***************************************************************************** * 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: St001903 ----------------------------------------------------------------------------- Collection: Parking functions ----------------------------------------------------------------------------- Description: The number of fixed points of a parking function. If $(a_1,\dots,a_n)$ is a parking function, a fixed point is an index $i$ such that $a_i = i$. It can be shown [1] that the generating function for parking functions with respect to this statistic is $$ \frac{1}{(n+1)^2} \left((q+n)^{n+1} - (q-1)^{n+1}\right). $$ ----------------------------------------------------------------------------- References: [1] Richard Stanley and Mei Yin, 2023, private communication. ----------------------------------------------------------------------------- Code: def statistic(p): return sum(1 for i, e in enumerate(p, 1) if i == e) def generating_function(n): R. = ZZ[] return 1/(n+1)^2*((q+n)^(n+1) - (q-1)^(n+1)) ----------------------------------------------------------------------------- Statistic values: [1] => 1 [1,1] => 1 [1,2] => 2 [2,1] => 0 [1,1,1] => 1 [1,1,2] => 1 [1,2,1] => 2 [2,1,1] => 0 [1,1,3] => 2 [1,3,1] => 1 [3,1,1] => 0 [1,2,2] => 2 [2,1,2] => 0 [2,2,1] => 1 [1,2,3] => 3 [1,3,2] => 1 [2,1,3] => 1 [2,3,1] => 0 [3,1,2] => 0 [3,2,1] => 1 [1,1,1,1] => 1 [1,1,1,2] => 1 [1,1,2,1] => 1 [1,2,1,1] => 2 [2,1,1,1] => 0 [1,1,1,3] => 1 [1,1,3,1] => 2 [1,3,1,1] => 1 [3,1,1,1] => 0 [1,1,1,4] => 2 [1,1,4,1] => 1 [1,4,1,1] => 1 [4,1,1,1] => 0 [1,1,2,2] => 1 [1,2,1,2] => 2 [1,2,2,1] => 2 [2,1,1,2] => 0 [2,1,2,1] => 0 [2,2,1,1] => 1 [1,1,2,3] => 1 [1,1,3,2] => 2 [1,2,1,3] => 2 [1,2,3,1] => 3 [1,3,1,2] => 1 [1,3,2,1] => 1 [2,1,1,3] => 0 [2,1,3,1] => 1 [2,3,1,1] => 0 [3,1,1,2] => 0 [3,1,2,1] => 0 [3,2,1,1] => 1 [1,1,2,4] => 2 [1,1,4,2] => 1 [1,2,1,4] => 3 [1,2,4,1] => 2 [1,4,1,2] => 1 [1,4,2,1] => 1 [2,1,1,4] => 1 [2,1,4,1] => 0 [2,4,1,1] => 0 [4,1,1,2] => 0 [4,1,2,1] => 0 [4,2,1,1] => 1 [1,1,3,3] => 2 [1,3,1,3] => 1 [1,3,3,1] => 2 [3,1,1,3] => 0 [3,1,3,1] => 1 [3,3,1,1] => 0 [1,1,3,4] => 3 [1,1,4,3] => 1 [1,3,1,4] => 2 [1,3,4,1] => 1 [1,4,1,3] => 1 [1,4,3,1] => 2 [3,1,1,4] => 1 [3,1,4,1] => 0 [3,4,1,1] => 0 [4,1,1,3] => 0 [4,1,3,1] => 1 [4,3,1,1] => 0 [1,2,2,2] => 2 [2,1,2,2] => 0 [2,2,1,2] => 1 [2,2,2,1] => 1 [1,2,2,3] => 2 [1,2,3,2] => 3 [1,3,2,2] => 1 [2,1,2,3] => 0 [2,1,3,2] => 1 [2,2,1,3] => 1 [2,2,3,1] => 2 [2,3,1,2] => 0 [2,3,2,1] => 0 [3,1,2,2] => 0 [3,2,1,2] => 1 [3,2,2,1] => 1 [1,2,2,4] => 3 [1,2,4,2] => 2 [1,4,2,2] => 1 [2,1,2,4] => 1 [2,1,4,2] => 0 [2,2,1,4] => 2 [2,2,4,1] => 1 [2,4,1,2] => 0 [2,4,2,1] => 0 [4,1,2,2] => 0 [4,2,1,2] => 1 [4,2,2,1] => 1 [1,2,3,3] => 3 [1,3,2,3] => 1 [1,3,3,2] => 2 [2,1,3,3] => 1 [2,3,1,3] => 0 [2,3,3,1] => 1 [3,1,2,3] => 0 [3,1,3,2] => 1 [3,2,1,3] => 1 [3,2,3,1] => 2 [3,3,1,2] => 0 [3,3,2,1] => 0 [1,2,3,4] => 4 [1,2,4,3] => 2 [1,3,2,4] => 2 [1,3,4,2] => 1 [1,4,2,3] => 1 [1,4,3,2] => 2 [2,1,3,4] => 2 [2,1,4,3] => 0 [2,3,1,4] => 1 [2,3,4,1] => 0 [2,4,1,3] => 0 [2,4,3,1] => 1 [3,1,2,4] => 1 [3,1,4,2] => 0 [3,2,1,4] => 2 [3,2,4,1] => 1 [3,4,1,2] => 0 [3,4,2,1] => 0 [4,1,2,3] => 0 [4,1,3,2] => 1 [4,2,1,3] => 1 [4,2,3,1] => 2 [4,3,1,2] => 0 [4,3,2,1] => 0 [1,1,1,1,1] => 1 [1,1,1,1,2] => 1 [1,1,1,2,1] => 1 [1,1,2,1,1] => 1 [1,2,1,1,1] => 2 [2,1,1,1,1] => 0 [1,1,1,1,3] => 1 [1,1,1,3,1] => 1 [1,1,3,1,1] => 2 [1,3,1,1,1] => 1 [3,1,1,1,1] => 0 [1,1,1,1,4] => 1 [1,1,1,4,1] => 2 [1,1,4,1,1] => 1 [1,4,1,1,1] => 1 [4,1,1,1,1] => 0 [1,1,1,1,5] => 2 [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] => 1 [1,1,2,1,2] => 1 [1,1,2,2,1] => 1 [1,2,1,1,2] => 2 [1,2,1,2,1] => 2 [1,2,2,1,1] => 2 [2,1,1,1,2] => 0 [2,1,1,2,1] => 0 [2,1,2,1,1] => 0 [2,2,1,1,1] => 1 [1,1,1,2,3] => 1 [1,1,1,3,2] => 1 [1,1,2,1,3] => 1 [1,1,2,3,1] => 1 [1,1,3,1,2] => 2 [1,1,3,2,1] => 2 [1,2,1,1,3] => 2 [1,2,1,3,1] => 2 [1,2,3,1,1] => 3 [1,3,1,1,2] => 1 [1,3,1,2,1] => 1 [1,3,2,1,1] => 1 [2,1,1,1,3] => 0 [2,1,1,3,1] => 0 [2,1,3,1,1] => 1 [2,3,1,1,1] => 0 [3,1,1,1,2] => 0 [3,1,1,2,1] => 0 [3,1,2,1,1] => 0 [3,2,1,1,1] => 1 [1,1,1,2,4] => 1 [1,1,1,4,2] => 2 [1,1,2,1,4] => 1 [1,1,2,4,1] => 2 [1,1,4,1,2] => 1 [1,1,4,2,1] => 1 [1,2,1,1,4] => 2 [1,2,1,4,1] => 3 [1,2,4,1,1] => 2 [1,4,1,1,2] => 1 [1,4,1,2,1] => 1 [1,4,2,1,1] => 1 [2,1,1,1,4] => 0 [2,1,1,4,1] => 1 [2,1,4,1,1] => 0 [2,4,1,1,1] => 0 [4,1,1,1,2] => 0 [4,1,1,2,1] => 0 [4,1,2,1,1] => 0 [4,2,1,1,1] => 1 [1,1,1,2,5] => 2 [1,1,1,5,2] => 1 [1,1,2,1,5] => 2 [1,1,2,5,1] => 1 [1,1,5,1,2] => 1 [1,1,5,2,1] => 1 [1,2,1,1,5] => 3 [1,2,1,5,1] => 2 [1,2,5,1,1] => 2 [1,5,1,1,2] => 1 [1,5,1,2,1] => 1 [1,5,2,1,1] => 1 [2,1,1,1,5] => 1 [2,1,1,5,1] => 0 [2,1,5,1,1] => 0 [2,5,1,1,1] => 0 [5,1,1,1,2] => 0 [5,1,1,2,1] => 0 [5,1,2,1,1] => 0 [5,2,1,1,1] => 1 [1,1,1,3,3] => 1 [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] => 2 [3,1,1,1,3] => 0 [3,1,1,3,1] => 0 [3,1,3,1,1] => 1 [3,3,1,1,1] => 0 [1,1,1,3,4] => 1 [1,1,1,4,3] => 2 [1,1,3,1,4] => 2 [1,1,3,4,1] => 3 [1,1,4,1,3] => 1 [1,1,4,3,1] => 1 [1,3,1,1,4] => 1 [1,3,1,4,1] => 2 [1,3,4,1,1] => 1 [1,4,1,1,3] => 1 [1,4,1,3,1] => 1 [1,4,3,1,1] => 2 [3,1,1,1,4] => 0 [3,1,1,4,1] => 1 [3,1,4,1,1] => 0 [3,4,1,1,1] => 0 [4,1,1,1,3] => 0 [4,1,1,3,1] => 0 [4,1,3,1,1] => 1 [4,3,1,1,1] => 0 [1,1,1,3,5] => 2 [1,1,1,5,3] => 1 [1,1,3,1,5] => 3 [1,1,3,5,1] => 