***************************************************************************** * 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: St000759 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The smallest missing part in an integer partition. In [3], this is referred to as the mex, the minimal excluded part of the partition. For compositions, this is studied in [sec.3.2., 1]. ----------------------------------------------------------------------------- References: [1] Hitczenko, Paweł, Louchard, G. Distinctness of compositions of an integer: a probabilistic analysis [[MathSciNet:1871561]] [2] Triangle read by rows: T(n,k) is the number of partitions of n having least gap k. [[OEIS:A264401]] [3] Hopkins, B., Sellers, J. A., Yee, A. J. Combinatorial Perspectives on the Crank and Mex Partition Statistics [[arXiv:2108.09414]] ----------------------------------------------------------------------------- Code: def statistic(pi): return min(set(range(1,len(pi)+2)).difference(set(pi))) ----------------------------------------------------------------------------- Statistic values: [] => 1 [1] => 2 [2] => 1 [1,1] => 2 [3] => 1 [2,1] => 3 [1,1,1] => 2 [4] => 1 [3,1] => 2 [2,2] => 1 [2,1,1] => 3 [1,1,1,1] => 2 [5] => 1 [4,1] => 2 [3,2] => 1 [3,1,1] => 2 [2,2,1] => 3 [2,1,1,1] => 3 [1,1,1,1,1] => 2 [6] => 1 [5,1] => 2 [4,2] => 1 [4,1,1] => 2 [3,3] => 1 [3,2,1] => 4 [3,1,1,1] => 2 [2,2,2] => 1 [2,2,1,1] => 3 [2,1,1,1,1] => 3 [1,1,1,1,1,1] => 2 [7] => 1 [6,1] => 2 [5,2] => 1 [5,1,1] => 2 [4,3] => 1 [4,2,1] => 3 [4,1,1,1] => 2 [3,3,1] => 2 [3,2,2] => 1 [3,2,1,1] => 4 [3,1,1,1,1] => 2 [2,2,2,1] => 3 [2,2,1,1,1] => 3 [2,1,1,1,1,1] => 3 [1,1,1,1,1,1,1] => 2 [8] => 1 [7,1] => 2 [6,2] => 1 [6,1,1] => 2 [5,3] => 1 [5,2,1] => 3 [5,1,1,1] => 2 [4,4] => 1 [4,3,1] => 2 [4,2,2] => 1 [4,2,1,1] => 3 [4,1,1,1,1] => 2 [3,3,2] => 1 [3,3,1,1] => 2 [3,2,2,1] => 4 [3,2,1,1,1] => 4 [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] => 3 [1,1,1,1,1,1,1,1] => 2 [9] => 1 [8,1] => 2 [7,2] => 1 [7,1,1] => 2 [6,3] => 1 [6,2,1] => 3 [6,1,1,1] => 2 [5,4] => 1 [5,3,1] => 2 [5,2,2] => 1 [5,2,1,1] => 3 [5,1,1,1,1] => 2 [4,4,1] => 2 [4,3,2] => 1 [4,3,1,1] => 2 [4,2,2,1] => 3 [4,2,1,1,1] => 3 [4,1,1,1,1,1] => 2 [3,3,3] => 1 [3,3,2,1] => 