***************************************************************************** * 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: St001209 ----------------------------------------------------------------------------- Collection: Parking functions ----------------------------------------------------------------------------- Description: The pmaj statistic of a parking function. This is the parking function analogue of the bounce statistic, see [1, Section 1.3]. The definition given there is equivalent to the following: One again and again scans the parking function from right to left, keeping track of how often one has started over again. At step $i$, one marks the next position whose value is at most $i$ with the number of restarts from the end of the parking function. The pmaj statistic is then the sum of the markings. For example, consider the parking function $[6,2,4,1,4,1,7,3]$. In the first round, we mark positions $6,4,2$ with $0$'s. In the second round, we mark positions $8,5,3,1$ with $1$'s. In the third round, we mark position $7$ with a $2$. In total, we obtain that $$\operatorname{pmaj}([6,2,4,1,4,1,7,3]) = 6.$$ This statistic is equidistributed with the area statistic [[St000188]] and the dinv statistic [[St000136]]. ----------------------------------------------------------------------------- References: [1] Loehr, N. A. Combinatorics of $q$, $t$-parking functions [[MathSciNet:2110560]] ----------------------------------------------------------------------------- Code: def statistic(PF): PF = list(PF) PF.reverse() n = len(PF) count = 1 run = 0 pm = 0 while any( not x is Infinity for x in PF ): for i in range(n): if PF[i] <= count: count += 1 PF[i] = Infinity pm += run run += 1 return pm ----------------------------------------------------------------------------- Statistic values: [1] => 0 [1,1] => 0 [1,2] => 1 [2,1] => 0 [1,1,1] => 0 [1,1,2] => 1 [1,2,1] => 0 [2,1,1] => 0 [1,1,3] => 1 [1,3,1] => 1 [3,1,1] => 0 [1,2,2] => 2 [2,1,2] => 1 [2,2,1] => 0 [1,2,3] => 3 [1,3,2] => 2 [2,1,3] => 1 [2,3,1] => 1 [3,1,2] => 2 [3,2,1] => 0 [1,1,1,1] => 0 [1,1,1,2] => 1 [1,1,2,1] => 0 [1,2,1,1] => 0 [2,1,1,1] => 0 [1,1,1,3] => 1 [1,1,3,1] => 1 [1,3,1,1] => 0 [3,1,1,1] => 0 [1,1,1,4] => 1 [1,1,4,1] => 1 [1,4,1,1] => 1 [4,1,1,1] => 0 [1,1,2,2] => 2 [1,2,1,2] => 1 [1,2,2,1] => 0 [2,1,1,2] => 1 [2,1,2,1] => 0 [2,2,1,1] => 0 [1,1,2,3] => 2 [1,1,3,2] => 2 [1,2,1,3] => 1 [1,2,3,1] => 1 [1,3,1,2] => 2 [1,3,2,1] => 0 [2,1,1,3] => 1 [2,1,3,1] => 1 [2,3,1,1] => 0 [3,1,1,2] => 1 [3,1,2,1] => 0 [3,2,1,1] => 0 [1,1,2,4] => 3 [1,1,4,2] => 2 [1,2,1,4] => 1 [1,2,4,1] => 1 [1,4,1,2] => 2 [1,4,2,1] => 1 [2,1,1,4] => 1 [2,1,4,1] => 1 [2,4,1,1] => 1 [4,1,1,2] => 2 [4,1,2,1] => 0 [4,2,1,1] => 0 [1,1,3,3] => 2 [1,3,1,3] => 2 [1,3,3,1] => 2 [3,1,1,3] => 1 [3,1,3,1] => 1 [3,3,1,1] => 0 [1,1,3,4] => 3 [1,1,4,3] => 2 [1,3,1,4] => 3 [1,3,4,1] => 3 [1,4,1,3] => 2 [1,4,3,1] => 2 [3,1,1,4] => 1 [3,1,4,1] => 1 [3,4,1,1] => 1 [4,1,1,3] => 2 [4,1,3,1] => 2 [4,3,1,1] => 0 [1,2,2,2] => 3 [2,1,2,2] => 2 [2,2,1,2] => 1 [2,2,2,1] => 0 [1,2,2,3] => 4 [1,2,3,2] => 3 [1,3,2,2] => 3 [2,1,2,3] => 2 [2,1,3,2] => 2 [2,2,1,3] => 1 [2,2,3,1] => 1 [2,3,1,2] => 2 [2,3,2,1] => 0 [3,1,2,2] => 3 [3,2,1,2] => 1 [3,2,2,1] => 0 [1,2,2,4] => 4 [1,2,4,2] => 4 [1,4,2,2] => 3 [2,1,2,4] => 3 [2,1,4,2] => 2 [2,2,1,4] => 1 [2,2,4,1] => 1 [2,4,1,2] => 2 [2,4,2,1] => 1 [4,1,2,2] => 3 [4,2,1,2] => 2 [4,2,2,1] => 0 [1,2,3,3] => 5 [1,3,2,3] => 4 [1,3,3,2] => 3 [2,1,3,3] => 2 [2,3,1,3] => 2 [2,3,3,1] => 2 [3,1,2,3] => 4 [3,1,3,2] => 3 [3,2,1,3] => 1 [3,2,3,1] => 1 [3,3,1,2] => 3 [3,3,2,1] => 0 [1,2,3,4] => 6 [1,2,4,3] => 5 [1,3,2,4] => 4 [1,3,4,2] => 4 [1,4,2,3] => 5 [1,4,3,2] => 3 [2,1,3,4] => 3 [2,1,4,3] => 2 [2,3,1,4] => 3 [2,3,4,1] => 3 [2,4,1,3] => 2 [2,4,3,1] => 2 [3,1,2,4] => 4 [3,1,4,2] => 4 [3,2,1,4] => 1 [3,2,4,1] => 1 [3,4,1,2] => 4 [3,4,2,1] => 1 [4,1,2,3] => 5 [4,1,3,2] => 3 [4,2,1,3] => 2 [4,2,3,1] => 2 [4,3,1,2] => 3 [4,3,2,1] => 0 [1,1,1,1,1] => 0 [1,1,1,1,2] => 1 [1,1,1,2,1] => 0 [1,1,2,1,1] => 0 [1,2,1,1,1] => 0 [2,1,1,1,1] => 0 [1,1,1,1,3] => 1 [1,1,1,3,1] => 1 [1,1,3,1,1] => 0 [1,3,1,1,1] => 0 [3,1,1,1,1] => 0 [1,1,1,1,4] => 1 [1,1,1,4,1] => 1 [1,1,4,1,1] => 1 [1,4,1,1,1] => 0 [4,1,1,1,1] => 0 [1,1,1,1,5] => 1 [1,1,1,5,1] => 1 [1,1,5,1,1] => 1 [1,5,1,1,1] => 1 [5,1,1,1,1] => 0 [1,1,1,2,2] => 2 [1,1,2,1,2] => 1 [1,1,2,2,1] => 0 [1,2,1,1,2] => 1 [1,2,1,2,1] => 0 [1,2,2,1,1] => 0 [2,1,1,1,2] => 1 [2,1,1,2,1] => 0 [2,1,2,1,1] => 0 [2,2,1,1,1] => 0 [1,1,1,2,3] => 2 [1,1,1,3,2] => 2 [1,1,2,1,3] => 1 [1,1,2,3,1] => 1 [1,1,3,1,2] => 2 [1,1,3,2,1] => 0 [1,2,1,1,3] => 1 [1,2,1,3,1] => 1 [1,2,3,1,1] => 0 [1,3,1,1,2] => 1 [1,3,1,2,1] => 0 [1,3,2,1,1] => 0 [2,1,1,1,3] => 1 [2,1,1,3,1] => 1 [2,1,3,1,1] => 0 [2,3,1,1,1] => 0 [3,1,1,1,2] => 1 [3,1,1,2,1] => 0 [3,1,2,1,1] => 0 [3,2,1,1,1] => 0 [1,1,1,2,4] => 2 [1,1,1,4,2] => 2 [1,1,2,1,4] => 1 [1,1,2,4,1] => 1 [1,1,4,1,2] => 2 [1,1,4,2,1] => 1 [1,2,1,1,4] => 1 [1,2,1,4,1] => 1 [1,2,4,1,1] => 1 [1,4,1,1,2] => 2 [1,4,1,2,1] => 0 [1,4,2,1,1] => 0 [2,1,1,1,4] => 1 [2,1,1,4,1] => 1 [2,1,4,1,1] => 1 [2,4,1,1,1] => 0 [4,1,1,1,2] => 1 [4,1,1,2,1] => 0 [4,1,2,1,1] => 0 [4,2,1,1,1] => 0 [1,1,1,2,5] => 3 [1,1,1,5,2] => 2 [1,1,2,1,5] => 1 [1,1,2,5,1] => 1 [1,1,5,1,2] => 2 [1,1,5,2,1] => 1 [1,2,1,1,5] => 1 [1,2,1,5,1] => 1 [1,2,5,1,1] => 1 [1,5,1,1,2] => 2 [1,5,1,2,1] => 1 [1,5,2,1,1] => 1 [2,1,1,1,5] => 1 [2,1,1,5,1] => 1 [2,1,5,1,1] => 1 [2,5,1,1,1] => 1 [5,1,1,1,2] => 2 [5,1,1,2,1] => 0 [5,1,2,1,1] => 0 [5,2,1,1,1] => 0 [1,1,1,3,3] => 2 [1,1,3,1,3] => 2 [1,1,3,3,1] => 2 [1,3,1,1,3] => 1 [1,3,1,3,1] => 1 [1,3,3,1,1] => 0 [3,1,1,1,3] => 1 [3,1,1,3,1] => 1 [3,1,3,1,1] => 0 [3,3,1,1,1] => 0 [1,1,1,3,4] => 2 [1,1,1,4,3] => 2 [1,1,3,1,4] => 2 [1,1,3,4,1] => 2 [1,1,4,1,3] => 2 [1,1,4,3,1] => 2 [1,3,1,1,4] => 1 [1,3,1,4,1] => 1 [1,3,4,1,1] => 1 [1,4,1,1,3] => 2 [1,4,1,3,1] => 2 [1,4,3,1,1] => 0 [3,1,1,1,4] => 1 [3,1,1,4,1] => 1 [3,1,4,1,1] => 1 [3,4,1,1,1] => 0 [4,1,1,1,3] => 1 [4,1,1,3,1] => 1 [4,1,3,1,1] => 0 [4,3,1,1,1] => 0 [1,1,1,3,5] => 3 [1,1,1,5,3] => 2 [1,1,3,1,5] => 3 [1,1,3,5,1] => 3 [1,1,5,1,3] => 2 [1,1,5,3,1] => 2 [1,3,1,1,5] => 1 [1,3,1,5,1] => 1 [1,3,5,1,1] => 1 [1,5,1,1,3] => 2 [1,5,1,3,1] => 2 [1,5,3,1,1] => 1 [3,1,1,1,5] => 1 [3,1,1,5,1] => 1 [3,1,5,1,1] => 1 [3,5,1,1,1] => 1 [5,1,1,1,3] => 2 [5,1,1,3,1] => 2 [5,1,3,1,1] => 0 [5,3,1,1,1] => 0 [1,1,1,4,4] => 2 [1,1,4,1,4] => 2 [1,1,4,4,1] => 2 [1,4,1,1,4] => 2 [1,4,1,4,1] => 2 [1,4,4,1,1] => 2 [4,1,1,1,4] => 1 [4,1,1,4,1] => 1 [4,1,4,1,1] => 1 [4,4,1,1,1] => 0 [1,1,1,4,5] => 3 [1,1,1,5,4] => 2 [1,1,4,1,5] => 3 [1,1,4,5,1] => 3 [1,1,5,1,4] => 2 [1,1,5,4,1] => 2 [1,4,1,1,5] => 3 [1,4,1,5,1] => 3 [1,4,5,1,1] => 3 [1,5,1,1,4] => 2 [1,5,1,4,1] => 2 [1,5,4,1,1] => 2 [4,1,1,1,5] => 1 [4,1,1,5,1] => 1 [4,1,5,1,1] => 1 [4,5,1,1,1] => 1 [5,1,1,1,4] => 2 [5,1,1,4,1] => 2 [5,1,4,1,1] => 2 [5,4,1,1,1] => 0 [1,1,2,2,2] => 3 [1,2,1,2,2] => 2 [1,2,2,1,2] => 1 [1,2,2,2,1] => 0 [2,1,1,2,2] => 2 [2,1,2,1,2] => 1 [2,1,2,2,1] => 0 [2,2,1,1,2] => 1 [2,2,1,2,1] => 0 [2,2,2,1,1] => 0 [1,1,2,2,3] => 3 [1,1,2,3,2] => 3 [1,1,3,2,2] => 3 [1,2,1,2,3] => 2 [1,2,1,3,2] => 2 [1,2,2,1,3] => 1 [1,2,2,3,1] => 1 [1,2,3,1,2] => 2 [1,2,3,2,1] => 0 [1,3,1,2,2] => 3 [1,3,2,1,2] => 1 [1,3,2,2,1] => 0 [2,1,1,2,3] => 2 [2,1,1,3,2] => 2 [2,1,2,1,3] => 1 [2,1,2,3,1] => 1 [2,1,3,1,2] => 2 [2,1,3,2,1] => 0 [2,2,1,1,3] => 1 [2,2,1,3,1] => 1 [2,2,3,1,1] => 0 [2,3,1,1,2] => 1 [2,3,1,2,1] => 0 [2,3,2,1,1] => 0 [3,1,1,2,2] => 2 [3,1,2,1,2] => 1 [3,1,2,2,1] => 0 [3,2,1,1,2] => 1 [3,2,1,2,1] => 0 [3,2,2,1,1] => 0 [1,1,2,2,4] => 4 [1,1,2,4,2] => 3 [1,1,4,2,2] => 3 [1,2,1,2,4] => 2 [1,2,1,4,2] => 2 [1,2,2,1,4] => 1 [1,2,2,4,1] => 1 [1,2,4,1,2] => 2 [1,2,4,2,1] => 1 [1,4,1,2,2] => 3 [1,4,2,1,2] => 2 [1,4,2,2,1] => 0 [2,1,1,2,4] => 2 [2,1,1,4,2] => 2 [2,1,2,1,4] => 1 [2,1,2,4,1] => 1 [2,1,4,1,2] => 2 [2,1,4,2,1] => 1 [2,2,1,1,4] => 1 [2,2,1,4,1] => 1 [2,2,4,1,1] => 1 [2,4,1,1,2] => 2 [2,4,1,2,1] => 0 [2,4,2,1,1] => 0 [4,1,1,2,2] => 3 [4,1,2,1,2] => 1 [4,1,2,2,1] => 0 [4,2,1,1,2] => 1 [4,2,1,2,1] => 0 [4,2,2,1,1] => 0 [1,1,2,2,5] => 4 [1,1,2,5,2] => 4 [1,1,5,2,2] => 3 [1,2,1,2,5] => 3 [1,2,1,5,2] => 2 [1,2,2,1,5] => 1 [1,2,2,5,1] => 1 [1,2,5,1,2] => 2 [1,2,5,2,1] => 1 [1,5,1,2,2] => 3 [1,5,2,1,2] => 2 [1,5,2,2,1] => 1 [2,1,1,2,5] => 3 [2,1,1,5,2] => 