2 [1,1,5,1,3] => 1 [1,1,5,3,1] => 1 [1,3,1,1,5] => 2 [1,3,1,5,1] => 1 [1,3,5,1,1] => 1 [1,5,1,1,3] => 1 [1,5,1,3,1] => 1 [1,5,3,1,1] => 2 [3,1,1,1,5] => 1 [3,1,1,5,1] => 0 [3,1,5,1,1] => 0 [3,5,1,1,1] => 0 [5,1,1,1,3] => 0 [5,1,1,3,1] => 0 [5,1,3,1,1] => 1 [5,3,1,1,1] => 0 [1,1,1,4,4] => 2 [1,1,4,1,4] => 1 [1,1,4,4,1] => 2 [1,4,1,1,4] => 1 [1,4,1,4,1] => 2 [1,4,4,1,1] => 1 [4,1,1,1,4] => 0 [4,1,1,4,1] => 1 [4,1,4,1,1] => 0 [4,4,1,1,1] => 0 [1,1,1,4,5] => 3 [1,1,1,5,4] => 1 [1,1,4,1,5] => 2 [1,1,4,5,1] => 1 [1,1,5,1,4] => 1 [1,1,5,4,1] => 2 [1,4,1,1,5] => 2 [1,4,1,5,1] => 1 [1,4,5,1,1] => 1 [1,5,1,1,4] => 1 [1,5,1,4,1] => 2 [1,5,4,1,1] => 1 [4,1,1,1,5] => 1 [4,1,1,5,1] => 0 [4,1,5,1,1] => 0 [4,5,1,1,1] => 0 [5,1,1,1,4] => 0 [5,1,1,4,1] => 1 [5,1,4,1,1] => 0 [5,4,1,1,1] => 0 [1,1,2,2,2] => 1 [1,2,1,2,2] => 2 [1,2,2,1,2] => 2 [1,2,2,2,1] => 2 [2,1,1,2,2] => 0 [2,1,2,1,2] => 0 [2,1,2,2,1] => 0 [2,2,1,1,2] => 1 [2,2,1,2,1] => 1 [2,2,2,1,1] => 1 [1,1,2,2,3] => 1 [1,1,2,3,2] => 1 [1,1,3,2,2] => 2 [1,2,1,2,3] => 2 [1,2,1,3,2] => 2 [1,2,2,1,3] => 2 [1,2,2,3,1] => 2 [1,2,3,1,2] => 3 [1,2,3,2,1] => 3 [1,3,1,2,2] => 1 [1,3,2,1,2] => 1 [1,3,2,2,1] => 1 [2,1,1,2,3] => 0 [2,1,1,3,2] => 0 [2,1,2,1,3] => 0 [2,1,2,3,1] => 0 [2,1,3,1,2] => 1 [2,1,3,2,1] => 1 [2,2,1,1,3] => 1 [2,2,1,3,1] => 1 [2,2,3,1,1] => 2 [2,3,1,1,2] => 0 [2,3,1,2,1] => 0 [2,3,2,1,1] => 0 [3,1,1,2,2] => 0 [3,1,2,1,2] => 0 [3,1,2,2,1] => 0 [3,2,1,1,2] => 1 [3,2,1,2,1] => 1 [3,2,2,1,1] => 1 [1,1,2,2,4] => 1 [1,1,2,4,2] => 2 [1,1,4,2,2] => 1 [1,2,1,2,4] => 2 [1,2,1,4,2] => 3 [1,2,2,1,4] => 2 [1,2,2,4,1] => 3 [1,2,4,1,2] => 2 [1,2,4,2,1] => 2 [1,4,1,2,2] => 1 [1,4,2,1,2] => 1 [1,4,2,2,1] => 1 [2,1,1,2,4] => 0 [2,1,1,4,2] => 1 [2,1,2,1,4] => 0 [2,1,2,4,1] => 1 [2,1,4,1,2] => 0 [2,1,4,2,1] => 0 [2,2,1,1,4] => 1 [2,2,1,4,1] => 2 [2,2,4,1,1] => 1 [2,4,1,1,2] => 0 [2,4,1,2,1] => 0 [2,4,2,1,1] => 0 [4,1,1,2,2] => 0 [4,1,2,1,2] => 0 [4,1,2,2,1] => 0 [4,2,1,1,2] => 1 [4,2,1,2,1] => 1 [4,2,2,1,1] => 1 [1,1,2,2,5] => 2 [1,1,2,5,2] => 1 [1,1,5,2,2] => 1 [1,2,1,2,5] => 3 [1,2,1,5,2] => 2 [1,2,2,1,5] => 3 [1,2,2,5,1] => 2 [1,2,5,1,2] => 2 [1,2,5,2,1] => 2 [1,5,1,2,2] => 1 [1,5,2,1,2] => 1 [1,5,2,2,1] => 1 [2,1,1,2,5] => 1 [2,1,1,5,2] => 0 [2,1,2,1,5] => 1 [2,1,2,5,1] => 0 [2,1,5,1,2] => 0 [2,1,5,2,1] => 0 [2,2,1,1,5] => 2 [2,2,1,5,1] => 1 [2,2,5,1,1] => 1 [2,5,1,1,2] => 0 [2,5,1,2,1] => 0 [2,5,2,1,1] => 0 [5,1,1,2,2] => 0 [5,1,2,1,2] => 0 [5,1,2,2,1] => 0 [5,2,1,1,2] => 1 [5,2,1,2,1] => 1 [5,2,2,1,1] => 1 [1,1,2,3,3] => 1 [1,1,3,2,3] => 2 [1,1,3,3,2] => 2 [1,2,1,3,3] => 2 [1,2,3,1,3] => 3 [1,2,3,3,1] => 3 [1,3,1,2,3] => 1 [1,3,1,3,2] => 1 [1,3,2,1,3] => 1 [1,3,2,3,1] => 1 [1,3,3,1,2] => 