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* *
* 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. *
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-----------------------------------------------------------------------------
Statistic identifier: St000165
-----------------------------------------------------------------------------
Collection: Parking functions
-----------------------------------------------------------------------------
Description: The sum of the entries of a parking function.
The generating function for parking functions by sum is the evaluation at $x=1$ and $y=1/q$ of the Tutte polynomial of the complete graph, multiplied by $q^\binom{n}{2}$.
-----------------------------------------------------------------------------
References: [1] Merino López, C. Chip firing and the Tutte polynomial [[MathSciNet:1630779]]
-----------------------------------------------------------------------------
Code:
def statistic(p):
return sum(p)
def generating_function(n):
R.<q> = ZZ[]
return graphs.CompleteGraph(n).tutte_polynomial().subs(x=1, y=1/q)*q^binomial(n, 2)
-----------------------------------------------------------------------------
Statistic values:
[1] => 1
[1,1] => 2
[1,2] => 3
[2,1] => 3
[1,1,1] => 3
[1,1,2] => 4
[1,2,1] => 4
[2,1,1] => 4
[1,1,3] => 5
[1,3,1] => 5
[3,1,1] => 5
[1,2,2] => 5
[2,1,2] => 5
[2,2,1] => 5
[1,2,3] => 6
[1,3,2] => 6
[2,1,3] => 6
[2,3,1] => 6
[3,1,2] => 6
[3,2,1] => 6
[1,1,1,1] => 4
[1,1,1,2] => 5
[1,1,2,1] => 5
[1,2,1,1] => 5
[2,1,1,1] => 5
[1,1,1,3] => 6
[1,1,3,1] => 6
[1,3,1,1] => 6
[3,1,1,1] => 6
[1,1,1,4] => 7
[1,1,4,1] => 7
[1,4,1,1] => 7
[4,1,1,1] => 7
[1,1,2,2] => 6
[1,2,1,2] => 6
[1,2,2,1] => 6
[2,1,1,2] => 6
[2,1,2,1] => 6
[2,2,1,1] => 6
[1,1,2,3] => 7
[1,1,3,2] => 7
[1,2,1,3] => 7
[1,2,3,1] => 7
[1,3,1,2] => 7
[1,3,2,1] => 7
[2,1,1,3] => 7
[2,1,3,1] => 7
[2,3,1,1] => 7
[3,1,1,2] => 7
[3,1,2,1] => 7
[3,2,1,1] => 7
[1,1,2,4] => 8
[1,1,4,2] => 8
[1,2,1,4] => 8
[1,2,4,1] => 8
[1,4,1,2] => 8
[1,4,2,1] => 8
[2,1,1,4] => 8
[2,1,4,1] => 8
[2,4,1,1] => 8
[4,1,1,2] => 8
[4,1,2,1] => 8
[4,2,1,1] => 8
[1,1,3,3] => 8
[1,3,1,3] => 8
[1,3,3,1] => 8
[3,1,1,3] => 8
[3,1,3,1] => 8
[3,3,1,1] => 8
[1,1,3,4] => 9
[1,1,4,3] => 9
[1,3,1,4] => 9
[1,3,4,1] => 9
[1,4,1,3] => 9
[1,4,3,1] => 9
[3,1,1,4] => 9
[3,1,4,1] => 9
[3,4,1,1] => 9
[4,1,1,3] => 9
[4,1,3,1] => 9
[4,3,1,1] => 9
[1,2,2,2] => 7
[2,1,2,2] => 7
[2,2,1,2] => 7
[2,2,2,1] => 7
[1,2,2,3] => 8
[1,2,3,2] => 8
[1,3,2,2] => 8
[2,1,2,3] => 8
[2,1,3,2] => 8
[2,2,1,3] => 8
[2,2,3,1] => 8
[2,3,1,2] => 8
[2,3,2,1] => 8
[3,1,2,2] => 8
[3,2,1,2] => 8
[3,2,2,1] => 8
[1,2,2,4] => 9
[1,2,4,2] => 9
[1,4,2,2] => 9
[2,1,2,4] => 9
[2,1,4,2] => 9
[2,2,1,4] => 9
[2,2,4,1] => 9
[2,4,1,2] => 9
[2,4,2,1] => 9
[4,1,2,2] => 9
[4,2,1,2] => 9
[4,2,2,1] => 9
[1,2,3,3] => 9
