10
[2,2,2,2,2,1,1,1,1] => 5
[2,2,2,2,1,1,1,1,1,1] => 6
[2,2,2,1,1,1,1,1,1,1,1] => 7
[2,2,1,1,1,1,1,1,1,1,1,1] => 8
[2,1,1,1,1,1,1,1,1,1,1,1,1] => 9
[1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 10
[15] => 10
[14,1] => 9
[13,2] => 8
[13,1,1] => 14
[12,3] => 13
[12,2,1] => 7
[12,1,1,1] => 7
[11,4] => 6
[11,3,1] => 6
[11,2,2] => 12
[11,2,1,1] => 17
[11,1,1,1,1] => 12
[10,5] => 11
[10,4,1] => 16
[10,3,2] => 5
[10,3,1,1] => 11
[10,2,2,1] => 17
[10,2,1,1,1] => 17
[10,1,1,1,1,1] => 5
[9,6] => 4
[9,5,1] => 16
[9,4,2] => 10
[9,4,1,1] => 16
[9,3,3] => 15
[9,3,2,1] => 4
[9,3,1,1,1] => 4
[9,2,2,2] => 4
[9,2,2,1,1] => 7
[9,2,1,1,1,1] => 4
[9,1,1,1,1,1,1] => 15
[8,7] => 14
[8,6,1] => 3
[8,5,2] => 3
[8,5,1,1] => 6
[8,4,3] => 15
[8,4,2,1] => 3
[8,4,1,1,1] => 3
[8,3,3,1] => 9
[8,3,2,2] => 14
[8,3,2,1,1] => 19
[8,3,1,1,1,1] => 14
[8,2,2,2,1] => 19
[8,2,2,1,1,1] => 19
[8,2,1,1,1,1,1] => 9
[8,1,1,1,1,1,1,1] => 15
[7,7,1] => 8
[7,6,2] => 13
[7,6,1,1] => 18
[7,5,3] => 2
[7,5,2,1] => 18
[7,5,1,1,1] => 18
[7,4,4] => 5
[7,4,3,1] => 2
[7,4,2,2] => 8
[7,4,2,1,1] => 13
[7,4,1,1,1,1] => 8
[7,3,3,2] => 14
[7,3,3,1,1] => 19
[7,3,2,2,1] => 19
[7,3,2,1,1,1] => 19
[7,3,1,1,1,1,1] => 14
[7,2,2,2,2] => 19
[7,2,2,2,1,1] => 19
[7,2,2,1,1,1,1] => 19
[7,2,1,1,1,1,1,1] => 4
[7,1,1,1,1,1,1,1,1] => 5
[6,6,3] => 7
[6,6,2,1] => 18
[6,6,1,1,1] => 18
[6,5,4] => 17
[6,5,3,1] => 12
[6,5,2,2] => 18
[6,5,2,1,1] => 18
[6,5,1,1,1,1] => 18
[6,4,4,1] => 17
[6,4,3,2] => 1
[6,4,3,1,1] => 7
[6,4,2,2,1] => 13
[6,4,2,1,1,1] => 13
[6,4,1,1,1,1,1] => 3
[6,3,3,3] => 4
[6,3,3,2,1] => 19
[6,3,3,1,1,1] => 19
[6,3,2,2,2] => 19
[6,3,2,2,1,1] => 19
[6,3,2,1,1,1,1] => 19
[6,3,1,1,1,1,1,1] => 4
[6,2,2,2,2,1] => 19
[6,2,2,2,1,1,1] => 19
[6,2,2,1,1,1,1,1] => 7
[6,2,1,1,1,1,1,1,1] => 17
[6,1,1,1,1,1,1,1,1,1] => 12
[5,5,5] => 17