2 [1,3,3,2,1] => 2 [2,1,1,3,3] => 0 [2,1,3,1,3] => 1 [2,1,3,3,1] => 1 [2,3,1,1,3] => 0 [2,3,1,3,1] => 0 [2,3,3,1,1] => 1 [3,1,1,2,3] => 0 [3,1,1,3,2] => 0 [3,1,2,1,3] => 0 [3,1,2,3,1] => 0 [3,1,3,1,2] => 1 [3,1,3,2,1] => 1 [3,2,1,1,3] => 1 [3,2,1,3,1] => 1 [3,2,3,1,1] => 2 [3,3,1,1,2] => 0 [3,3,1,2,1] => 0 [3,3,2,1,1] => 0 [1,1,2,3,4] => 1 [1,1,2,4,3] => 2 [1,1,3,2,4] => 2 [1,1,3,4,2] => 3 [1,1,4,2,3] => 1 [1,1,4,3,2] => 1 [1,2,1,3,4] => 2 [1,2,1,4,3] => 3 [1,2,3,1,4] => 3 [1,2,3,4,1] => 4 [1,2,4,1,3] => 2 [1,2,4,3,1] => 2 [1,3,1,2,4] => 1 [1,3,1,4,2] => 2 [1,3,2,1,4] => 1 [1,3,2,4,1] => 2 [1,3,4,1,2] => 1 [1,3,4,2,1] => 1 [1,4,1,2,3] => 1 [1,4,1,3,2] => 1 [1,4,2,1,3] => 1 [1,4,2,3,1] => 1 [1,4,3,1,2] => 2 [1,4,3,2,1] => 2 [2,1,1,3,4] => 0 [2,1,1,4,3] => 1 [2,1,3,1,4] => 1 [2,1,3,4,1] => 2 [2,1,4,1,3] => 0 [2,1,4,3,1] => 0 [2,3,1,1,4] => 0 [2,3,1,4,1] => 1 [2,3,4,1,1] => 0 [2,4,1,1,3] => 0 [2,4,1,3,1] => 0 [2,4,3,1,1] => 1 [3,1,1,2,4] => 0 [3,1,1,4,2] => 1 [3,1,2,1,4] => 0 [3,1,2,4,1] => 1 [3,1,4,1,2] => 0 [3,1,4,2,1] => 0 [3,2,1,1,4] => 1 [3,2,1,4,1] => 2 [3,2,4,1,1] => 1 [3,4,1,1,2] => 0 [3,4,1,2,1] => 0 [3,4,2,1,1] => 0 [4,1,1,2,3] => 0 [4,1,1,3,2] => 0 [4,1,2,1,3] => 0 [4,1,2,3,1] => 0 [4,1,3,1,2] => 1 [4,1,3,2,1] => 1 [4,2,1,1,3] => 1 [4,2,1,3,1] => 1 [4,2,3,1,1] => 2 [4,3,1,1,2] => 0 [4,3,1,2,1] => 0 [4,3,2,1,1] => 0 [1,1,2,3,5] => 2 [1,1,2,5,3] => 1 [1,1,3,2,5] => 3 [1,1,3,5,2] => 2 [1,1,5,2,3] => 1 [1,1,5,3,2] => 1 [1,2,1,3,5] => 3 [1,2,1,5,3] => 2 [1,2,3,1,5] => 4 [1,2,3,5,1] => 3 [1,2,5,1,3] => 2 [1,2,5,3,1] => 2 [1,3,1,2,5] => 2 [1,3,1,5,2] => 1 [1,3,2,1,5] => 2 [1,3,2,5,1] => 1 [1,3,5,1,2] => 1 [1,3,5,2,1] => 1 [1,5,1,2,3] => 1 [1,5,1,3,2] => 1 [1,5,2,1,3] => 1 [1,5,2,3,1] => 1 [1,5,3,1,2] => 2 [1,5,3,2,1] => 2 [2,1,1,3,5] => 1 [2,1,1,5,3] => 0 [2,1,3,1,5] => 2 [2,1,3,5,1] => 1 [2,1,5,1,3] => 0 [2,1,5,3,1] => 0 [2,3,1,1,5] => 1 [2,3,1,5,1] => 0 [2,3,5,1,1] => 0 [2,5,1,1,3] => 0 [2,5,1,3,1] => 0 [2,5,3,1,1] => 1 [3,1,1,2,5] => 1 [3,1,1,5,2] => 0 [3,1,2,1,5] => 1 [3,1,2,5,1] => 0 [3,1,5,1,2] => 0 [3,1,5,2,1] => 0 [3,2,1,1,5] => 2 [3,2,1,5,1] => 1 [3,2,5,1,1] => 1 [3,5,1,1,2] => 0 [3,5,1,2,1] => 0 [3,5,2,1,1] => 0 [5,1,1,2,3] => 0 [5,1,1,3,2] => 0 [5,1,2,1,3] => 0 [5,1,2,3,1] => 0 [5,1,3,1,2] => 1 [5,1,3,2,1] => 1 [5,2,1,1,3] => 1 [5,2,1,3,1] => 1 [5,2,3,1,1] => 2 [5,3,1,1,2] => 0 [5,3,1,2,1] => 0 [5,3,2,1,1] => 0 [1,1,2,4,4] => 2 [1,1,4,2,4] => 1 [1,1,4,4,2] => 2 [1,2,1,4,4] => 3 [1,2,4,1,4] => 2 [1,2,4,4,1] => 3 [1,4,1,2,4] => 1 [1,4,1,4,2] => 2 [1,4,2,1,4] => 1 [1,4,2,4,1] => 2 [1,4,4,1,2] => 1 [1,4,4,2,1] => 1 [2,1,1,4,4] => 1 [2,1,4,1,4] => 0 [2,1,4,4,1] => 1 [2,4,1,1,4] => 0 [2,4,1,4,1] => 1 [2,4,4,1,1] => 