[1,3,2,3] => 9
[1,3,3,2] => 9
[2,1,3,3] => 9
[2,3,1,3] => 9
[2,3,3,1] => 9
[3,1,2,3] => 9
[3,1,3,2] => 9
[3,2,1,3] => 9
[3,2,3,1] => 9
[3,3,1,2] => 9
[3,3,2,1] => 9
[1,2,3,4] => 10
[1,2,4,3] => 10
[1,3,2,4] => 10
[1,3,4,2] => 10
[1,4,2,3] => 10
[1,4,3,2] => 10
[2,1,3,4] => 10
[2,1,4,3] => 10
[2,3,1,4] => 10
[2,3,4,1] => 10
[2,4,1,3] => 10
[2,4,3,1] => 10
[3,1,2,4] => 10
[3,1,4,2] => 10
[3,2,1,4] => 10
[3,2,4,1] => 10
[3,4,1,2] => 10
[3,4,2,1] => 10
[4,1,2,3] => 10
[4,1,3,2] => 10
[4,2,1,3] => 10
[4,2,3,1] => 10
[4,3,1,2] => 10
[4,3,2,1] => 10
[1,1,1,1,1] => 5
[1,1,1,1,2] => 6
[1,1,1,2,1] => 6
[1,1,2,1,1] => 6
[1,2,1,1,1] => 6
[2,1,1,1,1] => 6
[1,1,1,1,3] => 7
[1,1,1,3,1] => 7
[1,1,3,1,1] => 7
[1,3,1,1,1] => 7
[3,1,1,1,1] => 7
[1,1,1,1,4] => 8
[1,1,1,4,1] => 8
[1,1,4,1,1] => 8
[1,4,1,1,1] => 8
[4,1,1,1,1] => 8
[1,1,1,1,5] => 9
[1,1,1,5,1] => 9
[1,1,5,1,1] => 9
[1,5,1,1,1] => 9
[5,1,1,1,1] => 9
[1,1,1,2,2] => 7
[1,1,2,1,2] => 7
[1,1,2,2,1] => 7
[1,2,1,1,2] => 7
[1,2,1,2,1] => 7
[1,2,2,1,1] => 7
[2,1,1,1,2] => 7
[2,1,1,2,1] => 7
[2,1,2,1,1] => 7
[2,2,1,1,1] => 7
[1,1,1,2,3] => 8
[1,1,1,3,2] => 8
[1,1,2,1,3] => 8
[1,1,2,3,1] => 8
[1,1,3,1,2] => 8
[1,1,3,2,1] => 8
[1,2,1,1,3] => 8
[1,2,1,3,1] => 8
[1,2,3,1,1] => 8
[1,3,1,1,2] => 8
[1,3,1,2,1] => 8
[1,3,2,1,1] => 8
[2,1,1,1,3] => 8
[2,1,1,3,1] => 8
[2,1,3,1,1] => 8
[2,3,1,1,1] => 8
[3,1,1,1,2] => 8
[3,1,1,2,1] => 8
[3,1,2,1,1] => 8
[3,2,1,1,1] => 8
[1,1,1,2,4] => 9
[1,1,1,4,2] => 9
[1,1,2,1,4] => 9
[1,1,2,4,1] => 9
[1,1,4,1,2] => 9
[1,1,4,2,1] => 9
[1,2,1,1,4] => 9
[1,2,1,4,1] => 9
[1,2,4,1,1] => 9
[1,4,1,1,2] => 9
[1,4,1,2,1] => 9
[1,4,2,1,1] => 9
[2,1,1,1,4] => 9
[2,1,1,4,1] => 9
[2,1,4,1,1] => 9
[2,4,1,1,1] => 9
[4,1,1,1,2] => 9
[4,1,1,2,1] => 9
[4,1,2,1,1] => 9
[4,2,1,1,1] => 9
[1,1,1,2,5] => 10
[1,1,1,5,2] => 10
[1,1,2,1,5] => 10
[1,1,2,5,1] => 10
[1,1,5,1,2] => 10
[1,1,5,2,1] => 10
[1,2,1,1,5] => 10
[1,2,1,5,1] => 10
[1,2,5,1,1] => 10
[1,5,1,1,2] => 10
[1,5,1,2,1] => 10
[1,5,2,1,1] => 10
[2,1,1,1,5] => 10
[2,1,1,5,1] => 10
[2,1,5,1,1] => 10
[2,5,1,1,1] => 10
[5,1,1,1,2] => 10
[5,1,1,2,1] => 10
[5,1,2,1,1] => 10
[5,2,1,1,1] => 10
[1,1,1,3,3] => 9
[1,1,3,1,3] => 9
[1,1,3,3,1] => 9
[1,3,1,1,3] => 9
[1,3,1,3,1] => 9
[1,3,3,1,1] => 9
[3,1,1,1,3] => 9
[3,1,1,3,1] => 9
[3,1,3,1,1] => 9
[3,3,1,1,1] => 9
[1,1,1,3,4] => 10
[1,1,1,4,3] => 10
[1,1,3,1,4] => 10
[1,1,3,4,1] => 10
[1,1,4,1,3] => 10
[1,1,4,3,1] => 10
[1,3,1,1,4] => 10
[1,3,1,4,1] => 10
[1,3,4,1,1] => 10
[1,4,1,1,3] => 10
[1,4,1,3,1] => 10
[1,4,3,1,1] => 10
[3,1,1,1,4] => 10
[3,1,1,4,1] => 10
[3,1,4,1,1] => 10
[3,4,1,1,1] => 10
[4,1,1,1,3] => 10
[4,1,1,3,1] => 10
[4,1,3,1,1] => 10
[4,3,1,1,1] => 10
[1,1,1,3,5] => 11
[1,1,1,5,3] => 11
[1,1,3,1,5] => 11
[1,1,3,5,1] => 11
[1,1,5,1,3] => 11
[1,1,5,3,1] => 11
[1,3,1,1,5] => 11
[1,3,1,5,1] => 11
[1,3,5,1,1] => 11