[5,5,4,1] => 17
[5,5,3,2] => 6
[5,5,3,1,1] => 12
[5,5,2,2,1] => 18
[5,5,2,1,1,1] => 18
[5,5,1,1,1,1,1] => 6
[5,4,4,2] => 11
[5,4,4,1,1] => 17
[5,4,3,3] => 16
[5,4,3,2,1] => 0
[5,4,3,1,1,1] => 2
[5,4,2,2,2] => 3
[5,4,2,2,1,1] => 8
[5,4,2,1,1,1,1] => 3
[5,4,1,1,1,1,1,1] => 16
[5,3,3,3,1] => 4
[5,3,3,2,2] => 9
[5,3,3,2,1,1] => 14
[5,3,3,1,1,1,1] => 9
[5,3,2,2,2,1] => 14
[5,3,2,2,1,1,1] => 14
[5,3,2,1,1,1,1,1] => 4
[5,3,1,1,1,1,1,1,1] => 11
[5,2,2,2,2,2] => 4
[5,2,2,2,2,1,1] => 9
[5,2,2,2,1,1,1,1] => 4
[5,2,2,1,1,1,1,1,1] => 17
[5,2,1,1,1,1,1,1,1,1] => 17
[5,1,1,1,1,1,1,1,1,1,1] => 7
[4,4,4,3] => 16
[4,4,4,2,1] => 5
[4,4,4,1,1,1] => 5
[4,4,3,3,1] => 10
[4,4,3,2,2] => 15
[4,4,3,2,1,1] => 20
[4,4,3,1,1,1,1] => 15
[4,4,2,2,2,1] => 20
[4,4,2,2,1,1,1] => 20
[4,4,2,1,1,1,1,1] => 10
[4,4,1,1,1,1,1,1,1] => 16
[4,3,3,3,2] => 15
[4,3,3,3,1,1] => 20
[4,3,3,2,2,1] => 20
[4,3,3,2,1,1,1] => 20
[4,3,3,1,1,1,1,1] => 15
[4,3,2,2,2,2] => 20
[4,3,2,2,2,1,1] => 20
[4,3,2,2,1,1,1,1] => 20
[4,3,2,1,1,1,1,1,1] => 5
[4,3,1,1,1,1,1,1,1,1] => 6
[4,2,2,2,2,2,1] => 15
[4,2,2,2,2,1,1,1] => 15
[4,2,2,2,1,1,1,1,1] => 5
[4,2,2,1,1,1,1,1,1,1] => 12
[4,2,1,1,1,1,1,1,1,1,1] => 7
[4,1,1,1,1,1,1,1,1,1,1,1] => 14
[3,3,3,3,3] => 5
[3,3,3,3,2,1] => 20
[3,3,3,3,1,1,1] => 20
[3,3,3,2,2,2] => 20
[3,3,3,2,2,1,1] => 20
[3,3,3,2,1,1,1,1] => 20
[3,3,3,1,1,1,1,1,1] => 5
[3,3,2,2,2,2,1] => 20
[3,3,2,2,2,1,1,1] => 20
[3,3,2,2,1,1,1,1,1] => 8
[3,3,2,1,1,1,1,1,1,1] => 18
[3,3,1,1,1,1,1,1,1,1,1] => 13
[3,2,2,2,2,2,2] => 5
[3,2,2,2,2,2,1,1] => 10
[3,2,2,2,2,1,1,1,1] => 5
[3,2,2,2,1,1,1,1,1,1] => 18
[3,2,2,1,1,1,1,1,1,1,1] => 18
[3,2,1,1,1,1,1,1,1,1,1,1] => 8
[3,1,1,1,1,1,1,1,1,1,1,1,1] => 9
[2,2,2,2,2,2,2,1] => 16
[2,2,2,2,2,2,1,1,1] => 16
[2,2,2,2,2,1,1,1,1,1] => 6
[2,2,2,2,1,1,1,1,1,1,1] => 13
[2,2,2,1,1,1,1,1,1,1,1,1] => 8
