***************************************************************************** * 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: St001568 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The smallest positive integer that does not appear twice in the partition. ----------------------------------------------------------------------------- References: [1] Ballantine, C., Merca, M. Combinatorial Proof of the Minimal Excludant Theorem [[arXiv:1908.06789]] ----------------------------------------------------------------------------- Code: def statistic(x): x = list(x) for i in range(1, x[0]+2): if x.count(i) < 2: return i ----------------------------------------------------------------------------- Statistic values: [2] => 1 [1,1] => 2 [3] => 1 [2,1] => 1 [1,1,1] => 2 [4] => 1 [3,1] => 1 [2,2] => 1 [2,1,1] => 2 [1,1,1,1] => 2 [5] => 1 [4,1] => 1 [3,2] => 1 [3,1,1] => 2 [2,2,1] => 1 [2,1,1,1] => 2 [1,1,1,1,1] => 2 [6] => 1 [5,1] => 1 [4,2] => 1 [4,1,1] => 2 [3,3] => 1 [3,2,1] => 1 [3,1,1,1] => 2 [2,2,2] => 1 [2,2,1,1] => 3 [2,1,1,1,1] => 2 [1,1,1,1,1,1] => 2 [7] => 1 [6,1] => 1 [5,2] => 1 [5,1,1] => 2 [4,3] => 1 [4,2,1] => 1 [4,1,1,1] => 2 [3,3,1] => 1 [3,2,2] => 1 [3,2,1,1] => 2 [3,1,1,1,1] => 2 [2,2,2,1] => 1 [2,2,1,1,1] => 3 [2,1,1,1,1,1] => 2 [1,1,1,1,1,1,1] => 2 [8] => 1 [7,1] => 1 [6,2] => 1 [6,1,1] => 2 [5,3] => 1 [5,2,1] => 1 [5,1,1,1] => 2 [4,4] => 1 [4,3,1] => 1 [4,2,2] => 1 [4,2,1,1] => 2 [4,1,1,1,1] => 2 [3,3,2] => 1 [3,3,1,1] => 2 [3,2,2,1] => 1 [3,2,1,1,1] => 2 [3,1,1,1,1,1] => 2 [2,2,2,2] => 1 [2,2,2,1,1] => 3 [2,2,1,1,1,1] => 3 [2,1,1,1,1,1,1] => 2 [1,1,1,1,1,1,1,1] => 2 [9] => 1 [8,1] => 1 [7,2] => 1 [7,1,1] => 2 [6,3] => 1 [6,2,1] => 1 [6,1,1,1] => 2 [5,4] => 1 [5,3,1] => 1 [5,2,2] => 1 [5,2,1,1] => 2 [5,1,1,1,1] => 2 [4,4,1] => 1 [4,3,2] => 1 [4,3,1,1] => 2 [4,2,2,1] => 1 [4,2,1,1,1] => 2 [4,1,1,1,1,1] => 2 [3,3,3] => 1 [3,3,2,1] => 1 [3,3,1,1,1] => 2 [3,2,2,2] => 1 [3,2,2,1,1] => 3 [3,2,1,1,1,1] => 2 [3,1,1,1,1,1,1] => 2 [2,2,2,2,1] => 1 [2,2,2,1,1,1] => 3 [2,2,1,1,1,1,1] => 3 [2,1,1,1,1,1,1,1] => 2 [1,1,1,1,1,1,1,1,1] => 2 [10] => 1 [9,1] => 1 [8,2] => 1 [8,1,1] => 2 [7,3] => 1 [7,2,1] => 1 [7,1,1,1] => 2 [6,4] => 1 [6,3,1] => 1 [6,2,2] => 1 [6,2,1,1] => 2 [6,1,1,1,1] => 2 [5,5] => 1 [5,4,1] => 1 [5,3,2] => 1 [5,3,1,1] => 2 [5,2,2,1] => 1 [5,2,1,1,1] => 2 [5,1,1,1,1,1] => 2 [4,4,2] => 1 [4,4,1,1] => 2 [4,3,3] => 1 [4,3,2,1] => 1 [4,3,1,1,1] => 2 [4,2,2,2] => 1 [4,2,2,1,1] => 3 [4,2,1,1,1,1] => 2 [4,1,1,1,1,1,1] => 2 [3,3,3,1] => 1 [3,3,2,2] => 1 [3,3,2,1,1] => 2 [3,3,1,1,1,1] => 2 [3,2,2,2,1] => 1 [3,2,2,1,1,1] => 3 [3,2,1,1,1,1,1] => 2 [3,1,1,1,1,1,1,1] => 2 [2,2,2,2,2] => 1 [2,2,2,2,1,1] => 3 [2,2,2,1,1,1,1] => 3 [2,2,1,1,1,1,1,1] => 3 [2,1,1,1,1,1,1,1,1] => 2 [1,1,1,1,1,1,1,1,1,1] => 2 ----------------------------------------------------------------------------- Created: Jul 15, 2020 at 15:01 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Jul 15, 2020 at 15:01 by Martin Rubey