4 [3,3,1,1,1] => 2 [3,2,2,2] => 1 [3,2,2,1,1] => 4 [3,2,1,1,1,1] => 4 [3,1,1,1,1,1,1] => 2 [2,2,2,2,1] => 3 [2,2,2,1,1,1] => 3 [2,2,1,1,1,1,1] => 3 [2,1,1,1,1,1,1,1] => 3 [1,1,1,1,1,1,1,1,1] => 2 [10] => 1 [9,1] => 2 [8,2] => 1 [8,1,1] => 2 [7,3] => 1 [7,2,1] => 3 [7,1,1,1] => 2 [6,4] => 1 [6,3,1] => 2 [6,2,2] => 1 [6,2,1,1] => 3 [6,1,1,1,1] => 2 [5,5] => 1 [5,4,1] => 2 [5,3,2] => 1 [5,3,1,1] => 2 [5,2,2,1] => 3 [5,2,1,1,1] => 3 [5,1,1,1,1,1] => 2 [4,4,2] => 1 [4,4,1,1] => 2 [4,3,3] => 1 [4,3,2,1] => 5 [4,3,1,1,1] => 2 [4,2,2,2] => 1 [4,2,2,1,1] => 3 [4,2,1,1,1,1] => 3 [4,1,1,1,1,1,1] => 2 [3,3,3,1] => 2 [3,3,2,2] => 1 [3,3,2,1,1] => 4 [3,3,1,1,1,1] => 2 [3,2,2,2,1] => 4 [3,2,2,1,1,1] => 4 [3,2,1,1,1,1,1] => 4 [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] => 3 [1,1,1,1,1,1,1,1,1,1] => 2 [11] => 1 [10,1] => 2 [9,2] => 1 [9,1,1] => 2 [8,3] => 1 [8,2,1] => 3 [8,1,1,1] => 2 [7,4] => 1 [7,3,1] => 2 [7,2,2] => 1 [7,2,1,1] => 3 [7,1,1,1,1] => 2 [6,5] => 1 [6,4,1] => 2 [6,3,2] => 1 [6,3,1,1] => 2 [6,2,2,1] => 3 [6,2,1,1,1] => 3 [6,1,1,1,1,1] => 2 [5,5,1] => 2 [5,4,2] => 1 [5,4,1,1] => 2 [5,3,3] => 1 [5,3,2,1] => 4 [5,3,1,1,1] => 2 [5,2,2,2] => 1 [5,2,2,1,1] => 3 [5,2,1,1,1,1] => 3 [5,1,1,1,1,1,1] => 2 [4,4,3] => 1 [4,4,2,1] => 3 [4,4,1,1,1] => 2 [4,3,3,1] => 2 [4,3,2,2] => 1 [4,3,2,1,1] => 5 [4,3,1,1,1,1] => 2 [4,2,2,2,1] => 3 [4,2,2,1,1,1] => 3 [4,2,1,1,1,1,1] => 3 [4,1,1,1,1,1,1,1] => 2 [3,3,3,2] => 1 [3,3,3,1,1] => 2 [3,3,2,2,1] => 4 [3,3,2,1,1,1] => 4 [3,3,1,1,1,1,1] => 2 [3,2,2,2,2] => 1 [3,2,2,2,1,1] => 4 [3,2,2,1,1,1,1] => 4 [3,2,1,1,1,1,1,1] => 4 [3,1,1,1,1,1,1,1,1] => 2 [2,2,2,2,2,1] => 3 [2,2,2,2,1,1,1] => 3 [2,2,2,1,1,1,1,1] => 3 [2,2,1,1,1,1,1,1,1] => 3 [2,1,1,1,1,1,1,1,1,1] => 3 [1,1,1,1,1,1,1,1,1,1,1] => 2 [12] => 1 [11,1] => 2 [10,2] => 1 [10,1,1] => 2 [9,3] => 1 [9,2,1] => 3 [9,1,1,1] => 2 [8,4] => 1 [8,3,1] => 2 [8,2,2] => 1 [8,2,1,1] => 3 [8,1,1,1,1] => 2 [7,5] => 1 [7,4,1] => 2 [7,3,2] => 1 [7,3,1,1] => 2 [7,2,2,1] => 3 [7,2,1,1,1] => 3 [7,1,1,1,1,1] => 2 [6,6] => 1 [6,5,1] => 2 [6,4,2] => 1 [6,4,1,1] => 2 [6,3,3] => 1 [6,3,2,1] => 4 [6,3,1,1,1] => 2 [6,2,2,2] => 1 [6,2,2,1,1] => 3 [6,2,1,1,1,1] => 3 [6,1,1,1,1,1,1] => 2 [5,5,2] => 1 [5,5,1,1] => 2 [5,4,3] => 1 [5,4,2,1] => 