[2,2,1,1,1,1,1,1,1,1,1,1,1] => 15
[2,1,1,1,1,1,1,1,1,1,1,1,1,1] => 10
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 11
[16] => 10
[15,1] => 9
[14,2] => 8
[14,1,1] => 9
[13,3] => 8
[13,2,1] => 7
[13,1,1,1] => 7
[12,4] => 6
[12,3,1] => 6
[12,2,2] => 7
[12,2,1,1] => 7
[12,1,1,1,1] => 7
[11,5] => 6
[11,4,1] => 6
[11,3,2] => 5
[11,3,1,1] => 6
[11,2,2,1] => 12
[11,2,1,1,1] => 12
[11,1,1,1,1,1] => 5
[10,6] => 4
[10,5,1] => 11
[10,4,2] => 5
[10,4,1,1] => 11
[10,3,3] => 5
[10,3,2,1] => 4
[10,3,1,1,1] => 4
[10,2,2,2] => 4
[10,2,2,1,1] => 7
[10,2,1,1,1,1] => 4
[10,1,1,1,1,1,1] => 5
[9,7] => 4
[9,6,1] => 3
[9,5,2] => 3
[9,5,1,1] => 6
[9,4,3] => 10
[9,4,2,1] => 3
[9,4,1,1,1] => 3
[9,3,3,1] => 4
[9,3,2,2] => 4
[9,3,2,1,1] => 4
[9,3,1,1,1,1] => 4
[9,2,2,2,1] => 4
[9,2,2,1,1,1] => 4
[9,2,1,1,1,1,1] => 4
[9,1,1,1,1,1,1,1] => 10
[8,8] => 9
[8,7,1] => 3
[8,6,2] => 3
[8,6,1,1] => 3
[8,5,3] => 2
[8,5,2,1] => 3
[8,5,1,1,1] => 3
[8,4,4] => 5
[8,4,3,1] => 2
[8,4,2,2] => 3
[8,4,2,1,1] => 3
[8,4,1,1,1,1] => 3
[8,3,3,2] => 9
[8,3,3,1,1] => 9
[8,3,2,2,1] => 14
[8,3,2,1,1,1] => 14
[8,3,1,1,1,1,1] => 9
[8,2,2,2,2] => 9
[8,2,2,2,1,1] => 14
[8,2,2,1,1,1,1] => 9
[8,2,1,1,1,1,1,1] => 4
[8,1,1,1,1,1,1,1,1] => 5
[7,7,2] => 8
[7,7,1,1] => 8
[7,6,3] => 2
[7,6,2,1] => 13
[7,6,1,1,1] => 13
[7,5,4] => 2
[7,5,3,1] => 2
[7,5,2,2] => 8
[7,5,2,1,1] => 13
[7,5,1,1,1,1] => 8
[7,4,4,1] => 2
[7,4,3,2] => 1
[7,4,3,1,1] => 2
[7,4,2,2,1] => 8
[7,4,2,1,1,1] => 8
[7,4,1,1,1,1,1] => 3
[7,3,3,3] => 4
[7,3,3,2,1] => 14
[7,3,3,1,1,1] => 14
[7,3,2,2,2] => 14
[7,3,2,2,1,1] => 9
[7,3,2,1,1,1,1] => 14
[7,3,1,1,1,1,1,1] => 4
[7,2,2,2,2,1] => 14
[7,2,2,2,1,1,1] => 14
[7,2,2,1,1,1,1,1] => 4
[7,2,1,1,1,1,1,1,1] => 4
[7,1,1,1,1,1,1,1,1,1] => 5
[6,6,4] => 7
[6,6,3,1] => 7
[6,6,2,2] => 13
[6,6,2,1,1] => 8
[6,6,1,1,1,1] => 13