[1,5,1,1,3] => 11
[1,5,1,3,1] => 11
[1,5,3,1,1] => 11
[3,1,1,1,5] => 11
[3,1,1,5,1] => 11
[3,1,5,1,1] => 11
[3,5,1,1,1] => 11
[5,1,1,1,3] => 11
[5,1,1,3,1] => 11
[5,1,3,1,1] => 11
[5,3,1,1,1] => 11
[1,1,1,4,4] => 11
[1,1,4,1,4] => 11
[1,1,4,4,1] => 11
[1,4,1,1,4] => 11
[1,4,1,4,1] => 11
[1,4,4,1,1] => 11
[4,1,1,1,4] => 11
[4,1,1,4,1] => 11
[4,1,4,1,1] => 11
[4,4,1,1,1] => 11
[1,1,1,4,5] => 12
[1,1,1,5,4] => 12
[1,1,4,1,5] => 12
[1,1,4,5,1] => 12
[1,1,5,1,4] => 12
[1,1,5,4,1] => 12
[1,4,1,1,5] => 12
[1,4,1,5,1] => 12
[1,4,5,1,1] => 12
[1,5,1,1,4] => 12
[1,5,1,4,1] => 12
[1,5,4,1,1] => 12
[4,1,1,1,5] => 12
[4,1,1,5,1] => 12
[4,1,5,1,1] => 12
[4,5,1,1,1] => 12
[5,1,1,1,4] => 12
[5,1,1,4,1] => 12
[5,1,4,1,1] => 12
[5,4,1,1,1] => 12
[1,1,2,2,2] => 8
[1,2,1,2,2] => 8
[1,2,2,1,2] => 8
[1,2,2,2,1] => 8
[2,1,1,2,2] => 8
[2,1,2,1,2] => 8
[2,1,2,2,1] => 8
[2,2,1,1,2] => 8
[2,2,1,2,1] => 8
[2,2,2,1,1] => 8
[1,1,2,2,3] => 9
[1,1,2,3,2] => 9
[1,1,3,2,2] => 9
[1,2,1,2,3] => 9
[1,2,1,3,2] => 9
[1,2,2,1,3] => 9
[1,2,2,3,1] => 9
[1,2,3,1,2] => 9
[1,2,3,2,1] => 9
[1,3,1,2,2] => 9
[1,3,2,1,2] => 9
[1,3,2,2,1] => 9
[2,1,1,2,3] => 9
[2,1,1,3,2] => 9
[2,1,2,1,3] => 9
[2,1,2,3,1] => 9
[2,1,3,1,2] => 9
[2,1,3,2,1] => 9
[2,2,1,1,3] => 9
[2,2,1,3,1] => 9
[2,2,3,1,1] => 9
[2,3,1,1,2] => 9
[2,3,1,2,1] => 9
[2,3,2,1,1] => 9
[3,1,1,2,2] => 9
[3,1,2,1,2] => 9
[3,1,2,2,1] => 9
[3,2,1,1,2] => 9
[3,2,1,2,1] => 9
[3,2,2,1,1] => 9
[1,1,2,2,4] => 10
[1,1,2,4,2] => 10
[1,1,4,2,2] => 10
[1,2,1,2,4] => 10
[1,2,1,4,2] => 10
[1,2,2,1,4] => 10
[1,2,2,4,1] => 10
[1,2,4,1,2] => 10
[1,2,4,2,1] => 10
[1,4,1,2,2] => 10
[1,4,2,1,2] => 10
[1,4,2,2,1] => 10
[2,1,1,2,4] => 10
[2,1,1,4,2] => 10
[2,1,2,1,4] => 10
[2,1,2,4,1] => 10
[2,1,4,1,2] => 10
[2,1,4,2,1] => 10
[2,2,1,1,4] => 10
[2,2,1,4,1] => 10
[2,2,4,1,1] => 10
[2,4,1,1,2] => 10
[2,4,1,2,1] => 10
[2,4,2,1,1] => 10
[4,1,1,2,2] => 10
[4,1,2,1,2] => 10
[4,1,2,2,1] => 10
[4,2,1,1,2] => 10
[4,2,1,2,1] => 10
[4,2,2,1,1] => 10
[1,1,2,2,5] => 11
[1,1,2,5,2] => 11
[1,1,5,2,2] => 11
[1,2,1,2,5] => 11
[1,2,1,5,2] => 11
[1,2,2,1,5] => 11
[1,2,2,5,1] => 11
[1,2,5,1,2] => 11
[1,2,5,2,1] => 11
[1,5,1,2,2] => 11
[1,5,2,1,2] => 11
[1,5,2,2,1] => 11
[2,1,1,2,5] => 11
[2,1,1,5,2] => 11
[2,1,2,1,5] => 11
[2,1,2,5,1] => 11
[2,1,5,1,2] => 11
[2,1,5,2,1] => 11
[2,2,1,1,5] => 11
[2,2,1,5,1] => 11
[2,2,5,1,1] => 11
[2,5,1,1,2] => 11
[2,5,1,2,1] => 11
[2,5,2,1,1] => 11
[5,1,1,2,2] => 11
[5,1,2,1,2] => 11
[5,1,2,2,1] => 11
[5,2,1,1,2] => 11
[5,2,1,2,1] => 11
[5,2,2,1,1] => 11
[1,1,2,3,3] => 10
[1,1,3,2,3] => 10
[1,1,3,3,2] => 10
[1,2,1,3,3] => 10
[1,2,3,1,3] => 10
[1,2,3,3,1] => 10
[1,3,1,2,3] => 10
[1,3,1,3,2] => 10
[1,3,2,1,3] => 10
[1,3,2,3,1] => 10
[1,3,3,1,2] => 10
[1,3,3,2,1] => 10
[2,1,1,3,3] => 10
[2,1,3,1,3] => 10
[2,1,3,3,1] => 10
[2,3,1,1,3] => 10
[2,3,1,3,1] => 10
[2,3,3,1,1] => 10