0 [4,1,1,2,4] => 0 [4,1,1,4,2] => 1 [4,1,2,1,4] => 0 [4,1,2,4,1] => 1 [4,1,4,1,2] => 0 [4,1,4,2,1] => 0 [4,2,1,1,4] => 1 [4,2,1,4,1] => 2 [4,2,4,1,1] => 1 [4,4,1,1,2] => 0 [4,4,1,2,1] => 0 [4,4,2,1,1] => 0 [1,1,2,4,5] => 3 [1,1,2,5,4] => 1 [1,1,4,2,5] => 2 [1,1,4,5,2] => 1 [1,1,5,2,4] => 1 [1,1,5,4,2] => 2 [1,2,1,4,5] => 4 [1,2,1,5,4] => 2 [1,2,4,1,5] => 3 [1,2,4,5,1] => 2 [1,2,5,1,4] => 2 [1,2,5,4,1] => 3 [1,4,1,2,5] => 2 [1,4,1,5,2] => 1 [1,4,2,1,5] => 2 [1,4,2,5,1] => 1 [1,4,5,1,2] => 1 [1,4,5,2,1] => 1 [1,5,1,2,4] => 1 [1,5,1,4,2] => 2 [1,5,2,1,4] => 1 [1,5,2,4,1] => 2 [1,5,4,1,2] => 1 [1,5,4,2,1] => 1 [2,1,1,4,5] => 2 [2,1,1,5,4] => 0 [2,1,4,1,5] => 1 [2,1,4,5,1] => 0 [2,1,5,1,4] => 0 [2,1,5,4,1] => 1 [2,4,1,1,5] => 1 [2,4,1,5,1] => 0 [2,4,5,1,1] => 0 [2,5,1,1,4] => 0 [2,5,1,4,1] => 1 [2,5,4,1,1] => 0 [4,1,1,2,5] => 1 [4,1,1,5,2] => 0 [4,1,2,1,5] => 1 [4,1,2,5,1] => 0 [4,1,5,1,2] => 0 [4,1,5,2,1] => 0 [4,2,1,1,5] => 2 [4,2,1,5,1] => 1 [4,2,5,1,1] => 1 [4,5,1,1,2] => 0 [4,5,1,2,1] => 0 [4,5,2,1,1] => 0 [5,1,1,2,4] => 0 [5,1,1,4,2] => 1 [5,1,2,1,4] => 0 [5,1,2,4,1] => 1 [5,1,4,1,2] => 0 [5,1,4,2,1] => 0 [5,2,1,1,4] => 1 [5,2,1,4,1] => 2 [5,2,4,1,1] => 1 [5,4,1,1,2] => 0 [5,4,1,2,1] => 0 [5,4,2,1,1] => 0 [1,1,3,3,3] => 2 [1,3,1,3,3] => 1 [1,3,3,1,3] => 2 [1,3,3,3,1] => 2 [3,1,1,3,3] => 0 [3,1,3,1,3] => 1 [3,1,3,3,1] => 1 [3,3,1,1,3] => 0 [3,3,1,3,1] => 0 [3,3,3,1,1] => 1 [1,1,3,3,4] => 2 [1,1,3,4,3] => 3 [1,1,4,3,3] => 1 [1,3,1,3,4] => 1 [1,3,1,4,3] => 2 [1,3,3,1,4] => 2 [1,3,3,4,1] => 3 [1,3,4,1,3] => 1 [1,3,4,3,1] => 1 [1,4,1,3,3] => 1 [1,4,3,1,3] => 2 [1,4,3,3,1] => 2 [3,1,1,3,4] => 0 [3,1,1,4,3] => 1 [3,1,3,1,4] => 1 [3,1,3,4,1] => 2 [3,1,4,1,3] => 0 [3,1,4,3,1] => 0 [3,3,1,1,4] => 0 [3,3,1,4,1] => 1 [3,3,4,1,1] => 0 [3,4,1,1,3] => 0 [3,4,1,3,1] => 0 [3,4,3,1,1] => 1 [4,1,1,3,3] => 0 [4,1,3,1,3] => 1 [4,1,3,3,1] => 1 [4,3,1,1,3] => 0 [4,3,1,3,1] => 0 [4,3,3,1,1] => 1 [1,1,3,3,5] => 3 [1,1,3,5,3] => 2 [1,1,5,3,3] => 1 [1,3,1,3,5] => 2 [1,3,1,5,3] => 1 [1,3,3,1,5] => 3 [1,3,3,5,1] => 2 [1,3,5,1,3] => 1 [1,3,5,3,1] => 1 [1,5,1,3,3] => 1 [1,5,3,1,3] => 2 [1,5,3,3,1] => 2 [3,1,1,3,5] => 1 [3,1,1,5,3] => 0 [3,1,3,1,5] => 2 [3,1,3,5,1] => 1 [3,1,5,1,3] => 0 [3,1,5,3,1] => 0 [3,3,1,1,5] => 1 [3,3,1,5,1] => 0 [3,3,5,1,1] => 0 [3,5,1,1,3] => 0 [3,5,1,3,1] => 0 [3,5,3,1,1] => 1 [5,1,1,3,3] => 0 [5,1,3,1,3] => 1 [5,1,3,3,1] => 1 [5,3,1,1,3] => 0 [5,3,1,3,1] => 0 [5,3,3,1,1] => 1 [1,1,3,4,4] => 3 [1,1,4,3,4] => 1 [1,1,4,4,3] => 2 [1,3,1,4,4] => 2 [1,3,4,1,4] => 1 [1,3,4,4,1] => 2 [1,4,1,3,4] => 1 [1,4,1,4,3] => 2 [1,4,3,1,4] => 2 [1,4,3,4,1] => 3 [1,4,4,1,3] => 1 [1,4,4,3,1] => 1 [3,1,1,4,4] => 1 [3,1,4,1,4] => 0 [3,1,4,4,1] => 