[6,5,5] => 12
[6,5,4,1] => 12
[6,5,3,2] => 1
[6,5,3,1,1] => 7
[6,5,2,2,1] => 13
[6,5,2,1,1,1] => 13
[6,5,1,1,1,1,1] => 3
[6,4,4,2] => 1
[6,4,4,1,1] => 7
[6,4,3,3] => 1
[6,4,3,2,1] => 0
[6,4,3,1,1,1] => 2
[6,4,2,2,2] => 3
[6,4,2,2,1,1] => 8
[6,4,2,1,1,1,1] => 3
[6,4,1,1,1,1,1,1] => 3
[6,3,3,3,1] => 4
[6,3,3,2,2] => 9
[6,3,3,2,1,1] => 14
[6,3,3,1,1,1,1] => 9
[6,3,2,2,2,1] => 14
[6,3,2,2,1,1,1] => 14
[6,3,2,1,1,1,1,1] => 4
[6,3,1,1,1,1,1,1,1] => 4
[6,2,2,2,2,2] => 4
[6,2,2,2,2,1,1] => 9
[6,2,2,2,1,1,1,1] => 4
[6,2,2,1,1,1,1,1,1] => 7
[6,2,1,1,1,1,1,1,1,1] => 12
[6,1,1,1,1,1,1,1,1,1,1] => 7
[5,5,5,1] => 7
[5,5,4,2] => 6
[5,5,4,1,1] => 12
[5,5,3,3] => 6
[5,5,3,2,1] => 0
[5,5,3,1,1,1] => 2
[5,5,2,2,2] => 3
[5,5,2,2,1,1] => 8
[5,5,2,1,1,1,1] => 3
[5,5,1,1,1,1,1,1] => 6
[5,4,4,3] => 11
[5,4,4,2,1] => 0
[5,4,4,1,1,1] => 2
[5,4,3,3,1] => 0
[5,4,3,2,2] => 0
[5,4,3,2,1,1] => 0
[5,4,3,1,1,1,1] => 2
[5,4,2,2,2,1] => 3
[5,4,2,2,1,1,1] => 3
[5,4,2,1,1,1,1,1] => 3
[5,4,1,1,1,1,1,1,1] => 11
[5,3,3,3,2] => 4
[5,3,3,3,1,1] => 4
[5,3,3,2,2,1] => 9
[5,3,3,2,1,1,1] => 9
[5,3,3,1,1,1,1,1] => 4
[5,3,2,2,2,2] => 4
[5,3,2,2,2,1,1] => 9
[5,3,2,2,1,1,1,1] => 4
[5,3,2,1,1,1,1,1,1] => 4
[5,3,1,1,1,1,1,1,1,1] => 6
[5,2,2,2,2,2,1] => 4
[5,2,2,2,2,1,1,1] => 4
[5,2,2,2,1,1,1,1,1] => 4
[5,2,2,1,1,1,1,1,1,1] => 12
[5,2,1,1,1,1,1,1,1,1,1] => 7
[5,1,1,1,1,1,1,1,1,1,1,1] => 7
[4,4,4,4] => 6
[4,4,4,3,1] => 5
[4,4,4,2,2] => 5
[4,4,4,2,1,1] => 5
[4,4,4,1,1,1,1] => 5
[4,4,3,3,2] => 10
[4,4,3,3,1,1] => 10
[4,4,3,2,2,1] => 15
[4,4,3,2,1,1,1] => 15
[4,4,3,1,1,1,1,1] => 10
[4,4,2,2,2,2] => 10
[4,4,2,2,2,1,1] => 15
[4,4,2,2,1,1,1,1] => 10
[4,4,2,1,1,1,1,1,1] => 5
[4,4,1,1,1,1,1,1,1,1] => 6
[4,3,3,3,3] => 5
[4,3,3,3,2,1] => 15
[4,3,3,3,1,1,1] => 15
[4,3,3,2,2,2] => 15
[4,3,3,2,2,1,1] => 10
[4,3,3,2,1,1,1,1] => 15