[3,1,1,2,3] => 10
[3,1,1,3,2] => 10
[3,1,2,1,3] => 10
[3,1,2,3,1] => 10
[3,1,3,1,2] => 10
[3,1,3,2,1] => 10
[3,2,1,1,3] => 10
[3,2,1,3,1] => 10
[3,2,3,1,1] => 10
[3,3,1,1,2] => 10
[3,3,1,2,1] => 10
[3,3,2,1,1] => 10
[1,1,2,3,4] => 11
[1,1,2,4,3] => 11
[1,1,3,2,4] => 11
[1,1,3,4,2] => 11
[1,1,4,2,3] => 11
[1,1,4,3,2] => 11
[1,2,1,3,4] => 11
[1,2,1,4,3] => 11
[1,2,3,1,4] => 11
[1,2,3,4,1] => 11
[1,2,4,1,3] => 11
[1,2,4,3,1] => 11
[1,3,1,2,4] => 11
[1,3,1,4,2] => 11
[1,3,2,1,4] => 11
[1,3,2,4,1] => 11
[1,3,4,1,2] => 11
[1,3,4,2,1] => 11
[1,4,1,2,3] => 11
[1,4,1,3,2] => 11
[1,4,2,1,3] => 11
[1,4,2,3,1] => 11
[1,4,3,1,2] => 11
[1,4,3,2,1] => 11
[2,1,1,3,4] => 11
[2,1,1,4,3] => 11
[2,1,3,1,4] => 11
[2,1,3,4,1] => 11
[2,1,4,1,3] => 11
[2,1,4,3,1] => 11
[2,3,1,1,4] => 11
[2,3,1,4,1] => 11
[2,3,4,1,1] => 11
[2,4,1,1,3] => 11
[2,4,1,3,1] => 11
[2,4,3,1,1] => 11
[3,1,1,2,4] => 11
[3,1,1,4,2] => 11
[3,1,2,1,4] => 11
[3,1,2,4,1] => 11
[3,1,4,1,2] => 11
[3,1,4,2,1] => 11
[3,2,1,1,4] => 11
[3,2,1,4,1] => 11
[3,2,4,1,1] => 11
[3,4,1,1,2] => 11
[3,4,1,2,1] => 11
[3,4,2,1,1] => 11
[4,1,1,2,3] => 11
[4,1,1,3,2] => 11
[4,1,2,1,3] => 11
[4,1,2,3,1] => 11
[4,1,3,1,2] => 11
[4,1,3,2,1] => 11
[4,2,1,1,3] => 11
[4,2,1,3,1] => 11
[4,2,3,1,1] => 11
[4,3,1,1,2] => 11
[4,3,1,2,1] => 11
[4,3,2,1,1] => 11
[1,1,2,3,5] => 12
[1,1,2,5,3] => 12
[1,1,3,2,5] => 12
[1,1,3,5,2] => 12
[1,1,5,2,3] => 12
[1,1,5,3,2] => 12
[1,2,1,3,5] => 12
[1,2,1,5,3] => 12
[1,2,3,1,5] => 12
[1,2,3,5,1] => 12
[1,2,5,1,3] => 12
[1,2,5,3,1] => 12
[1,3,1,2,5] => 12
[1,3,1,5,2] => 12
[1,3,2,1,5] => 12
[1,3,2,5,1] => 12
[1,3,5,1,2] => 12
[1,3,5,2,1] => 12
[1,5,1,2,3] => 12
[1,5,1,3,2] => 12
[1,5,2,1,3] => 12
[1,5,2,3,1] => 12
[1,5,3,1,2] => 12
[1,5,3,2,1] => 12
[2,1,1,3,5] => 12
[2,1,1,5,3] => 12
[2,1,3,1,5] => 12
[2,1,3,5,1] => 12
[2,1,5,1,3] => 12
[2,1,5,3,1] => 12
[2,3,1,1,5] => 12
[2,3,1,5,1] => 12
[2,3,5,1,1] => 12
[2,5,1,1,3] => 12
[2,5,1,3,1] => 12
[2,5,3,1,1] => 12
[3,1,1,2,5] => 12
[3,1,1,5,2] => 12
[3,1,2,1,5] => 12
[3,1,2,5,1] => 12
[3,1,5,1,2] => 12
[3,1,5,2,1] => 12
[3,2,1,1,5] => 12
[3,2,1,5,1] => 12
[3,2,5,1,1] => 12
[3,5,1,1,2] => 12
[3,5,1,2,1] => 12
[3,5,2,1,1] => 12
[5,1,1,2,3] => 12
[5,1,1,3,2] => 12
[5,1,2,1,3] => 12
[5,1,2,3,1] => 12
[5,1,3,1,2] => 12
[5,1,3,2,1] => 12
[5,2,1,1,3] => 12
[5,2,1,3,1] => 12
[5,2,3,1,1] => 12
[5,3,1,1,2] => 12
[5,3,1,2,1] => 12
[5,3,2,1,1] => 12
[1,1,2,4,4] => 12
[1,1,4,2,4] => 12
[1,1,4,4,2] => 12
[1,2,1,4,4] => 12
[1,2,4,1,4] => 12
[1,2,4,4,1] => 12
[1,4,1,2,4] => 12
[1,4,1,4,2] => 12
[1,4,2,1,4] => 12
[1,4,2,4,1] => 12
[1,4,4,1,2] => 12
[1,4,4,2,1] => 12
[2,1,1,4,4] => 12
[2,1,4,1,4] => 12
[2,1,4,4,1] => 12
[2,4,1,1,4] => 12
[2,4,1,4,1] => 12
[2,4,4,1,1] => 12
[4,1,1,2,4] => 12
[4,1,1,4,2] => 12
[4,1,2,1,4] => 12
[4,1,2,4,1] => 12
[4,1,4,1,2] => 12
[4,1,4,2,1] => 12