1 [3,4,1,1,4] => 0 [3,4,1,4,1] => 1 [3,4,4,1,1] => 0 [4,1,1,3,4] => 0 [4,1,1,4,3] => 1 [4,1,3,1,4] => 1 [4,1,3,4,1] => 2 [4,1,4,1,3] => 0 [4,1,4,3,1] => 0 [4,3,1,1,4] => 0 [4,3,1,4,1] => 1 [4,3,4,1,1] => 0 [4,4,1,1,3] => 0 [4,4,1,3,1] => 0 [4,4,3,1,1] => 1 [1,1,3,4,5] => 4 [1,1,3,5,4] => 2 [1,1,4,3,5] => 2 [1,1,4,5,3] => 1 [1,1,5,3,4] => 1 [1,1,5,4,3] => 2 [1,3,1,4,5] => 3 [1,3,1,5,4] => 1 [1,3,4,1,5] => 2 [1,3,4,5,1] => 1 [1,3,5,1,4] => 1 [1,3,5,4,1] => 2 [1,4,1,3,5] => 2 [1,4,1,5,3] => 1 [1,4,3,1,5] => 3 [1,4,3,5,1] => 2 [1,4,5,1,3] => 1 [1,4,5,3,1] => 1 [1,5,1,3,4] => 1 [1,5,1,4,3] => 2 [1,5,3,1,4] => 2 [1,5,3,4,1] => 3 [1,5,4,1,3] => 1 [1,5,4,3,1] => 1 [3,1,1,4,5] => 2 [3,1,1,5,4] => 0 [3,1,4,1,5] => 1 [3,1,4,5,1] => 0 [3,1,5,1,4] => 0 [3,1,5,4,1] => 1 [3,4,1,1,5] => 1 [3,4,1,5,1] => 0 [3,4,5,1,1] => 0 [3,5,1,1,4] => 0 [3,5,1,4,1] => 1 [3,5,4,1,1] => 0 [4,1,1,3,5] => 1 [4,1,1,5,3] => 0 [4,1,3,1,5] => 2 [4,1,3,5,1] => 1 [4,1,5,1,3] => 0 [4,1,5,3,1] => 0 [4,3,1,1,5] => 1 [4,3,1,5,1] => 0 [4,3,5,1,1] => 0 [4,5,1,1,3] => 0 [4,5,1,3,1] => 0 [4,5,3,1,1] => 1 [5,1,1,3,4] => 0 [5,1,1,4,3] => 1 [5,1,3,1,4] => 1 [5,1,3,4,1] => 2 [5,1,4,1,3] => 0 [5,1,4,3,1] => 0 [5,3,1,1,4] => 0 [5,3,1,4,1] => 1 [5,3,4,1,1] => 0 [5,4,1,1,3] => 0 [5,4,1,3,1] => 0 [5,4,3,1,1] => 1 [1,2,2,2,2] => 2 [2,1,2,2,2] => 0 [2,2,1,2,2] => 1 [2,2,2,1,2] => 1 [2,2,2,2,1] => 1 [1,2,2,2,3] => 2 [1,2,2,3,2] => 2 [1,2,3,2,2] => 3 [1,3,2,2,2] => 1 [2,1,2,2,3] => 0 [2,1,2,3,2] => 0 [2,1,3,2,2] => 1 [2,2,1,2,3] => 1 [2,2,1,3,2] => 1 [2,2,2,1,3] => 1 [2,2,2,3,1] => 1 [2,2,3,1,2] => 2 [2,2,3,2,1] => 2 [2,3,1,2,2] => 0 [2,3,2,1,2] => 0 [2,3,2,2,1] => 0 [3,1,2,2,2] => 0 [3,2,1,2,2] => 1 [3,2,2,1,2] => 1 [3,2,2,2,1] => 1 [1,2,2,2,4] => 2 [1,2,2,4,2] => 3 [1,2,4,2,2] => 2 [1,4,2,2,2] => 1 [2,1,2,2,4] => 0 [2,1,2,4,2] => 1 [2,1,4,2,2] => 0 [2,2,1,2,4] => 1 [2,2,1,4,2] => 2 [2,2,2,1,4] => 1 [2,2,2,4,1] => 2 [2,2,4,1,2] => 1 [2,2,4,2,1] => 1 [2,4,1,2,2] => 0 [2,4,2,1,2] => 0 [2,4,2,2,1] => 0 [4,1,2,2,2] => 0 [4,2,1,2,2] => 1 [4,2,2,1,2] => 1 [4,2,2,2,1] => 1 [1,2,2,2,5] => 3 [1,2,2,5,2] => 2 [1,2,5,2,2] => 2 [1,5,2,2,2] => 1 [2,1,2,2,5] => 1 [2,1,2,5,2] => 0 [2,1,5,2,2] => 0 [2,2,1,2,5] => 2 [2,2,1,5,2] => 1 [2,2,2,1,5] => 2 [2,2,2,5,1] => 1 [2,2,5,1,2] => 1 [2,2,5,2,1] => 1 [2,5,1,2,2] => 0 [2,5,2,1,2] => 0 [2,5,2,2,1] => 0 [5,1,2,2,2] => 0 [5,2,1,2,2] => 1 [5,2,2,1,2] => 1 [5,2,2,2,1] => 1 [1,2,2,3,3] => 2 [1,2,3,2,3] => 3 [1,2,3,3,2] => 3 [1,3,2,2,3] => 1 [1,3,2,3,2] => 1 [1,3,3,2,2] => 2 [2,1,2,3,3] => 0 [2,1,3,2,3] => 1 [2,1,3,3,2] => 1 [2,2,1,3,3] => 1 [2,2,3,1,3] => 2 [2,2,3,3,1] => 2 [2,3,1,2,3] => 0 [2,3,1,3,2] => 0 [2,3,2,1,3] => 0 [2,3,2,3,1] => 0 [2,3,3,1,2] => 