[4,3,3,1,1,1,1,1,1] => 5
[4,3,2,2,2,2,1] => 15
[4,3,2,2,2,1,1,1] => 15
[4,3,2,2,1,1,1,1,1] => 5
[4,3,2,1,1,1,1,1,1,1] => 5
[4,3,1,1,1,1,1,1,1,1,1] => 6
[4,2,2,2,2,2,2] => 5
[4,2,2,2,2,2,1,1] => 10
[4,2,2,2,2,1,1,1,1] => 5
[4,2,2,2,1,1,1,1,1,1] => 5
[4,2,2,1,1,1,1,1,1,1,1] => 7
[4,2,1,1,1,1,1,1,1,1,1,1] => 7
[4,1,1,1,1,1,1,1,1,1,1,1,1] => 9
[3,3,3,3,3,1] => 5
[3,3,3,3,2,2] => 10
[3,3,3,3,2,1,1] => 15
[3,3,3,3,1,1,1,1] => 10
[3,3,3,2,2,2,1] => 15
[3,3,3,2,2,1,1,1] => 15
[3,3,3,2,1,1,1,1,1] => 5
[3,3,3,1,1,1,1,1,1,1] => 5
[3,3,2,2,2,2,2] => 5
[3,3,2,2,2,2,1,1] => 10
[3,3,2,2,2,1,1,1,1] => 5
[3,3,2,2,1,1,1,1,1,1] => 8
[3,3,2,1,1,1,1,1,1,1,1] => 13
[3,3,1,1,1,1,1,1,1,1,1,1] => 8
[3,2,2,2,2,2,2,1] => 5
[3,2,2,2,2,2,1,1,1] => 5
[3,2,2,2,2,1,1,1,1,1] => 5
[3,2,2,2,1,1,1,1,1,1,1] => 13
[3,2,2,1,1,1,1,1,1,1,1,1] => 8
[3,2,1,1,1,1,1,1,1,1,1,1,1] => 8
[3,1,1,1,1,1,1,1,1,1,1,1,1,1] => 9
[2,2,2,2,2,2,2,2] => 6
[2,2,2,2,2,2,2,1,1] => 11
[2,2,2,2,2,2,1,1,1,1] => 6
[2,2,2,2,2,1,1,1,1,1,1] => 6
[2,2,2,2,1,1,1,1,1,1,1,1] => 8
[2,2,2,1,1,1,1,1,1,1,1,1,1] => 8
[2,2,1,1,1,1,1,1,1,1,1,1,1,1] => 10
[2,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 10
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 11
[17] => 11
[16,1] => 10
[15,2] => 9
[15,1,1] => 9
[14,3] => 8
[14,2,1] => 8
[14,1,1,1] => 8
[13,4] => 7
[13,3,1] => 7
[13,2,2] => 7
[13,2,1,1] => 7
[13,1,1,1,1] => 7
[12,5] => 6
[12,4,1] => 6
[12,3,2] => 6
[12,3,1,1] => 6
[12,2,2,1] => 7
[12,2,1,1,1] => 7
[12,1,1,1,1,1] => 6
[11,6] => 5
[11,5,1] => 6
[11,4,2] => 5
[11,4,1,1] => 6
[11,3,3] => 5
[11,3,2,1] => 5
[11,3,1,1,1] => 5
[11,2,2,2] => 5
[11,2,2,1,1] => 7
[11,2,1,1,1,1] => 5
[11,1,1,1,1,1,1] => 5
[10,7] => 4
[10,6,1] => 4
[10,5,2] => 4
[10,5,1,1] => 6
[10,4,3] => 5
[10,4,2,1] => 4
[10,4,1,1,1] => 4
[10,3,3,1] => 4