[4,2,1,1,4] => 12
[4,2,1,4,1] => 12
[4,2,4,1,1] => 12
[4,4,1,1,2] => 12
[4,4,1,2,1] => 12
[4,4,2,1,1] => 12
[1,1,2,4,5] => 13
[1,1,2,5,4] => 13
[1,1,4,2,5] => 13
[1,1,4,5,2] => 13
[1,1,5,2,4] => 13
[1,1,5,4,2] => 13
[1,2,1,4,5] => 13
[1,2,1,5,4] => 13
[1,2,4,1,5] => 13
[1,2,4,5,1] => 13
[1,2,5,1,4] => 13
[1,2,5,4,1] => 13
[1,4,1,2,5] => 13
[1,4,1,5,2] => 13
[1,4,2,1,5] => 13
[1,4,2,5,1] => 13
[1,4,5,1,2] => 13
[1,4,5,2,1] => 13
[1,5,1,2,4] => 13
[1,5,1,4,2] => 13
[1,5,2,1,4] => 13
[1,5,2,4,1] => 13
[1,5,4,1,2] => 13
[1,5,4,2,1] => 13
[2,1,1,4,5] => 13
[2,1,1,5,4] => 13
[2,1,4,1,5] => 13
[2,1,4,5,1] => 13
[2,1,5,1,4] => 13
[2,1,5,4,1] => 13
[2,4,1,1,5] => 13
[2,4,1,5,1] => 13
[2,4,5,1,1] => 13
[2,5,1,1,4] => 13
[2,5,1,4,1] => 13
[2,5,4,1,1] => 13
[4,1,1,2,5] => 13
[4,1,1,5,2] => 13
[4,1,2,1,5] => 13
[4,1,2,5,1] => 13
[4,1,5,1,2] => 13
[4,1,5,2,1] => 13
[4,2,1,1,5] => 13
[4,2,1,5,1] => 13
[4,2,5,1,1] => 13
[4,5,1,1,2] => 13
[4,5,1,2,1] => 13
[4,5,2,1,1] => 13
[5,1,1,2,4] => 13
[5,1,1,4,2] => 13
[5,1,2,1,4] => 13
[5,1,2,4,1] => 13
[5,1,4,1,2] => 13
[5,1,4,2,1] => 13
[5,2,1,1,4] => 13
[5,2,1,4,1] => 13
[5,2,4,1,1] => 13
[5,4,1,1,2] => 13
[5,4,1,2,1] => 13
[5,4,2,1,1] => 13
[1,1,3,3,3] => 11
[1,3,1,3,3] => 11
[1,3,3,1,3] => 11
[1,3,3,3,1] => 11
[3,1,1,3,3] => 11
[3,1,3,1,3] => 11
[3,1,3,3,1] => 11
[3,3,1,1,3] => 11
[3,3,1,3,1] => 11
[3,3,3,1,1] => 11
[1,1,3,3,4] => 12
[1,1,3,4,3] => 12
[1,1,4,3,3] => 12
[1,3,1,3,4] => 12
[1,3,1,4,3] => 12
[1,3,3,1,4] => 12
[1,3,3,4,1] => 12
[1,3,4,1,3] => 12
[1,3,4,3,1] => 12
[1,4,1,3,3] => 12
[1,4,3,1,3] => 12
[1,4,3,3,1] => 12
[3,1,1,3,4] => 12
[3,1,1,4,3] => 12
[3,1,3,1,4] => 12
[3,1,3,4,1] => 12
[3,1,4,1,3] => 12
[3,1,4,3,1] => 12
[3,3,1,1,4] => 12
[3,3,1,4,1] => 12
[3,3,4,1,1] => 12
[3,4,1,1,3] => 12
[3,4,1,3,1] => 12
[3,4,3,1,1] => 12
[4,1,1,3,3] => 12
[4,1,3,1,3] => 12
[4,1,3,3,1] => 12
[4,3,1,1,3] => 12
[4,3,1,3,1] => 12
[4,3,3,1,1] => 12
[1,1,3,3,5] => 13
[1,1,3,5,3] => 13
[1,1,5,3,3] => 13
[1,3,1,3,5] => 13
[1,3,1,5,3] => 13
[1,3,3,1,5] => 13
[1,3,3,5,1] => 13
[1,3,5,1,3] => 13
[1,3,5,3,1] => 13
[1,5,1,3,3] => 13
[1,5,3,1,3] => 13
[1,5,3,3,1] => 13
[3,1,1,3,5] => 13
[3,1,1,5,3] => 13
[3,1,3,1,5] => 13
[3,1,3,5,1] => 13
[3,1,5,1,3] => 13
[3,1,5,3,1] => 13
[3,3,1,1,5] => 13
[3,3,1,5,1] => 13
[3,3,5,1,1] => 13
[3,5,1,1,3] => 13
[3,5,1,3,1] => 13
[3,5,3,1,1] => 13
[5,1,1,3,3] => 13
[5,1,3,1,3] => 13
[5,1,3,3,1] => 13
[5,3,1,1,3] => 13
[5,3,1,3,1] => 13
[5,3,3,1,1] => 13
[1,1,3,4,4] => 13
[1,1,4,3,4] => 13
[1,1,4,4,3] => 13
[1,3,1,4,4] => 13
[1,3,4,1,4] => 13
[1,3,4,4,1] => 13
[1,4,1,3,4] => 13
[1,4,1,4,3] => 13
[1,4,3,1,4] => 13
[1,4,3,4,1] => 13
[1,4,4,1,3] => 13
[1,4,4,3,1] => 13
[3,1,1,4,4] => 13
[3,1,4,1,4] => 13
[3,1,4,4,1] => 13
[3,4,1,1,4] => 13
[3,4,1,4,1] => 13
[3,4,4,1,1] => 13
[4,1,1,3,4] => 13
[4,1,1,4,3] => 13