3 [5,4,1,1,1] => 2 [5,3,3,1] => 2 [5,3,2,2] => 1 [5,3,2,1,1] => 4 [5,3,1,1,1,1] => 2 [5,2,2,2,1] => 3 [5,2,2,1,1,1] => 3 [5,2,1,1,1,1,1] => 3 [5,1,1,1,1,1,1,1] => 2 [4,4,4] => 1 [4,4,3,1] => 2 [4,4,2,2] => 1 [4,4,2,1,1] => 3 [4,4,1,1,1,1] => 2 [4,3,3,2] => 1 [4,3,3,1,1] => 2 [4,3,2,2,1] => 5 [4,3,2,1,1,1] => 5 [4,3,1,1,1,1,1] => 2 [4,2,2,2,2] => 1 [4,2,2,2,1,1] => 3 [4,2,2,1,1,1,1] => 3 [4,2,1,1,1,1,1,1] => 3 [4,1,1,1,1,1,1,1,1] => 2 [3,3,3,3] => 1 [3,3,3,2,1] => 4 [3,3,3,1,1,1] => 2 [3,3,2,2,2] => 1 [3,3,2,2,1,1] => 4 [3,3,2,1,1,1,1] => 4 [3,3,1,1,1,1,1,1] => 2 [3,2,2,2,2,1] => 4 [3,2,2,2,1,1,1] => 4 [3,2,2,1,1,1,1,1] => 4 [3,2,1,1,1,1,1,1,1] => 4 [3,1,1,1,1,1,1,1,1,1] => 2 [2,2,2,2,2,2] => 1 [2,2,2,2,2,1,1] => 3 [2,2,2,2,1,1,1,1] => 3 [2,2,2,1,1,1,1,1,1] => 3 [2,2,1,1,1,1,1,1,1,1] => 3 [2,1,1,1,1,1,1,1,1,1,1] => 3 [1,1,1,1,1,1,1,1,1,1,1,1] => 2 [8,5] => 1 [7,5,1] => 2 [7,4,2] => 1 [5,5,3] => 1 [5,4,4] => 1 [5,4,3,1] => 2 [5,4,2,2] => 1 [5,4,2,1,1] => 3 [5,4,1,1,1,1] => 2 [5,3,3,2] => 1 [5,3,3,1,1] => 2 [5,3,2,2,1] => 4 [5,3,2,1,1,1] => 4 [4,4,4,1] => 2 [4,4,3,2] => 1 [4,4,3,1,1] => 2 [4,4,2,2,1] => 3 [4,3,3,3] => 1 [4,3,3,2,1] => 5 [3,3,3,3,1] => 2 [3,3,3,2,2] => 1 [3,3,2,2,2,1] => 4 [9,5] => 1 [8,5,1] => 2 [7,5,2] => 1 [7,4,3] => 1 [6,4,4] => 1 [6,2,2,2,2] => 1 [5,5,4] => 1 [5,5,1,1,1,1] => 2 [5,4,3,2] => 1 [5,4,3,1,1] => 2 [5,4,2,2,1] => 3 [5,4,2,1,1,1] => 3 [5,3,3,2,1] => 4 [5,3,2,2,2] => 1 [5,2,2,2,2,1] => 3 [4,4,4,2] => 1 [4,4,3,3] => 1 [4,4,3,2,1] => 5 [4,3,2,2,2,1] => 5 [3,3,3,3,2] => 1 [3,3,3,3,1,1] => 2 [9,5,1] => 2 [8,5,2] => 1 [7,5,3] => 1 [6,5,4] => 1 [6,5,1,1,1,1] => 2 [6,3,3,3] => 1 [6,2,2,2,2,1] => 3 [5,5,5] => 1 [5,4,3,2,1] => 6 [5,4,3,1,1,1] => 2 [5,3,2,2,2,1] => 4 [4,4,4,3] => 1 [4,4,4,1,1,1] => 2 [3,3,3,3,3] => 1 [3,3,3,3,2,1] => 4 [8,5,3] => 1 [7,5,3,1] => 2 [5,5,3,3] => 1 [5,5,2,2,2] => 1 [5,4,3,2,1,1] => 6 [5,4,2,2,2,1] => 3 [4,4,4,4] => 1 [4,4,4,2,2] => 1 [4,3,3,3,2,1] => 5 [8,6,3] => 1 [6,5,3,3] => 1 [6,5,2,2,2] => 1 [6,4,4,3] => 1 [6,4,4,1,1,1] => 2 [6,3,3,3,2] => 1 [6,3,3,3,1,1] => 2 [5,5,4,3] => 1 [5,5,4,1,1,1] => 2 [5,5,2,2,2,1] => 3 [5,4,3,2,2,1] => 6 [5,3,3,3,2,1] => 4 [4,4,4,3,2] => 1 [4,4,4,3,1,1] => 2 [4,4,4,2,2,1] => 3 [4,4,4,3,2,1] => 5 [5,4,3,3,2,1] => 6 [6,3,3,3,2,1] => 4 [6,5,2,2,2,1] => 3 [5,5,3,3,1,1] => 2 [6,5,4,1,1,1] => 2 [5,5,3,3,2] => 1 [5,5,4,2,2] => 1 [6,4,4,2,2] => 1 [6,5,4,3] => 1 [9,6,3] => 1 [8,6,4] => 1 [5,4,4,3,2,1] => 6 [5,5,3,3,2,1] => 4 [5,5,4,2,2,1] => 3 [6,4,4,2,2,1] => 3 [5,5,4,3,1,1] => 2 [6,4,4,3,1,1] => 2 [6,5,3,3,1,1] => 2 [5,5,4,3,2] => 1 [6,4,4,3,2] => 1 [6,5,3,3,2] => 1 [6,5,4,2,2] => 1 [6,5,4,3,1] => 2 [6,5,4,1,1,1,1] => 2 [9,6,4] => 1 [8,5,4,2] => 1 [8,5,5,1] => 2 [5,5,4,3,2,1] => 6 [6,4,4,3,2,1] => 5 [6,5,3,3,2,1] => 4 [6,5,4,2,2,1] => 3 [6,5,4,3,1,1] => 2 [6,5,4,3,2] => 1 [6,5,2,2,2,2,1] => 3 [6,5,4,2,1,1,1] => 3 [7,5,4,3,1] => 2 [8,6,4,2] => 1 [10,6,4] => 1 [10,7,3] => 1 [9,7,4] => 1 [9,5,5,1] => 2 [6,5,4,3,2,1] => 7 [6,3,3,3,3,2,1] => 4 [6,5,3,2,2,2,1] => 4 [6,5,4,3,1,1,1] => 2 [11,7,3] => 1 [4,4,4,4,3,2,1] => 5 [6,4,3,3,3,2,1] => 5 [6,5,4,2,2,2,1] => 3 [6,5,4,3,2,1,1] => 7 [9,6,4,3] => 1 [5,4,4,4,3,2,1] => 6 [6,5,3,3,3,2,1] => 4 [6,5,4,3,2,2,1] => 7 [9,6,5,3] => 1 [8,6,5,3,1] => 2 [6,4,4,4,3,2,1] => 5 [6,5,4,3,3,2,1] => 7 [11,7,5,1] => 2 [9,7,5,3] => 1 [5,5,5,4,3,2,1] => 6 [6,5,4,4,3,2,1] => 7 [9,7,5,3,1] => 2 [10,7,5,3] => 1 [6,5,5,4,3,2,1] => 7 [9,7,5,4,1] => 2 [6,6,5,4,3,2,1] => 7 [7,6,5,4,3,2] => 1 [7,6,5,4,3,2,1] => 8 [7,6,5,4,3,1,1,1] => 2 [10,7,6,4,1] => 2 [9,7,6,4,2] => 1 [10,8,5,4,1] => 2 [7,6,5,4,3,2,1,1] => 8 [7,6,5,4,2,2,2,1] => 3 [10,8,6,4,1] => 2 [9,7,5,5,3,1] => 2 [7,6,5,4,3,2,2,1] => 8 [7,6,5,3,3,3,2,1] => 4 [11,8,6,4,1] => 2 [10,8,6,4,2] => 1 [7,6,5,4,3,3,2,1] => 8 [7,6,4,4,4,3,2,1] => 5 [11,8,6,5,1] => 2 [7,6,5,4,4,3,2,1] => 8 [7,5,5,5,4,3,2,1] => 6 [7,6,5,5,4,3,2,1] => 8 [6,6,6,5,4,3,2,1] => 7 [7,6,6,5,4,3,2,1] => 8 [12,9,7,5,1] => 2 [7,7,6,5,4,3,2,1] => 8 [13,9,7,5,1] => 2 [11,9,7,5,3,1] => 2 [11,8,7,5,4,1] => 2 [8,7,6,5,4,3,2,1] => 9 [8,7,6,5,4,3,2,1,1] => 9 [8,7,6,5,4,3,2,2,1] => 9 [8,7,6,5,4,3,3,2,1] => 9 [8,7,6,5,4,4,3,2,1] => 9 [11,9,7,5,5,3] => 1 [8,7,6,5,5,4,3,2,1] => 9 [8,7,6,6,5,4,3,2,1] => 9 [8,7,7,6,5,4,3,2,1] => 9 [8,8,7,6,5,4,3,2,1] => 9 [9,8,7,6,5,4,3,2,1] => 10 [11,9,7,7,5,3,3] => 1 ----------------------------------------------------------------------------- Created: Apr 08, 2017 at 22:40 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Aug 24, 2021 at 12:59 by Martin Rubey