1 [2,3,3,2,1] => 1 [3,1,2,2,3] => 0 [3,1,2,3,2] => 0 [3,1,3,2,2] => 1 [3,2,1,2,3] => 1 [3,2,1,3,2] => 1 [3,2,2,1,3] => 1 [3,2,2,3,1] => 1 [3,2,3,1,2] => 2 [3,2,3,2,1] => 2 [3,3,1,2,2] => 0 [3,3,2,1,2] => 0 [3,3,2,2,1] => 0 [1,2,2,3,4] => 2 [1,2,2,4,3] => 3 [1,2,3,2,4] => 3 [1,2,3,4,2] => 4 [1,2,4,2,3] => 2 [1,2,4,3,2] => 2 [1,3,2,2,4] => 1 [1,3,2,4,2] => 2 [1,3,4,2,2] => 1 [1,4,2,2,3] => 1 [1,4,2,3,2] => 1 [1,4,3,2,2] => 2 [2,1,2,3,4] => 0 [2,1,2,4,3] => 1 [2,1,3,2,4] => 1 [2,1,3,4,2] => 2 [2,1,4,2,3] => 0 [2,1,4,3,2] => 0 [2,2,1,3,4] => 1 [2,2,1,4,3] => 2 [2,2,3,1,4] => 2 [2,2,3,4,1] => 3 [2,2,4,1,3] => 1 [2,2,4,3,1] => 1 [2,3,1,2,4] => 0 [2,3,1,4,2] => 1 [2,3,2,1,4] => 0 [2,3,2,4,1] => 1 [2,3,4,1,2] => 0 [2,3,4,2,1] => 0 [2,4,1,2,3] => 0 [2,4,1,3,2] => 0 [2,4,2,1,3] => 0 [2,4,2,3,1] => 0 [2,4,3,1,2] => 1 [2,4,3,2,1] => 1 [3,1,2,2,4] => 0 [3,1,2,4,2] => 1 [3,1,4,2,2] => 0 [3,2,1,2,4] => 1 [3,2,1,4,2] => 2 [3,2,2,1,4] => 1 [3,2,2,4,1] => 2 [3,2,4,1,2] => 1 [3,2,4,2,1] => 1 [3,4,1,2,2] => 0 [3,4,2,1,2] => 0 [3,4,2,2,1] => 0 [4,1,2,2,3] => 0 [4,1,2,3,2] => 0 [4,1,3,2,2] => 1 [4,2,1,2,3] => 1 [4,2,1,3,2] => 1 [4,2,2,1,3] => 1 [4,2,2,3,1] => 1 [4,2,3,1,2] => 2 [4,2,3,2,1] => 2 [4,3,1,2,2] => 0 [4,3,2,1,2] => 0 [4,3,2,2,1] => 0 [1,2,2,3,5] => 3 [1,2,2,5,3] => 2 [1,2,3,2,5] => 4 [1,2,3,5,2] => 3 [1,2,5,2,3] => 2 [1,2,5,3,2] => 2 [1,3,2,2,5] => 2 [1,3,2,5,2] => 1 [1,3,5,2,2] => 1 [1,5,2,2,3] => 1 [1,5,2,3,2] => 1 [1,5,3,2,2] => 2 [2,1,2,3,5] => 1 [2,1,2,5,3] => 0 [2,1,3,2,5] => 2 [2,1,3,5,2] => 1 [2,1,5,2,3] => 0 [2,1,5,3,2] => 0 [2,2,1,3,5] => 2 [2,2,1,5,3] => 1 [2,2,3,1,5] => 3 [2,2,3,5,1] => 2 [2,2,5,1,3] => 1 [2,2,5,3,1] => 1 [2,3,1,2,5] => 1 [2,3,1,5,2] => 0 [2,3,2,1,5] => 1 [2,3,2,5,1] => 0 [2,3,5,1,2] => 0 [2,3,5,2,1] => 0 [2,5,1,2,3] => 0 [2,5,1,3,2] => 0 [2,5,2,1,3] => 0 [2,5,2,3,1] => 0 [2,5,3,1,2] => 1 [2,5,3,2,1] => 1 [3,1,2,2,5] => 1 [3,1,2,5,2] => 0 [3,1,5,2,2] => 0 [3,2,1,2,5] => 2 [3,2,1,5,2] => 1 [3,2,2,1,5] => 2 [3,2,2,5,1] => 1 [3,2,5,1,2] => 1 [3,2,5,2,1] => 1 [3,5,1,2,2] => 0 [3,5,2,1,2] => 0 [3,5,2,2,1] => 0 [5,1,2,2,3] => 0 [5,1,2,3,2] => 0 [5,1,3,2,2] => 1 [5,2,1,2,3] => 1 [5,2,1,3,2] => 1 [5,2,2,1,3] => 1 [5,2,2,3,1] => 1 [5,2,3,1,2] => 2 [5,2,3,2,1] => 2 [5,3,1,2,2] => 0 [5,3,2,1,2] => 0 [5,3,2,2,1] => 0 [1,2,2,4,4] => 3 [1,2,4,2,4] => 2 [1,2,4,4,2] => 3 [1,4,2,2,4] => 1 [1,4,2,4,2] => 2 [1,4,4,2,2] => 1 [2,1,2,4,4] => 1 [2,1,4,2,4] => 0 [2,1,4,4,2] => 1 [2,2,1,4,4] => 2 [2,2,4,1,4] => 1 [2,2,4,4,1] => 2 [2,4,1,2,4] => 0 [2,4,1,4,2] => 1 [2,4,2,1,4] => 0 [2,4,2,4,1] => 1 [2,4,4,1,2] => 0 [2,4,4,2,1] => 0 [4,1,2,2,4] => 0 [4,1,2,4,2] => 1 [4,1,4,2,2] => 0 [4,2,1,2,4] => 1 [4,2,1,4,2] => 2 [4,2,2,1,4] => 