2 [2,1,2,1,5] => 1 [2,1,2,5,1] => 1 [2,1,5,1,2] => 2 [2,1,5,2,1] => 1 [2,2,1,1,5] => 1 [2,2,1,5,1] => 1 [2,2,5,1,1] => 1 [2,5,1,1,2] => 2 [2,5,1,2,1] => 1 [2,5,2,1,1] => 1 [5,1,1,2,2] => 3 [5,1,2,1,2] => 2 [5,1,2,2,1] => 0 [5,2,1,1,2] => 2 [5,2,1,2,1] => 0 [5,2,2,1,1] => 0 [1,1,2,3,3] => 3 [1,1,3,2,3] => 3 [1,1,3,3,2] => 3 [1,2,1,3,3] => 2 [1,2,3,1,3] => 2 [1,2,3,3,1] => 2 [1,3,1,2,3] => 3 [1,3,1,3,2] => 3 [1,3,2,1,3] => 1 [1,3,2,3,1] => 1 [1,3,3,1,2] => 3 [1,3,3,2,1] => 0 [2,1,1,3,3] => 2 [2,1,3,1,3] => 2 [2,1,3,3,1] => 2 [2,3,1,1,3] => 1 [2,3,1,3,1] => 1 [2,3,3,1,1] => 0 [3,1,1,2,3] => 2 [3,1,1,3,2] => 2 [3,1,2,1,3] => 1 [3,1,2,3,1] => 1 [3,1,3,1,2] => 2 [3,1,3,2,1] => 0 [3,2,1,1,3] => 1 [3,2,1,3,1] => 1 [3,2,3,1,1] => 0 [3,3,1,1,2] => 1 [3,3,1,2,1] => 0 [3,3,2,1,1] => 0 [1,1,2,3,4] => 4 [1,1,2,4,3] => 3 [1,1,3,2,4] => 4 [1,1,3,4,2] => 3 [1,1,4,2,3] => 3 [1,1,4,3,2] => 3 [1,2,1,3,4] => 2 [1,2,1,4,3] => 2 [1,2,3,1,4] => 2 [1,2,3,4,1] => 2 [1,2,4,1,3] => 2 [1,2,4,3,1] => 2 [1,3,1,2,4] => 4 [1,3,1,4,2] => 3 [1,3,2,1,4] => 1 [1,3,2,4,1] => 1 [1,3,4,1,2] => 3 [1,3,4,2,1] => 1 [1,4,1,2,3] => 3 [1,4,1,3,2] => 3 [1,4,2,1,3] => 2 [1,4,2,3,1] => 2 [1,4,3,1,2] => 3 [1,4,3,2,1] => 0 [2,1,1,3,4] => 2 [2,1,1,4,3] => 2 [2,1,3,1,4] => 2 [2,1,3,4,1] => 2 [2,1,4,1,3] => 2 [2,1,4,3,1] => 2 [2,3,1,1,4] => 1 [2,3,1,4,1] => 1 [2,3,4,1,1] => 1 [2,4,1,1,3] => 2 [2,4,1,3,1] => 2 [2,4,3,1,1] => 0 [3,1,1,2,4] => 2 [3,1,1,4,2] => 2 [3,1,2,1,4] => 1 [3,1,2,4,1] => 1 [3,1,4,1,2] => 2 [3,1,4,2,1] => 1 [3,2,1,1,4] => 1 [3,2,1,4,1] => 1 [3,2,4,1,1] => 1 [3,4,1,1,2] => 2 [3,4,1,2,1] => 0 [3,4,2,1,1] => 0 [4,1,1,2,3] => 3 [4,1,1,3,2] => 3 [4,1,2,1,3] => 1 [4,1,2,3,1] => 1 [4,1,3,1,2] => 3 [4,1,3,2,1] => 0 [4,2,1,1,3] => 1 [4,2,1,3,1] => 1 [4,2,3,1,1] => 0 [4,3,1,1,2] => 1 [4,3,1,2,1] => 0 [4,3,2,1,1] => 0 [1,1,2,3,5] => 4 [1,1,2,5,3] => 4 [1,1,3,2,5] => 4 [1,1,3,5,2] => 4 [1,1,5,2,3] => 3 [1,1,5,3,2] => 3 [1,2,1,3,5] => 3 [1,2,1,5,3] => 2 [1,2,3,1,5] => 3 [1,2,3,5,1] => 3 [1,2,5,1,3] => 2 [1,2,5,3,1] => 2 [1,3,1,2,5] => 4 [1,3,1,5,2] => 4 [1,3,2,1,5] => 1 [1,3,2,5,1] => 1 [1,3,5,1,2] => 4 [1,3,5,2,1] => 1 [1,5,1,2,3] => 3 [1,5,1,3,2] => 3 [1,5,2,1,3] => 2 [1,5,2,3,1] => 2 [1,5,3,1,2] => 3 [1,5,3,2,1] => 1 [2,1,1,3,5] => 3 [2,1,1,5,3] => 2 [2,1,3,1,5] => 3 [2,1,3,5,1] => 3 [2,1,5,1,3] => 2 [2,1,5,3,1] => 2 [2,3,1,1,5] => 1 [2,3,1,5,1] => 1 [2,3,5,1,1] => 1 [2,5,1,1,3] => 2 [2,5,1,3,1] => 2 [2,5,3,1,1] => 1 [3,1,1,2,5] => 3 [3,1,1,5,2] => 2 [3,1,2,1,5] => 1 [3,1,2,5,1] => 1 [3,1,5,1,2] => 2 [3,1,5,2,1] => 1 [3,2,1,1,5] => 1 [3,2,1,5,1] => 1 [3,2,5,1,1] => 1 [3,5,1,1,2] => 2 [3,5,1,2,1] => 1 [3,5,2,1,1] => 1 [5,1,1,2,3] => 3 [5,1,1,3,2] => 3 [5,1,2,1,3] => 2 [5,1,2,3,1] => 2 [5,1,3,1,2] => 3 [5,1,3,2,1] => 0 [5,2,1,1,3] => 2 [5,2,1,3,1] => 2 [5,2,3,1,1] => 0 [5,3,1,1,2] => 2 [5,3,1,2,1] => 0 [5,3,2,1,1] => 0 [1,1,2,4,4] => 5 [1,1,4,2,4] => 4 [1,1,4,4,2] => 3 [1,2,1,4,4] => 2 [1,2,4,1,4] => 2 [1,2,4,4,1] => 2 [1,4,1,2,4] => 4 [1,4,1,4,2] => 3 [1,4,2,1,4] => 2 [1,4,2,4,1] => 2 [1,4,4,1,2] => 3 [1,4,4,2,1] => 2 [2,1,1,4,4] => 2 [2,1,4,1,4] => 2 [2,1,4,4,1] => 2 [2,4,1,1,4] => 2 [2,4,1,4,1] => 2 [2,4,4,1,1] => 2 [4,1,1,2,4] => 4 [4,1,1,4,2] => 3 [4,1,2,1,4] => 1 [4,1,2,4,1] => 1 [4,1,4,1,2] => 3 [4,1,4,2,1] => 1 [4,2,1,1,4] => 1 [4,2,1,4,1] => 1 [4,2,4,1,1] => 1 [4,4,1,1,2] => 3 [4,4,1,2,1] => 0 [4,4,2,1,1] => 0 [1,1,2,4,5] => 6 [1,1,2,5,4] => 5 [1,1,4,2,5] => 4 [1,1,4,5,2] => 4 [1,1,5,2,4] => 5 [1,1,5,4,2] => 3 [1,2,1,4,5] => 3 [1,2,1,5,4] => 2 [1,2,4,1,5] => 3 [1,2,4,5,1] => 3 [1,2,5,1,4] => 2 [1,2,5,4,1] => 2 [1,4,1,2,5] => 4 [1,4,1,5,2] => 4 [1,4,2,1,5] => 3 [1,4,2,5,1] => 3 [1,4,5,1,2] => 4 [1,4,5,2,1] => 3 [1,5,1,2,4] => 5 [1,5,1,4,2] => 3 [1,5,2,1,4] => 2 [1,5,2,4,1] => 2 [1,5,4,1,2] => 3 [1,5,4,2,1] => 2 [2,1,1,4,5] => 3 [2,1,1,5,4] => 2 [2,1,4,1,5] => 3 [2,1,4,5,1] => 3 [2,1,5,1,4] => 2 [2,1,5,4,1] => 2 [2,4,1,1,5] => 3 [2,4,1,5,1] => 3 [2,4,5,1,1] => 3 [2,5,1,1,4] => 2 [2,5,1,4,1] => 2 [2,5,4,1,1] => 2 [4,1,1,2,5] => 4 [4,1,1,5,2] => 4 [4,1,2,1,5] => 1 [4,1,2,5,1] => 1 [4,1,5,1,2] => 4 [4,1,5,2,1] => 1 [4,2,1,1,5] => 1 [4,2,1,5,1] => 1 [4,2,5,1,1] => 1 [4,5,1,1,2] => 4 [4,5,1,2,1] => 1 [4,5,2,1,1] => 1 [5,1,1,2,4] => 5 [5,1,1,4,2] => 3 [5,1,2,1,4] => 2 [5,1,2,4,1] => 2 [5,1,4,1,2] => 3 [5,1,4,2,1] => 2 [5,2,1,1,4] => 2 [5,2,1,4,1] => 2 [5,2,4,1,1] => 2 [5,4,1,1,2] => 3 [5,4,1,2,1] => 0 [5,4,2,1,1] => 0 [1,1,3,3,3] => 3 [1,3,1,3,3] => 3 [1,3,3,1,3] => 3 [1,3,3,3,1] => 3 [3,1,1,3,3] => 2 [3,1,3,1,3] => 2 [3,1,3,3,1] => 2 [3,3,1,1,3] => 1 [3,3,1,3,1] => 1 [3,3,3,1,1] => 0 [1,1,3,3,4] => 4 [1,1,3,4,3] => 3 [1,1,4,3,3] => 3 [1,3,1,3,4] => 4 [1,3,1,4,3] => 3 [1,3,3,1,4] => 4 [1,3,3,4,1] => 4 [1,3,4,1,3] => 3 [1,3,4,3,1] => 3 [1,4,1,3,3] => 3 [1,4,3,1,3] => 3 [1,4,3,3,1] => 3 [3,1,1,3,4] => 2 [3,1,1,4,3] => 2 [3,1,3,1,4] => 2 [3,1,3,4,1] => 2 [3,1,4,1,3] => 2 [3,1,4,3,1] => 2 [3,3,1,1,4] => 1 [3,3,1,4,1] => 1 [3,3,4,1,1] => 1 [3,4,1,1,3] => 2 [3,4,1,3,1] => 2 [3,4,3,1,1] => 0 [4,1,1,3,3] => 3 [4,1,3,1,3] => 3 [4,1,3,3,1] => 3 [4,3,1,1,3] => 1 [4,3,1,3,1] => 1 [4,3,3,1,1] => 0 [1,1,3,3,5] => 4 [1,1,3,5,3] => 4 [1,1,5,3,3] => 3 [1,3,1,3,5] => 4 [1,3,1,5,3] => 4 [1,3,3,1,5] => 4 [1,3,3,5,1] => 4 [1,3,5,1,3] => 4 [1,3,5,3,1] => 4 [1,5,1,3,3] => 3 [1,5,3,1,3] => 3 [1,5,3,3,1] => 3 [3,1,1,3,5] => 3 [3,1,1,5,3] => 2 [3,1,3,1,5] => 3 [3,1,3,5,1] => 3 [3,1,5,1,3] => 2 [3,1,5,3,1] => 2 [3,3,1,1,5] => 1 [3,3,1,5,1] => 1 [3,3,5,1,1] => 1 [3,5,1,1,3] => 2 [3,5,1,3,1] => 2 [3,5,3,1,1] => 1 [5,1,1,3,3] => 3 [5,1,3,1,3] => 3 [5,1,3,3,1] => 3 [5,3,1,1,3] => 2 [5,3,1,3,1] => 