[10,3,2,2] => 4
[10,3,2,1,1] => 4
[10,3,1,1,1,1] => 4
[10,2,2,2,1] => 4
[10,2,2,1,1,1] => 4
[10,2,1,1,1,1,1] => 4
[10,1,1,1,1,1,1,1] => 5
[9,8] => 4
[9,7,1] => 3
[9,6,2] => 3
[9,6,1,1] => 3
[9,5,3] => 3
[9,5,2,1] => 3
[9,5,1,1,1] => 3
[9,4,4] => 5
[9,4,3,1] => 3
[9,4,2,2] => 3
[9,4,2,1,1] => 3
[9,4,1,1,1,1] => 3
[9,3,3,2] => 4
[9,3,3,1,1] => 4
[9,3,2,2,1] => 4
[9,3,2,1,1,1] => 4
[9,3,1,1,1,1,1] => 4
[9,2,2,2,2] => 4
[9,2,2,2,1,1] => 4
[9,2,2,1,1,1,1] => 4
[9,2,1,1,1,1,1,1] => 4
[9,1,1,1,1,1,1,1,1] => 5
[8,8,1] => 3
[8,7,2] => 3
[8,7,1,1] => 3
[8,6,3] => 2
[8,6,2,1] => 3
[8,6,1,1,1] => 3
[8,5,4] => 2
[8,5,3,1] => 2
[8,5,2,2] => 3
[8,5,2,1,1] => 3
[8,5,1,1,1,1] => 3
[8,4,4,1] => 2
[8,4,3,2] => 2
[8,4,3,1,1] => 2
[8,4,2,2,1] => 3
[8,4,2,1,1,1] => 3
[8,4,1,1,1,1,1] => 3
[8,3,3,3] => 4
[8,3,3,2,1] => 9
[8,3,3,1,1,1] => 9
[8,3,2,2,2] => 9
[8,3,2,2,1,1] => 9
[8,3,2,1,1,1,1] => 9
[8,3,1,1,1,1,1,1] => 4
[8,2,2,2,2,1] => 9
[8,2,2,2,1,1,1] => 9
[8,2,2,1,1,1,1,1] => 4
[8,2,1,1,1,1,1,1,1] => 4
[8,1,1,1,1,1,1,1,1,1] => 5
[7,7,3] => 2
[7,7,2,1] => 8
[7,7,1,1,1] => 8
[7,6,4] => 2
[7,6,3,1] => 2
[7,6,2,2] => 8
[7,6,2,1,1] => 8
[7,6,1,1,1,1] => 8
[7,5,5] => 2
[7,5,4,1] => 2
[7,5,3,2] => 1
[7,5,3,1,1] => 2
[7,5,2,2,1] => 8
[7,5,2,1,1,1] => 8
[7,5,1,1,1,1,1] => 3
[7,4,4,2] => 1
[7,4,4,1,1] => 2
[7,4,3,3] => 1
[7,4,3,2,1] => 1
[7,4,3,1,1,1] => 2
[7,4,2,2,2] => 3
[7,4,2,2,1,1] => 8
[7,4,2,1,1,1,1] => 3
[7,4,1,1,1,1,1,1] => 3
[7,3,3,3,1] => 4
[7,3,3,2,2] => 9
[7,3,3,2,1,1] => 9
[7,3,3,1,1,1,1] => 9
[7,3,2,2,2,1] => 9
[7,3,2,2,1,1,1] => 9
[7,3,2,1,1,1,1,1] => 4
[7,3,1,1,1,1,1,1,1] => 4
[7,2,2,2,2,2] => 4
[7,2,2,2,2,1,1] => 9
[7,2,2,2,1,1,1,1] => 4
[7,2,2,1,1,1,1,1,1] => 4
[7,2,1,1,1,1,1,1,1,1] => 5
[7,1,1,1,1,1,1,1,1,1,1] => 6
[6,6,5] => 7
[6,6,4,1] => 7
[6,6,3,2] => 1
[6,6,3,1,1] => 7