1 [4,2,2,4,1] => 2 [4,2,4,1,2] => 1 [4,2,4,2,1] => 1 [4,4,1,2,2] => 0 [4,4,2,1,2] => 0 [4,4,2,2,1] => 0 [1,2,2,4,5] => 4 [1,2,2,5,4] => 2 [1,2,4,2,5] => 3 [1,2,4,5,2] => 2 [1,2,5,2,4] => 2 [1,2,5,4,2] => 3 [1,4,2,2,5] => 2 [1,4,2,5,2] => 1 [1,4,5,2,2] => 1 [1,5,2,2,4] => 1 [1,5,2,4,2] => 2 [1,5,4,2,2] => 1 [2,1,2,4,5] => 2 [2,1,2,5,4] => 0 [2,1,4,2,5] => 1 [2,1,4,5,2] => 0 [2,1,5,2,4] => 0 [2,1,5,4,2] => 1 [2,2,1,4,5] => 3 [2,2,1,5,4] => 1 [2,2,4,1,5] => 2 [2,2,4,5,1] => 1 [2,2,5,1,4] => 1 [2,2,5,4,1] => 2 [2,4,1,2,5] => 1 [2,4,1,5,2] => 0 [2,4,2,1,5] => 1 [2,4,2,5,1] => 0 [2,4,5,1,2] => 0 [2,4,5,2,1] => 0 [2,5,1,2,4] => 0 [2,5,1,4,2] => 1 [2,5,2,1,4] => 0 [2,5,2,4,1] => 1 [2,5,4,1,2] => 0 [2,5,4,2,1] => 0 [4,1,2,2,5] => 1 [4,1,2,5,2] => 0 [4,1,5,2,2] => 0 [4,2,1,2,5] => 2 [4,2,1,5,2] => 1 [4,2,2,1,5] => 2 [4,2,2,5,1] => 1 [4,2,5,1,2] => 1 [4,2,5,2,1] => 1 [4,5,1,2,2] => 0 [4,5,2,1,2] => 0 [4,5,2,2,1] => 0 [5,1,2,2,4] => 0 [5,1,2,4,2] => 1 [5,1,4,2,2] => 0 [5,2,1,2,4] => 1 [5,2,1,4,2] => 2 [5,2,2,1,4] => 1 [5,2,2,4,1] => 2 [5,2,4,1,2] => 1 [5,2,4,2,1] => 1 [5,4,1,2,2] => 0 [5,4,2,1,2] => 0 [5,4,2,2,1] => 0 [1,2,3,3,3] => 3 [1,3,2,3,3] => 1 [1,3,3,2,3] => 2 [1,3,3,3,2] => 2 [2,1,3,3,3] => 1 [2,3,1,3,3] => 0 [2,3,3,1,3] => 1 [2,3,3,3,1] => 1 [3,1,2,3,3] => 0 [3,1,3,2,3] => 1 [3,1,3,3,2] => 1 [3,2,1,3,3] => 1 [3,2,3,1,3] => 2 [3,2,3,3,1] => 2 [3,3,1,2,3] => 0 [3,3,1,3,2] => 0 [3,3,2,1,3] => 0 [3,3,2,3,1] => 0 [3,3,3,1,2] => 1 [3,3,3,2,1] => 1 [1,2,3,3,4] => 3 [1,2,3,4,3] => 4 [1,2,4,3,3] => 2 [1,3,2,3,4] => 1 [1,3,2,4,3] => 2 [1,3,3,2,4] => 2 [1,3,3,4,2] => 3 [1,3,4,2,3] => 1 [1,3,4,3,2] => 1 [1,4,2,3,3] => 1 [1,4,3,2,3] => 2 [1,4,3,3,2] => 2 [2,1,3,3,4] => 1 [2,1,3,4,3] => 2 [2,1,4,3,3] => 0 [2,3,1,3,4] => 0 [2,3,1,4,3] => 1 [2,3,3,1,4] => 1 [2,3,3,4,1] => 2 [2,3,4,1,3] => 0 [2,3,4,3,1] => 0 [2,4,1,3,3] => 0 [2,4,3,1,3] => 1 [2,4,3,3,1] => 1 [3,1,2,3,4] => 0 [3,1,2,4,3] => 1 [3,1,3,2,4] => 1 [3,1,3,4,2] => 2 [3,1,4,2,3] => 0 [3,1,4,3,2] => 0 [3,2,1,3,4] => 1 [3,2,1,4,3] => 2 [3,2,3,1,4] => 2 [3,2,3,4,1] => 3 [3,2,4,1,3] => 1 [3,2,4,3,1] => 1 [3,3,1,2,4] => 0 [3,3,1,4,2] => 1 [3,3,2,1,4] => 0 [3,3,2,4,1] => 1 [3,3,4,1,2] => 0 [3,3,4,2,1] => 0 [3,4,1,2,3] => 0 [3,4,1,3,2] => 0 [3,4,2,1,3] => 0 [3,4,2,3,1] => 0 [3,4,3,1,2] => 1 [3,4,3,2,1] => 1 [4,1,2,3,3] => 0 [4,1,3,2,3] => 1 [4,1,3,3,2] => 1 [4,2,1,3,3] => 1 [4,2,3,1,3] => 2 [4,2,3,3,1] => 2 [4,3,1,2,3] => 0 [4,3,1,3,2] => 0 [4,3,2,1,3] => 0 [4,3,2,3,1] => 0 [4,3,3,1,2] => 1 ----------------------------------------------------------------------------- Created: Jun 08, 2023 at 09:00 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Jun 15, 2023 at 14:41 by Martin Rubey