[4,1,3,1,4] => 13
[4,1,3,4,1] => 13
[4,1,4,1,3] => 13
[4,1,4,3,1] => 13
[4,3,1,1,4] => 13
[4,3,1,4,1] => 13
[4,3,4,1,1] => 13
[4,4,1,1,3] => 13
[4,4,1,3,1] => 13
[4,4,3,1,1] => 13
[1,1,3,4,5] => 14
[1,1,3,5,4] => 14
[1,1,4,3,5] => 14
[1,1,4,5,3] => 14
[1,1,5,3,4] => 14
[1,1,5,4,3] => 14
[1,3,1,4,5] => 14
[1,3,1,5,4] => 14
[1,3,4,1,5] => 14
[1,3,4,5,1] => 14
[1,3,5,1,4] => 14
[1,3,5,4,1] => 14
[1,4,1,3,5] => 14
[1,4,1,5,3] => 14
[1,4,3,1,5] => 14
[1,4,3,5,1] => 14
[1,4,5,1,3] => 14
[1,4,5,3,1] => 14
[1,5,1,3,4] => 14
[1,5,1,4,3] => 14
[1,5,3,1,4] => 14
[1,5,3,4,1] => 14
[1,5,4,1,3] => 14
[1,5,4,3,1] => 14
[3,1,1,4,5] => 14
[3,1,1,5,4] => 14
[3,1,4,1,5] => 14
[3,1,4,5,1] => 14
[3,1,5,1,4] => 14
[3,1,5,4,1] => 14
[3,4,1,1,5] => 14
[3,4,1,5,1] => 14
[3,4,5,1,1] => 14
[3,5,1,1,4] => 14
[3,5,1,4,1] => 14
[3,5,4,1,1] => 14
[4,1,1,3,5] => 14
[4,1,1,5,3] => 14
[4,1,3,1,5] => 14
[4,1,3,5,1] => 14
[4,1,5,1,3] => 14
[4,1,5,3,1] => 14
[4,3,1,1,5] => 14
[4,3,1,5,1] => 14
[4,3,5,1,1] => 14
[4,5,1,1,3] => 14
[4,5,1,3,1] => 14
[4,5,3,1,1] => 14
[5,1,1,3,4] => 14
[5,1,1,4,3] => 14
[5,1,3,1,4] => 14
[5,1,3,4,1] => 14
[5,1,4,1,3] => 14
[5,1,4,3,1] => 14
[5,3,1,1,4] => 14
[5,3,1,4,1] => 14
[5,3,4,1,1] => 14
[5,4,1,1,3] => 14
[5,4,1,3,1] => 14
[5,4,3,1,1] => 14
[1,2,2,2,2] => 9
[2,1,2,2,2] => 9
[2,2,1,2,2] => 9
[2,2,2,1,2] => 9
[2,2,2,2,1] => 9
[1,2,2,2,3] => 10
[1,2,2,3,2] => 10
[1,2,3,2,2] => 10
[1,3,2,2,2] => 10
[2,1,2,2,3] => 10
[2,1,2,3,2] => 10
[2,1,3,2,2] => 10
[2,2,1,2,3] => 10
[2,2,1,3,2] => 10
[2,2,2,1,3] => 10
[2,2,2,3,1] => 10
[2,2,3,1,2] => 10
[2,2,3,2,1] => 10
[2,3,1,2,2] => 10
[2,3,2,1,2] => 10
[2,3,2,2,1] => 10
[3,1,2,2,2] => 10
[3,2,1,2,2] => 10
[3,2,2,1,2] => 10
[3,2,2,2,1] => 10
[1,2,2,2,4] => 11
[1,2,2,4,2] => 11
[1,2,4,2,2] => 11
[1,4,2,2,2] => 11
[2,1,2,2,4] => 11
[2,1,2,4,2] => 11
[2,1,4,2,2] => 11
[2,2,1,2,4] => 11
[2,2,1,4,2] => 11
[2,2,2,1,4] => 11
[2,2,2,4,1] => 11
[2,2,4,1,2] => 11
[2,2,4,2,1] => 11
[2,4,1,2,2] => 11
[2,4,2,1,2] => 11
[2,4,2,2,1] => 11
[4,1,2,2,2] => 11
[4,2,1,2,2] => 11
[4,2,2,1,2] => 11
[4,2,2,2,1] => 11
[1,2,2,2,5] => 12
[1,2,2,5,2] => 12
[1,2,5,2,2] => 12
[1,5,2,2,2] => 12
[2,1,2,2,5] => 12
[2,1,2,5,2] => 12
[2,1,5,2,2] => 12
[2,2,1,2,5] => 12
[2,2,1,5,2] => 12
[2,2,2,1,5] => 12
[2,2,2,5,1] => 12
[2,2,5,1,2] => 12
[2,2,5,2,1] => 12
[2,5,1,2,2] => 12
[2,5,2,1,2] => 12
[2,5,2,2,1] => 12
[5,1,2,2,2] => 12
[5,2,1,2,2] => 12
[5,2,2,1,2] => 12
[5,2,2,2,1] => 12
[1,2,2,3,3] => 11
[1,2,3,2,3] => 11
[1,2,3,3,2] => 11
[1,3,2,2,3] => 11
[1,3,2,3,2] => 11
[1,3,3,2,2] => 11
[2,1,2,3,3] => 11
[2,1,3,2,3] => 11
[2,1,3,3,2] => 11
[2,2,1,3,3] => 11
[2,2,3,1,3] => 11
[2,2,3,3,1] => 11
[2,3,1,2,3] => 11
[2,3,1,3,2] => 11
[2,3,2,1,3] => 11
[2,3,2,3,1] => 11
[2,3,3,1,2] => 11
[2,3,3,2,1] => 11
[3,1,2,2,3] => 11
[3,1,2,3,2] => 11
[3,1,3,2,2] => 11