[6,6,2,2,1] => 8
[6,6,2,1,1,1] => 8
[6,6,1,1,1,1,1] => 3
[6,5,5,1] => 7
[6,5,4,2] => 1
[6,5,4,1,1] => 7
[6,5,3,3] => 1
[6,5,3,2,1] => 0
[6,5,3,1,1,1] => 2
[6,5,2,2,2] => 3
[6,5,2,2,1,1] => 8
[6,5,2,1,1,1,1] => 3
[6,5,1,1,1,1,1,1] => 3
[6,4,4,3] => 1
[6,4,4,2,1] => 0
[6,4,4,1,1,1] => 2
[6,4,3,3,1] => 0
[6,4,3,2,2] => 0
[6,4,3,2,1,1] => 0
[6,4,3,1,1,1,1] => 2
[6,4,2,2,2,1] => 3
[6,4,2,2,1,1,1] => 3
[6,4,2,1,1,1,1,1] => 3
[6,4,1,1,1,1,1,1,1] => 4
[6,3,3,3,2] => 4
[6,3,3,3,1,1] => 4
[6,3,3,2,2,1] => 9
[6,3,3,2,1,1,1] => 9
[6,3,3,1,1,1,1,1] => 4
[6,3,2,2,2,2] => 4
[6,3,2,2,2,1,1] => 9
[6,3,2,2,1,1,1,1] => 4
[6,3,2,1,1,1,1,1,1] => 4
[6,3,1,1,1,1,1,1,1,1] => 5
[6,2,2,2,2,2,1] => 4
[6,2,2,2,2,1,1,1] => 4
[6,2,2,2,1,1,1,1,1] => 4
[6,2,2,1,1,1,1,1,1,1] => 7
[6,2,1,1,1,1,1,1,1,1,1] => 7
[6,1,1,1,1,1,1,1,1,1,1,1] => 7
[5,5,5,2] => 6
[5,5,5,1,1] => 7
[5,5,4,3] => 6
[5,5,4,2,1] => 0
[5,5,4,1,1,1] => 2
[5,5,3,3,1] => 0
[5,5,3,2,2] => 0
[5,5,3,2,1,1] => 0
[5,5,3,1,1,1,1] => 2
[5,5,2,2,2,1] => 3
[5,5,2,2,1,1,1] => 3
[5,5,2,1,1,1,1,1] => 3
[5,5,1,1,1,1,1,1,1] => 6
[5,4,4,4] => 6
[5,4,4,3,1] => 0
[5,4,4,2,2] => 0
[5,4,4,2,1,1] => 0
[5,4,4,1,1,1,1] => 2
[5,4,3,3,2] => 0
[5,4,3,3,1,1] => 0
[5,4,3,2,2,1] => 0
[5,4,3,2,1,1,1] => 2
[5,4,3,1,1,1,1,1] => 3
[5,4,2,2,2,2] => 3
[5,4,2,2,2,1,1] => 3
[5,4,2,2,1,1,1,1] => 3
[5,4,2,1,1,1,1,1,1] => 4
[5,4,1,1,1,1,1,1,1,1] => 6
[5,3,3,3,3] => 4
[5,3,3,3,2,1] => 4
[5,3,3,3,1,1,1] => 4
[5,3,3,2,2,2] => 4
[5,3,3,2,2,1,1] => 9
[5,3,3,2,1,1,1,1] => 4
[5,3,3,1,1,1,1,1,1] => 4
[5,3,2,2,2,2,1] => 4
[5,3,2,2,2,1,1,1] => 4
[5,3,2,2,1,1,1,1,1] => 4
[5,3,2,1,1,1,1,1,1,1] => 5
[5,3,1,1,1,1,1,1,1,1,1] => 6
[5,2,2,2,2,2,2] => 4
[5,2,2,2,2,2,1,1] => 4
[5,2,2,2,2,1,1,1,1] => 4
[5,2,2,2,1,1,1,1,1,1] => 5
[5,2,2,1,1,1,1,1,1,1,1] => 7
[5,2,1,1,1,1,1,1,1,1,1,1] => 7
