***************************************************************************** * www.FindStat.org - The Combinatorial Statistic Finder * * * * Copyright (C) 2019 The FindStatCrew * * * * 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: 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. = 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