[3,2,1,2,3] => 11
[3,2,1,3,2] => 11
[3,2,2,1,3] => 11
[3,2,2,3,1] => 11
[3,2,3,1,2] => 11
[3,2,3,2,1] => 11
[3,3,1,2,2] => 11
[3,3,2,1,2] => 11
[3,3,2,2,1] => 11
[1,2,2,3,4] => 12
[1,2,2,4,3] => 12
[1,2,3,2,4] => 12
[1,2,3,4,2] => 12
[1,2,4,2,3] => 12
[1,2,4,3,2] => 12
[1,3,2,2,4] => 12
[1,3,2,4,2] => 12
[1,3,4,2,2] => 12
[1,4,2,2,3] => 12
[1,4,2,3,2] => 12
[1,4,3,2,2] => 12
[2,1,2,3,4] => 12
[2,1,2,4,3] => 12
[2,1,3,2,4] => 12
[2,1,3,4,2] => 12
[2,1,4,2,3] => 12
[2,1,4,3,2] => 12
[2,2,1,3,4] => 12
[2,2,1,4,3] => 12
[2,2,3,1,4] => 12
[2,2,3,4,1] => 12
[2,2,4,1,3] => 12
[2,2,4,3,1] => 12
[2,3,1,2,4] => 12
[2,3,1,4,2] => 12
[2,3,2,1,4] => 12
[2,3,2,4,1] => 12
[2,3,4,1,2] => 12
[2,3,4,2,1] => 12
[2,4,1,2,3] => 12
[2,4,1,3,2] => 12
[2,4,2,1,3] => 12
[2,4,2,3,1] => 12
[2,4,3,1,2] => 12
[2,4,3,2,1] => 12
[3,1,2,2,4] => 12
[3,1,2,4,2] => 12
[3,1,4,2,2] => 12
[3,2,1,2,4] => 12
[3,2,1,4,2] => 12
[3,2,2,1,4] => 12
[3,2,2,4,1] => 12
[3,2,4,1,2] => 12
[3,2,4,2,1] => 12
[3,4,1,2,2] => 12
[3,4,2,1,2] => 12
[3,4,2,2,1] => 12
[4,1,2,2,3] => 12
[4,1,2,3,2] => 12
[4,1,3,2,2] => 12
[4,2,1,2,3] => 12
[4,2,1,3,2] => 12
[4,2,2,1,3] => 12
[4,2,2,3,1] => 12
[4,2,3,1,2] => 12
[4,2,3,2,1] => 12
[4,3,1,2,2] => 12
[4,3,2,1,2] => 12
[4,3,2,2,1] => 12
[1,2,2,3,5] => 13
[1,2,2,5,3] => 13
[1,2,3,2,5] => 13
[1,2,3,5,2] => 13
[1,2,5,2,3] => 13
[1,2,5,3,2] => 13
[1,3,2,2,5] => 13
[1,3,2,5,2] => 13
[1,3,5,2,2] => 13
[1,5,2,2,3] => 13
[1,5,2,3,2] => 13
[1,5,3,2,2] => 13
[2,1,2,3,5] => 13
[2,1,2,5,3] => 13
[2,1,3,2,5] => 13
[2,1,3,5,2] => 13
[2,1,5,2,3] => 13
[2,1,5,3,2] => 13
[2,2,1,3,5] => 13
[2,2,1,5,3] => 13
[2,2,3,1,5] => 13
[2,2,3,5,1] => 13
[2,2,5,1,3] => 13
[2,2,5,3,1] => 13
[2,3,1,2,5] => 13
[2,3,1,5,2] => 13
[2,3,2,1,5] => 13
[2,3,2,5,1] => 13
[2,3,5,1,2] => 13
[2,3,5,2,1] => 13
[2,5,1,2,3] => 13
[2,5,1,3,2] => 13
[2,5,2,1,3] => 13
[2,5,2,3,1] => 13
[2,5,3,1,2] => 13
[2,5,3,2,1] => 13
[3,1,2,2,5] => 13
[3,1,2,5,2] => 13
[3,1,5,2,2] => 13
[3,2,1,2,5] => 13
[3,2,1,5,2] => 13
[3,2,2,1,5] => 13
[3,2,2,5,1] => 13
[3,2,5,1,2] => 13
[3,2,5,2,1] => 13
[3,5,1,2,2] => 13
[3,5,2,1,2] => 13
[3,5,2,2,1] => 13
[5,1,2,2,3] => 13
[5,1,2,3,2] => 13
[5,1,3,2,2] => 13
[5,2,1,2,3] => 13
[5,2,1,3,2] => 13
[5,2,2,1,3] => 13
[5,2,2,3,1] => 13
[5,2,3,1,2] => 13
[5,2,3,2,1] => 13
[5,3,1,2,2] => 13
[5,3,2,1,2] => 13
[5,3,2,2,1] => 13
[1,2,2,4,4] => 13
[1,2,4,2,4] => 13
[1,2,4,4,2] => 13
[1,4,2,2,4] => 13
[1,4,2,4,2] => 13
[1,4,4,2,2] => 13
[2,1,2,4,4] => 13
[2,1,4,2,4] => 13
[2,1,4,4,2] => 13
[2,2,1,4,4] => 13
[2,2,4,1,4] => 13
[2,2,4,4,1] => 13
[2,4,1,2,4] => 13
[2,4,1,4,2] => 13
[2,4,2,1,4] => 13
[2,4,2,4,1] => 13
[2,4,4,1,2] => 13
[2,4,4,2,1] => 13
[4,1,2,2,4] => 13
[4,1,2,4,2] => 13
[4,1,4,2,2] => 13
[4,2,1,2,4] => 13
[4,2,1,4,2] => 13
[4,2,2,1,4] => 13
[4,2,2,4,1] => 13
[4,2,4,1,2] => 13
[4,2,4,2,1] => 13