2 [5,3,3,1,1] => 0 [1,1,3,4,4] => 5 [1,1,4,3,4] => 4 [1,1,4,4,3] => 3 [1,3,1,4,4] => 5 [1,3,4,1,4] => 5 [1,3,4,4,1] => 5 [1,4,1,3,4] => 4 [1,4,1,4,3] => 3 [1,4,3,1,4] => 4 [1,4,3,4,1] => 4 [1,4,4,1,3] => 3 [1,4,4,3,1] => 3 [3,1,1,4,4] => 2 [3,1,4,1,4] => 2 [3,1,4,4,1] => 2 [3,4,1,1,4] => 2 [3,4,1,4,1] => 2 [3,4,4,1,1] => 2 [4,1,1,3,4] => 4 [4,1,1,4,3] => 3 [4,1,3,1,4] => 4 [4,1,3,4,1] => 4 [4,1,4,1,3] => 3 [4,1,4,3,1] => 3 [4,3,1,1,4] => 1 [4,3,1,4,1] => 1 [4,3,4,1,1] => 1 [4,4,1,1,3] => 3 [4,4,1,3,1] => 3 [4,4,3,1,1] => 0 [1,1,3,4,5] => 6 [1,1,3,5,4] => 5 [1,1,4,3,5] => 4 [1,1,4,5,3] => 4 [1,1,5,3,4] => 5 [1,1,5,4,3] => 3 [1,3,1,4,5] => 6 [1,3,1,5,4] => 5 [1,3,4,1,5] => 6 [1,3,4,5,1] => 6 [1,3,5,1,4] => 5 [1,3,5,4,1] => 5 [1,4,1,3,5] => 4 [1,4,1,5,3] => 4 [1,4,3,1,5] => 4 [1,4,3,5,1] => 4 [1,4,5,1,3] => 4 [1,4,5,3,1] => 4 [1,5,1,3,4] => 5 [1,5,1,4,3] => 3 [1,5,3,1,4] => 5 [1,5,3,4,1] => 5 [1,5,4,1,3] => 3 [1,5,4,3,1] => 3 [3,1,1,4,5] => 3 [3,1,1,5,4] => 2 [3,1,4,1,5] => 3 [3,1,4,5,1] => 3 [3,1,5,1,4] => 2 [3,1,5,4,1] => 2 [3,4,1,1,5] => 3 [3,4,1,5,1] => 3 [3,4,5,1,1] => 3 [3,5,1,1,4] => 2 [3,5,1,4,1] => 2 [3,5,4,1,1] => 2 [4,1,1,3,5] => 4 [4,1,1,5,3] => 4 [4,1,3,1,5] => 4 [4,1,3,5,1] => 4 [4,1,5,1,3] => 4 [4,1,5,3,1] => 4 [4,3,1,1,5] => 1 [4,3,1,5,1] => 1 [4,3,5,1,1] => 1 [4,5,1,1,3] => 4 [4,5,1,3,1] => 4 [4,5,3,1,1] => 1 [5,1,1,3,4] => 5 [5,1,1,4,3] => 3 [5,1,3,1,4] => 5 [5,1,3,4,1] => 5 [5,1,4,1,3] => 3 [5,1,4,3,1] => 3 [5,3,1,1,4] => 2 [5,3,1,4,1] => 2 [5,3,4,1,1] => 2 [5,4,1,1,3] => 3 [5,4,1,3,1] => 3 [5,4,3,1,1] => 0 [1,2,2,2,2] => 4 [2,1,2,2,2] => 3 [2,2,1,2,2] => 2 [2,2,2,1,2] => 1 [2,2,2,2,1] => 0 [1,2,2,2,3] => 5 [1,2,2,3,2] => 4 [1,2,3,2,2] => 4 [1,3,2,2,2] => 4 [2,1,2,2,3] => 3 [2,1,2,3,2] => 3 [2,1,3,2,2] => 3 [2,2,1,2,3] => 2 [2,2,1,3,2] => 2 [2,2,2,1,3] => 1 [2,2,2,3,1] => 1 [2,2,3,1,2] => 2 [2,2,3,2,1] => 0 [2,3,1,2,2] => 3 [2,3,2,1,2] => 1 [2,3,2,2,1] => 0 [3,1,2,2,2] => 4 [3,2,1,2,2] => 2 [3,2,2,1,2] => 1 [3,2,2,2,1] => 0 [1,2,2,2,4] => 5 [1,2,2,4,2] => 5 [1,2,4,2,2] => 4 [1,4,2,2,2] => 4 [2,1,2,2,4] => 4 [2,1,2,4,2] => 3 [2,1,4,2,2] => 3 [2,2,1,2,4] => 2 [2,2,1,4,2] => 2 [2,2,2,1,4] => 1 [2,2,2,4,1] => 1 [2,2,4,1,2] => 2 [2,2,4,2,1] => 1 [2,4,1,2,2] => 3 [2,4,2,1,2] => 2 [2,4,2,2,1] => 0 [4,1,2,2,2] => 4 [4,2,1,2,2] => 3 [4,2,2,1,2] => 1 [4,2,2,2,1] => 0 [1,2,2,2,5] => 5 [1,2,2,5,2] => 5 [1,2,5,2,2] => 5 [1,5,2,2,2] => 4 [2,1,2,2,5] => 4 [2,1,2,5,2] => 4 [2,1,5,2,2] => 3 [2,2,1,2,5] => 3 [2,2,1,5,2] => 2 [2,2,2,1,5] => 1 [2,2,2,5,1] => 1 [2,2,5,1,2] => 2 [2,2,5,2,1] => 1 [2,5,1,2,2] => 3 [2,5,2,1,2] => 2 [2,5,2,2,1] => 1 [5,1,2,2,2] => 4 [5,2,1,2,2] => 3 [5,2,2,1,2] => 2 [5,2,2,2,1] => 0 [1,2,2,3,3] => 6 [1,2,3,2,3] => 5 [1,2,3,3,2] => 4 [1,3,2,2,3] => 5 [1,3,2,3,2] => 4 [1,3,3,2,2] => 4 [2,1,2,3,3] => 3 [2,1,3,2,3] => 3 [2,1,3,3,2] => 3 [2,2,1,3,3] => 2 [2,2,3,1,3] => 2 [2,2,3,3,1] => 2 [2,3,1,2,3] => 3 [2,3,1,3,2] => 3 [2,3,2,1,3] => 1 [2,3,2,3,1] => 1 [2,3,3,1,2] => 3 [2,3,3,2,1] => 0 [3,1,2,2,3] => 5 [3,1,2,3,2] => 4 [3,1,3,2,2] => 4 [3,2,1,2,3] => 2 [3,2,1,3,2] => 2 [3,2,2,1,3] => 1 [3,2,2,3,1] => 1 [3,2,3,1,2] => 2 [3,2,3,2,1] => 0 [3,3,1,2,2] => 4 [3,3,2,1,2] => 1 [3,3,2,2,1] => 0 [1,2,2,3,4] => 6 [1,2,2,4,3] => 6 [1,2,3,2,4] => 5 [1,2,3,4,2] => 5 [1,2,4,2,3] => 6 [1,2,4,3,2] => 4 [1,3,2,2,4] => 5 [1,3,2,4,2] => 5 [1,3,4,2,2] => 4 [1,4,2,2,3] => 5 [1,4,2,3,2] => 4 [1,4,3,2,2] => 4 [2,1,2,3,4] => 4 [2,1,2,4,3] => 3 [2,1,3,2,4] => 4 [2,1,3,4,2] => 3 [2,1,4,2,3] => 3 [2,1,4,3,2] => 3 [2,2,1,3,4] => 2 [2,2,1,4,3] => 2 [2,2,3,1,4] => 2 [2,2,3,4,1] => 2 [2,2,4,1,3] => 2 [2,2,4,3,1] => 2 [2,3,1,2,4] => 4 [2,3,1,4,2] => 3 [2,3,2,1,4] => 1 [2,3,2,4,1] => 1 [2,3,4,1,2] => 3 [2,3,4,2,1] => 1 [2,4,1,2,3] => 3 [2,4,1,3,2] => 3 [2,4,2,1,3] => 2 [2,4,2,3,1] => 2 [2,4,3,1,2] => 3 [2,4,3,2,1] => 0 [3,1,2,2,4] => 5 [3,1,2,4,2] => 5 [3,1,4,2,2] => 4 [3,2,1,2,4] => 2 [3,2,1,4,2] => 2 [3,2,2,1,4] => 1 [3,2,2,4,1] => 1 [3,2,4,1,2] => 2 [3,2,4,2,1] => 1 [3,4,1,2,2] => 4 [3,4,2,1,2] => 2 [3,4,2,2,1] => 0 [4,1,2,2,3] => 5 [4,1,2,3,2] => 4 [4,1,3,2,2] => 4 [4,2,1,2,3] => 3 [4,2,1,3,2] => 3 [4,2,2,1,3] => 1 [4,2,2,3,1] => 1 [4,2,3,1,2] => 3 [4,2,3,2,1] => 0 [4,3,1,2,2] => 4 [4,3,2,1,2] => 1 [4,3,2,2,1] => 0 [1,2,2,3,5] => 7 [1,2,2,5,3] => 6 [1,2,3,2,5] => 5 [1,2,3,5,2] => 5 [1,2,5,2,3] => 6 [1,2,5,3,2] => 5 [1,3,2,2,5] => 5 [1,3,2,5,2] => 5 [1,3,5,2,2] => 5 [1,5,2,2,3] => 6 [1,5,2,3,2] => 4 [1,5,3,2,2] => 4 [2,1,2,3,5] => 4 [2,1,2,5,3] => 4 [2,1,3,2,5] => 4 [2,1,3,5,2] => 4 [2,1,5,2,3] => 3 [2,1,5,3,2] => 3 [2,2,1,3,5] => 3 [2,2,1,5,3] => 2 [2,2,3,1,5] => 3 [2,2,3,5,1] => 3 [2,2,5,1,3] => 2 [2,2,5,3,1] => 2 [2,3,1,2,5] => 4 [2,3,1,5,2] => 4 [2,3,2,1,5] => 1 [2,3,2,5,1] => 1 [2,3,5,1,2] => 4 [2,3,5,2,1] => 1 [2,5,1,2,3] => 3 [2,5,1,3,2] => 3 [2,5,2,1,3] => 2 [2,5,2,3,1] => 2 [2,5,3,1,2] => 3 [2,5,3,2,1] => 1 [3,1,2,2,5] => 5 [3,1,2,5,2] => 5 [3,1,5,2,2] => 5 [3,2,1,2,5] => 3 [3,2,1,5,2] => 2 [3,2,2,1,5] => 1 [3,2,2,5,1] => 1 [3,2,5,1,2] => 2 [3,2,5,2,1] => 1 [3,5,1,2,2] => 5 [3,5,2,1,2] => 2 [3,5,2,2,1] => 1 [5,1,2,2,3] => 6 [5,1,2,3,2] => 4 [5,1,3,2,2] => 4 [5,2,1,2,3] => 3 [5,2,1,3,2] => 3 [5,2,2,1,3] => 2 [5,2,2,3,1] => 2 [5,2,3,1,2] => 3 [5,2,3,2,1] => 0 [5,3,1,2,2] => 4 [5,3,2,1,2] => 2 [5,3,2,2,1] => 0 [1,2,2,4,4] => 6 [1,2,4,2,4] => 6 [1,2,4,4,2] => 6 [1,4,2,2,4] => 5 [1,4,2,4,2] => 5 [1,4,4,2,2] => 4 [2,1,2,4,4] => 5 [2,1,4,2,4] => 4 [2,1,4,4,2] => 3 [2,2,1,4,4] => 2 [2,2,4,1,4] => 2 [2,2,4,4,1] => 2 [2,4,1,2,4] => 4 [2,4,1,4,2] => 3 [2,4,2,1,4] => 2 [2,4,2,4,1] => 2 [2,4,4,1,2] => 3 [2,4,4,2,1] => 2 [4,1,2,2,4] => 5 [4,1,2,4,2] => 5 [4,1,4,2,2] => 4 [4,2,1,2,4] => 4 [4,2,1,4,2] => 3 [4,2,2,1,4] => 1 [4,2,2,4,1] => 1 [4,2,4,1,2] => 3 [4,2,4,2,1] => 1 [4,4,1,2,2] => 4 [4,4,2,1,2] => 3 [4,4,2,2,1] => 0 [1,2,2,4,5] => 7 [1,2,2,5,4] => 6 [1,2,4,2,5] => 7 [1,2,4,5,2] => 7 [1,2,5,2,4] => 6 [1,2,5,4,2] => 6 [1,4,2,2,5] => 5 [1,4,2,5,2] => 5 [1,4,5,2,2] => 5 [1,5,2,2,4] => 6 [1,5,2,4,2] => 6 [1,5,4,2,2] => 4 [2,1,2,4,5] => 6 [2,1,2,5,4] => 5 [2,1,4,2,5] => 4 [2,1,4,5,2] => 4 [2,1,5,2,4] => 5 [2,1,5,4,2] => 3 [2,2,1,4,5] => 3 [2,2,1,5,4] => 2 [2,2,4,1,5] => 3 [2,2,4,5,1] => 3 [2,2,5,1,4] => 2 [2,2,5,4,1] => 2 [2,4,1,2,5] => 4 [2,4,1,5,2] => 4 [2,4,2,1,5] => 3 [2,4,2,5,1] => 3 [2,4,5,1,2] => 4 [2,4,5,2,1] => 3 [2,5,1,2,4] => 5 [2,5,1,4,2] => 3 [2,5,2,1,4] => 2 [2,5,2,4,1] => 2 [2,5,4,1,2] => 3 [2,5,4,2,1] => 2 [4,1,2,2,5] => 5 [4,1,2,5,2] => 5 [4,1,5,2,2] => 5 [4,2,1,2,5] => 4 [4,2,1,5,2] => 4 [4,2,2,1,5] => 1 [4,2,2,5,1] => 1 [4,2,5,1,2] => 4 [4,2,5,2,1] => 1 [4,5,1,2,2] => 5 [4,5,2,1,2] => 4 [4,5,2,2,1] => 1 [5,1,2,2,4] => 6 [5,1,2,4,2] => 6 [5,1,4,2,2] => 4 [5,2,1,2,4] => 5 [5,2,1,4,2] => 3 [5,2,2,1,4] => 2 [5,2,2,4,1] => 2 [5,2,4,1,2] => 3 [5,2,4,2,1] => 2 [5,4,1,2,2] => 4 [5,4,2,1,2] => 3 [5,4,2,2,1] => 0 [1,2,3,3,3] => 7 [1,3,2,3,3] => 6 [1,3,3,2,3] => 5 [1,3,3,3,2] => 4 [2,1,3,3,3] => 3 [2,3,1,3,3] => 3 [2,3,3,1,3] => 3 [2,3,3,3,1] => 3 [3,1,2,3,3] => 6 [3,1,3,2,3] => 5 [3,1,3,3,2] => 4 [3,2,1,3,3] => 2 [3,2,3,1,3] => 2 [3,2,3,3,1] => 2 [3,3,1,2,3] => 5 [3,3,1,3,2] => 4 [3,3,2,1,3] => 1 [3,3,2,3,1] => 1 [3,3,3,1,2] => 4 [3,3,3,2,1] => 0 [1,2,3,3,4] => 8 [1,2,3,4,3] => 7 [1,2,4,3,3] => 7 [1,3,2,3,4] => 6 [1,3,2,4,3] => 6 [1,3,3,2,4] => 5 [1,3,3,4,2] => 5 [1,3,4,2,3] => 6 [1,3,4,3,2] => 4 [1,4,2,3,3] => 7 [1,4,3,2,3] => 5 [1,4,3,3,2] => 4 [2,1,3,3,4] => 4 [2,1,3,4,3] => 3 [2,1,4,3,3] => 3 [2,3,1,3,4] => 4 [2,3,1,4,3] => 3 [2,3,3,1,4] => 4 [2,3,3,4,1] => 4 [2,3,4,1,3] => 3 [2,3,4,3,1] => 3 [2,4,1,3,3] => 3 [2,4,3,1,3] => 3 [2,4,3,3,1] => 3 [3,1,2,3,4] => 6 [3,1,2,4,3] => 6 [3,1,3,2,4] => 5 [3,1,3,4,2] => 5 [3,1,4,2,3] => 6 [3,1,4,3,2] => 4 [3,2,1,3,4] => 2 [3,2,1,4,3] => 2 [3,2,3,1,4] => 2 [3,2,3,4,1] => 2 [3,2,4,1,3] => 2 [3,2,4,3,1] => 2 [3,3,1,2,4] => 5 [3,3,1,4,2] => 5 [3,3,2,1,4] => 1 [3,3,2,4,1] => 1 [3,3,4,1,2] => 5 [3,3,4,2,1] => 1 [3,4,1,2,3] => 6 [3,4,1,3,2] => 4 [3,4,2,1,3] => 2 [3,4,2,3,1] => 2 [3,4,3,1,2] => 4 [3,4,3,2,1] => 0 [4,1,2,3,3] => 7 [4,1,3,2,3] => 5 [4,1,3,3,2] => 4 [4,2,1,3,3] => 3 [4,2,3,1,3] => 3 [4,2,3,3,1] => 3 [4,3,1,2,3] => 5 [4,3,1,3,2] => 4 [4,3,2,1,3] => 1 [4,3,2,3,1] => 1 [4,3,3,1,2] => 4 ----------------------------------------------------------------------------- Created: Jun 14, 2018 at 11:31 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Jun 14, 2018 at 11:31 by Christian Stump