[5,1,1,1,1,1,1,1,1,1,1,1,1] => 8
[4,4,4,4,1] => 5
[4,4,4,3,2] => 5
[4,4,4,3,1,1] => 5
[4,4,4,2,2,1] => 5
[4,4,4,2,1,1,1] => 5
[4,4,4,1,1,1,1,1] => 5
[4,4,3,3,3] => 5
[4,4,3,3,2,1] => 10
[4,4,3,3,1,1,1] => 10
[4,4,3,2,2,2] => 10
[4,4,3,2,2,1,1] => 10
[4,4,3,2,1,1,1,1] => 10
[4,4,3,1,1,1,1,1,1] => 5
[4,4,2,2,2,2,1] => 10
[4,4,2,2,2,1,1,1] => 10
[4,4,2,2,1,1,1,1,1] => 5
[4,4,2,1,1,1,1,1,1,1] => 5
[4,4,1,1,1,1,1,1,1,1,1] => 6
[4,3,3,3,3,1] => 5
[4,3,3,3,2,2] => 10
[4,3,3,3,2,1,1] => 10
[4,3,3,3,1,1,1,1] => 10
[4,3,3,2,2,2,1] => 10
[4,3,3,2,2,1,1,1] => 10
[4,3,3,2,1,1,1,1,1] => 5
[4,3,3,1,1,1,1,1,1,1] => 5
[4,3,2,2,2,2,2] => 5
[4,3,2,2,2,2,1,1] => 10
[4,3,2,2,2,1,1,1,1] => 5
[4,3,2,2,1,1,1,1,1,1] => 5
[4,3,2,1,1,1,1,1,1,1,1] => 6
[4,3,1,1,1,1,1,1,1,1,1,1] => 7
[4,2,2,2,2,2,2,1] => 5
[4,2,2,2,2,2,1,1,1] => 5
[4,2,2,2,2,1,1,1,1,1] => 5
[4,2,2,2,1,1,1,1,1,1,1] => 6
[4,2,2,1,1,1,1,1,1,1,1,1] => 7
[4,2,1,1,1,1,1,1,1,1,1,1,1] => 8
[4,1,1,1,1,1,1,1,1,1,1,1,1,1] => 9
[3,3,3,3,3,2] => 5
[3,3,3,3,3,1,1] => 5
[3,3,3,3,2,2,1] => 10
[3,3,3,3,2,1,1,1] => 10
[3,3,3,3,1,1,1,1,1] => 5
[3,3,3,2,2,2,2] => 5
[3,3,3,2,2,2,1,1] => 10
[3,3,3,2,2,1,1,1,1] => 5
[3,3,3,2,1,1,1,1,1,1] => 5
[3,3,3,1,1,1,1,1,1,1,1] => 6
[3,3,2,2,2,2,2,1] => 5
[3,3,2,2,2,2,1,1,1] => 5
[3,3,2,2,2,1,1,1,1,1] => 5
[3,3,2,2,1,1,1,1,1,1,1] => 8
[3,3,2,1,1,1,1,1,1,1,1,1] => 8
[3,3,1,1,1,1,1,1,1,1,1,1,1] => 8
[3,2,2,2,2,2,2,2] => 5
[3,2,2,2,2,2,2,1,1] => 5
[3,2,2,2,2,2,1,1,1,1] => 5
[3,2,2,2,2,1,1,1,1,1,1] => 6
[3,2,2,2,1,1,1,1,1,1,1,1] => 8
[3,2,2,1,1,1,1,1,1,1,1,1,1] => 8
-----------------------------------------------------------------------------
Created: Aug 11, 2023 at 10:48 by Martin Rubey
-----------------------------------------------------------------------------
Last Updated: Aug 11, 2023 at 10:48 by Martin Rubey