[4,4,1,2,2] => 13
[4,4,2,1,2] => 13
[4,4,2,2,1] => 13
[1,2,2,4,5] => 14
[1,2,2,5,4] => 14
[1,2,4,2,5] => 14
[1,2,4,5,2] => 14
[1,2,5,2,4] => 14
[1,2,5,4,2] => 14
[1,4,2,2,5] => 14
[1,4,2,5,2] => 14
[1,4,5,2,2] => 14
[1,5,2,2,4] => 14
[1,5,2,4,2] => 14
[1,5,4,2,2] => 14
[2,1,2,4,5] => 14
[2,1,2,5,4] => 14
[2,1,4,2,5] => 14
[2,1,4,5,2] => 14
[2,1,5,2,4] => 14
[2,1,5,4,2] => 14
[2,2,1,4,5] => 14
[2,2,1,5,4] => 14
[2,2,4,1,5] => 14
[2,2,4,5,1] => 14
[2,2,5,1,4] => 14
[2,2,5,4,1] => 14
[2,4,1,2,5] => 14
[2,4,1,5,2] => 14
[2,4,2,1,5] => 14
[2,4,2,5,1] => 14
[2,4,5,1,2] => 14
[2,4,5,2,1] => 14
[2,5,1,2,4] => 14
[2,5,1,4,2] => 14
[2,5,2,1,4] => 14
[2,5,2,4,1] => 14
[2,5,4,1,2] => 14
[2,5,4,2,1] => 14
[4,1,2,2,5] => 14
[4,1,2,5,2] => 14
[4,1,5,2,2] => 14
[4,2,1,2,5] => 14
[4,2,1,5,2] => 14
[4,2,2,1,5] => 14
[4,2,2,5,1] => 14
[4,2,5,1,2] => 14
[4,2,5,2,1] => 14
[4,5,1,2,2] => 14
[4,5,2,1,2] => 14
[4,5,2,2,1] => 14
[5,1,2,2,4] => 14
[5,1,2,4,2] => 14
[5,1,4,2,2] => 14
[5,2,1,2,4] => 14
[5,2,1,4,2] => 14
[5,2,2,1,4] => 14
[5,2,2,4,1] => 14
[5,2,4,1,2] => 14
[5,2,4,2,1] => 14
[5,4,1,2,2] => 14
[5,4,2,1,2] => 14
[5,4,2,2,1] => 14
[1,2,3,3,3] => 12
[1,3,2,3,3] => 12
[1,3,3,2,3] => 12
[1,3,3,3,2] => 12
[2,1,3,3,3] => 12
[2,3,1,3,3] => 12
[2,3,3,1,3] => 12
[2,3,3,3,1] => 12
[3,1,2,3,3] => 12
[3,1,3,2,3] => 12
[3,1,3,3,2] => 12
[3,2,1,3,3] => 12
[3,2,3,1,3] => 12
[3,2,3,3,1] => 12
[3,3,1,2,3] => 12
[3,3,1,3,2] => 12
[3,3,2,1,3] => 12
[3,3,2,3,1] => 12
[3,3,3,1,2] => 12
[3,3,3,2,1] => 12
[1,2,3,3,4] => 13
[1,2,3,4,3] => 13
[1,2,4,3,3] => 13
[1,3,2,3,4] => 13
[1,3,2,4,3] => 13
[1,3,3,2,4] => 13
[1,3,3,4,2] => 13
[1,3,4,2,3] => 13
[1,3,4,3,2] => 13
[1,4,2,3,3] => 13
[1,4,3,2,3] => 13
[1,4,3,3,2] => 13
[2,1,3,3,4] => 13
[2,1,3,4,3] => 13
[2,1,4,3,3] => 13
[2,3,1,3,4] => 13
[2,3,1,4,3] => 13
[2,3,3,1,4] => 13
[2,3,3,4,1] => 13
[2,3,4,1,3] => 13
[2,3,4,3,1] => 13
[2,4,1,3,3] => 13
[2,4,3,1,3] => 13
[2,4,3,3,1] => 13
[3,1,2,3,4] => 13
[3,1,2,4,3] => 13
[3,1,3,2,4] => 13
[3,1,3,4,2] => 13
[3,1,4,2,3] => 13
[3,1,4,3,2] => 13
[3,2,1,3,4] => 13
[3,2,1,4,3] => 13
[3,2,3,1,4] => 13
[3,2,3,4,1] => 13
[3,2,4,1,3] => 13
[3,2,4,3,1] => 13
[3,3,1,2,4] => 13
[3,3,1,4,2] => 13
[3,3,2,1,4] => 13
[3,3,2,4,1] => 13
[3,3,4,1,2] => 13
[3,3,4,2,1] => 13
[3,4,1,2,3] => 13
[3,4,1,3,2] => 13
[3,4,2,1,3] => 13
[3,4,2,3,1] => 13
[3,4,3,1,2] => 13
[3,4,3,2,1] => 13
[4,1,2,3,3] => 13
[4,1,3,2,3] => 13
[4,1,3,3,2] => 13
[4,2,1,3,3] => 13
[4,2,3,1,3] => 13
[4,2,3,3,1] => 13
[4,3,1,2,3] => 13
[4,3,1,3,2] => 13
[4,3,2,1,3] => 13
[4,3,2,3,1] => 13
[] => 0
-----------------------------------------------------------------------------
Created: Nov 05, 2013 at 18:44 by Jeremy L. Martin
-----------------------------------------------------------------------------
Last Updated: Mar 27, 2024 at 13:49 by Martin Rubey